Contributions to Control Law Designs and Stability Analysis in AC Microgrids

Proyecto Fin de Carrera Ingeniería de Telecomunicación Formato de Publicación de la Escuela Técnica Superior de Ingeniería Autor: F. Javier Payán Somet Tutor: Juan José Murillo Fuentes Dep. Teoría de la Señal y Comunicaciones Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2013 Doctoral Dissertation Ingeniería Automatica, Electrónica y de Telecomunicaciones Contributions to Control Law Designs and Stability Analysis in AC Microgrids Author: María Camila Merchán-Riveros Directors: Carolina Albea and Francisco Salas Ingeniería de Sistemas y Automática Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2024 Doctoral Dissertation Ingeniería Automatica, Electrónica y de Telecomunicaciones Contributions to Control Law Designs and Stability Analysis in AC Microgrids Author: María Camila Merchán-Riveros Directors: Carolina Albea and Francisco Salas Professors Ingenier´ıa de Sistemas y Autom´atica Escuela T´ecnica Superior de Ingenier´ıa Universidad de Sevilla 2024 Doctoral Dissertation: Contributions to Control Law Designs and Stability Analysis in AC Microgrids Author: María Camila Merchán-Riveros Directors: Carolina Albea and Francisco Salas El tribunal nombrado para juzgar la Tesis arriba indicada, compuesto por los siguientes doctores: Presidente: Vocales: Secretario: acuerdan otorgarle la calificación de: El Secretario del Tribunal Fecha: A mi familia y a Jose Aknowledgements T he period of my thesis development has been a turning point in my life. Since I moved to Spain and left practically all my life and loved ones in Colombia, I have learned and grown a lot both professionally and personally, and the support and help of many people around me in different areas of life have been key to do this thesis. First of all, I would like to express my gratitude to my supervisors, Carolina Albea and Paco Salas, for their support, guidance and teaching, as well as for their ideas and comments contributed to the development of the thesis and their revision and correction of this document. Carolina, I would also like to thank you not only for your tireless work but also for your advice and emotional support, which was very important to me. I would like to thank Professor Paco Gordillo, the first person who welcomed me when I arrived in Seville, for the opportunity of joining the research group, and for his trust and his encouraging guidance. Also I would like to thank Professor Alexandre Seuret for his ideas and contributions to this thesis. I would like to thank Professor Luca Zaccarian for accepting to review my work as well as for participating in my jury committee. I am also grateful to Professor Giovanni Garraffa for accepting to review my work and helping with the experimental tests in Palermo. I deeply appreciate their constructive suggestions on the thesis, and I am grateful they take the time to undertake these tasks despite their busy schedules. Many thanks to Professor Suzanne Lesecq, Professor Fabio Gómez-Estern, Professor Carlos Bordons, and Professor Ascensión Zafra, for their commitment to take part in my jury committee. I would like to thank Professor Antonino Sferlazza for inviting me to Palermo for doing my international period and for his guidance there. III XContents 3.1 Introduction 25 3.1.1 Problem formulation 26 3.2 Proposed control scheme 29 3.3 Control objectives 30 3.4 Dynamical models in the primary and secondary control loops 30 3.4.1 Consensus algorithm 30 3.4.2 Droop control 31 3.4.3 Continuous-time control law for the inverters 32 3.5 Complete control dynamic 34 3.5.1 Closed-loop dynamics for a BESSi35 3.5.2 Complete control dynamic 35 3.5.3 Three time-scale separation 37 3.6 Stability analysis of the complete continuous-time system 37 3.6.1 Singular perturbation form 37 3.6.2 The quasi-steady-state equilibrium ’manifold’ 38 3.6.3 Stability of the boundary layer system 38 3.6.4 Stability of the reduced system 39 3.7 Experimental validation 39 3.7.1 Scenario 1 43 3.7.2 Scenario 2 43 3.7.3 Comparison with [1] 46 3.8 Conclusion 49 4 Hybrid dynamical control scheme for discharging rate consensus 51 4.1 Control objectives 52 4.2 Inverter model and control 52 4.2.1 Voltage output regulation for inverters 53 4.3 Hybrid control structure 54 4.4 Global hybrid structure 56 4.4.1 Three time-scale separation 57 4.5 Stability analysis of the complete hybrid system 58 4.5.1 Singular perturbation form 58 4.5.2 Regularity of System’s Data 59 4.5.3 Regularity of the ‘manifold’ 59 4.5.4 Stability of the boundary layer system 60 4.5.5 Stability of the reduced system 61 4.6 Simulation results 62 4.6.1 Comparison with [1] 65 4.7 Conclusions 67 5 Hybrid dynamical control scheme for reactive power sharing 69 5.1 Introduction 69 5.2 Proposed control scheme 71 Contents XI 5.3 Control objectives 72 5.4 Dynamical models for reactive power sharing 72 5.4.1 Consensus algorithm 72 5.4.2 Droop control 72 5.5 Hybrid control structure 73 5.6 Global hybrid structure 74 5.6.1 Three time-scale separation 76 5.7 Stability analysis of the complete hybrid system 76 5.7.1 Regularity of System’s Data 77 5.7.2 Regularity of the ’manifold’ 78 5.7.3 Stability of the ‘boundary layer’ 78 5.7.4 Stability of the reduced system 79 5.8 Simulation results 80 5.8.1 Scenario 1 82 5.8.2 Scenario 2 85 5.9 Conclusion 88 6 Bifurcation analysis of islanded AC microgrids with Constant Power Loads 91 6.1 Introduction 91 6.2 Preliminary concepts 92 6.2.1 Some notions of bifurcation theory 92 Poincaré-Andronov-Hopf bifurcation 92 6.3 AC microgrid model 96 6.3.1 Modelling of a three phase inverter 96 6.3.2 Model of a CPL 99 Selected CPL model 101 6.3.3 Model of the microgrid with a CPL 102 6.4 Numerical stability analysis 104 6.5 Simulation results 106 6.6 Experimental results 109 6.6.1 CPL time constant test 109 6.6.2 Experimental results of Bifurcation analysis 110 6.7 Conclusions 116 7 Conclusions and future work 117 7.1 Conclusions and contribution summary 117 7.2 Future works 118 Appendix A Appendix to Chapter 1 121 List of Figures 127 List of Tables 131 Bibliography 133 Notation List of Symbols RThe set of real numbers R≥0The set of nonnegative real numbers RnThe n-dimensional euclidean space Rn×mThe real n×mmatrices space ZThe set of integer numbers NThe set of positive integer numbers A⊂B The set Ais subset of B A∪B The union of sets Aand B A∩B The intersection of sets Aand B A\B The difference between sets Aand B A×B The cartesian product between sets Aand B AThe closure of the set A ATThe transpose of matrix A A=Si∈N AiThe union of sets Aifor i∈N A=Ti∈N AiThe intersection of sets Aifor i∈N IThe identity matrix ∅The empty set 1and 0The vectors of ones and zeros, resp. of suited dimension eig(M)The eigenvalues of matrix M Re(a)The real part of the complex number a M≻0(M≺0) The eigenvalues of Mare strictly positive (negative) diag{a1,a2,...,aN}The diagonal matrix whose elements are a1,a2,...,aN ˙xThe state time derivative x+The state value after an instantaneous change (jump) |x|The absolute value of x ∥x∥The euclidean norm of vector x XIII XIV Chapter 0. Notation min(A)The minimum element of set A f:Rm→Rn This notation indicates that f is a function from a space Rm to a space Rn F:Rm⇒Rn This notation indicates that F is a set-valued mapping with F(x)⊂Rn for each x∈Rm argmin(f)stand for the argument that minimize the function f ρiBiThe closed ball of radius ρi>0 ν→0+A positive parameter νis small enough. Acronyms AC Alternating Current BESS Battery Energy Storage System CPL Constant Power Load CSI Current-Source Inverters DC Direct Current DDE Delay-Differential Equation DG Distributed Generation ESS Energy Storage Systems HB Half-Bridge HDS Hybrid Dynamical System LQR Linear-Quadratic Regulator MAS Multi-Agent Systems MG Microgrid MPPT Maximum Power Point Tracking MOSFET Metal-oxide semiconductor field-effect transistor OL Outer Loop PCC Point of Common Coupling PI Proportional Integral (controller) PLL Phase-Locked Loop PR Proportional Resonant (controller) PV Photo-Voltaic PWM Pulse-Width Modulation RES Renewable Energy Sources RMS Root-Mean Square SOC State of Charge SPAS Semi-Global Practical Stability VSI Voltage-Source Inverters 1 Introduction There is a driving force more powerful than steam, electricity and nuclear power: the will. attributed to Albert Einstein T his chapter presents the background and a brief summary of the context of this thesis’ contributions, with the objective of orientating the reader within the broader scope of the work’s field of study. Firstly, a general overview of AC microgrids is provided. Secondly, the main contributions are presented, followed by a description of the structure of the manuscript. Finally, a list of publications and contributions is provided. 1.1 Introduction to AC Microgrids In our rapidly changing world, energy efficiency and sustainability are crucial, especially for future generations’ well-being. Alternative energies are now vital, meeting the growing global demand for cleaner, sustainable power sources. As a result, the generation of green and clean energy through Renewable Energy Sources (RESs), such as photovoltaic (PV) and wind energy, has emerged as a prominent topic in recent decades. The concept of Microgrid has been widely accepted as a promising solution to integrate RESs into the conventional electricity grid in a flexible, reliable, and sustainable way, as is introduced by USA’s CERTS (Consortium for Electric Reliability Technology Solutions) in 2002 [2]. Microgrids (MGs) are defined as power distribution systems which consist of Distributed Generation (DG), Energy Storage Systems (ESSs), and loads operating as a single controlled subsystem. Generally, they can operate in parallel with the broader utility grid (grid-connected mode) with a grid-following (current dependent) control [3] or as autonomous power system (islanded mode) with a grid-forming (voltage dependent) control [4]. To operate in islanded mode allows to ensure energy supply in remote locations or in the event of electrical outages. In grid-connected mode, the microgrid is integrated into 1 2Chapter 1. Introduction Figure 1.1 Block diagram of an AC microgrid. the utility grid through a single point of connection called the Point of Common Coupling (PCC), typically with an electromechanical circuit breaker. The main advantages of microgrids are: improved reliability through distributed generation, higher efficiency via reduced transmission, and integration of alternative energy sources. To maximize these advantages, microgrids must ensure minimal distribution losses, high reliability, energy efficiency, blackout resilience, and scalability. In terms of the type of power topology, microgrids can be divided into three main groups: AC, DC, and hybrid. Because of their ability to provide a direct way to integrate DGs into the main grid, their simple structure, and cost-effectiveness, AC microgrids are now more in demand [5]. A typical diagram of an AC microgrid is shown in Fig. 1.1 [6]. The flow of energy from the DGs must be controlled according to the load demand. The control scheme, whether centralized or decentralized, must maintain the quality and reliability of the supply. DGs can be classified into two types depending on whether power output can be dispatched on demand or not [6]. A dispatchable unit, like a diesel generator, can be fully controlled. In contrast, nondispatchable units, such as DGs based on PV and wind energy, are generally intermittent and difficult to control. Consequently, microgrids must necessarily include power electronics converters to act as the interface medium between nondispatchable DGs and the microgrid. Furthermore, ESSs must be integrated not only to store the power produced by distributed generators (DGs), but also to enhance the overall 1.1 Introduction to AC Microgrids 3 performance and stability of the microgrid. This is achieved by allowing the DGs to run at a constant output while they compensate for fluctuations in load power. All these components are integrated into a complex system with several control levels, whose control laws must be designed to exploit the advantages of microgrids, guaranteeing minimal distribution losses, elevated reliability, energy efficiency, resilience against blackouts, and scalability. 1.1.1 Control techniques in AC microgrids As mentioned before, control structures play an important role in ensuring synchronization, voltage regulation, power balance, and load sharing across all buses in the network, contributing to a stable and robust operating point. The main aim of any control technique used in AC microgrids is to maintain a constant voltage and frequency, therefore some of the main control targets are listed below [7]: •To regulate voltage and frequency under any operating condition. •To obtain an optimal power sharing of active and reactive power. • To obtain a consensus on the charging/discharging state of batteries in islanded mode. • To achieve a smooth transition from grid-connected to islanded mode and vice-versa •To optimize operating cost of production and power exchanges with the utility grid. The control strategy employed depends on the operating mode: grid-connected, or islanded. During grid-connected mode, the voltage and frequency magnitudes are regulated by the utility grid. In islanded mode, a grid forming/supporting converter determines voltage and frequency, using either a centralized or decentralized control approach [7]. In this thesis, we will focus on control approaches applied to islanded microgrids. To accomplish these control requirements, different hierarchical layers are established [6]. The Union for the Coordination of Transmission of Electricity (UCTE, Continental Europe) has presented a hierarchical control scheme for larger power systems, where three main controls are defined: primary, secondary, and tertiary, which together are used to provide coordination among all DGs, ESSs, and loads through management strategies with and without communication networks [8]. The primary control layer is responsible for local voltage regulation and for ensuring power sharing between DGs. The secondary control focuses on mitigating voltage and frequency deviations caused by primary control, and the tertiary control is responsible for controlling the flow of energy between the utility grid and the microgrid at the PCC, and takes into account economic issues. 1.1.2 Primary control This is the first level of the hierarchical control scheme, which is an independent and locally executed strategy that allows an autonomous operation of each DG [9], i.e., the primary control does not need to communicate and only uses measured signals. This control layer aims to achieve the DG operating point, typically involving DC/AC power converters (inverters) controlled by the inner control. At the same time, appropriate power sharing is generally ensured by a droop control, which applies the well-known P/ω (active 4Chapter 1. Introduction Figure 1.2 Primary control loop. power/ frequency ) and Q/V (reactive power/ voltage) curves, which determine the voltage and frequency references of the inner control to enhance the performance of the system. A general scheme of a DG with primary control is shown in Fig. 1.2. Inner control (inverter output control) and power sharing (droop control) strategies are discussed below: Inner control The control objective of the inner control loop is to achieve the operating point of a power converter that works as an interface of a DG or ESS with the microgrid. Generally, these power converters are inverters that can be controlled as Current-Source Inverters (CSIs) or Voltage-Source Inverters (VSIs) [5]. The control of CSIs consists of an internal current control loop and a Phase-Locked Loop (PLL) to stay synchronized with the grid, while VSIs use an external voltage control loop and an internal current control loop. CSIs are used mainly in grid-connected operation as they work as a grid-feeding converter capable of injecting active and reactive power into the PCC, while the voltage at the PCC is governed by the utility grid or a grid-forming converter [7]. These converters are often connected to PV or wind turbines, which are controlled by a maximum power point tracking algorithm. In contrast, VSIs work as a grid-forming converter, which sets voltage and frequency references for the microgrid in islanded mode operation. These converters are often connected to ESS units as they set the voltage and frequency at the PCC. Both VSIs and CSIs can work together in a microgrid, so several VSIs and CSIs, or only VSIs, can be connected in parallel to form a microgrid. The use of VSIs is more flexible because they do not need an external reference to stay synchronized and can provide ride-through capability (resilience to short-duration faults or disturbances) and improve power quality [8]. In addition, it is the main type of converter that is used in islanded microgrids. The internal and external control loops of 1.1 Introduction to AC Microgrids 5 Figure 1.3 Inner control loop. VSIs are described in Fig. 1.3 [10]. The external voltage regulator must minimise the error between the reference and the measured output voltage and generate the current reference for the internal current loop which generates the desired output voltage to be modulated. Then, a modulator is used to convert the continuous control signal u into a discrete one for the switching of the converter, where the most common is the Pulse-Width Modulation (PWM), which changes the duty cycle of a fixed frequency square signal in response to the modulated signal (control signal). However, there are a number of control strategies that do not use the common scheme of a continuous controller plus a modulator. Instead, they simply calculate the discrete signals, taking into account all possible combinations of outputs. The controllers for both loops can be linear, such as Proportional-Integral (PI), ProportionalResonant (PR), predictive controllers, or nonlinear, such as hysteresis, sliding mode, fuzzy logic or hybrid dynamic controllers. A particular interest has emerged to apply hybrid control techniques to power converters with the aim of considering the subtlety of the hybrid nature of these systems [11 – 13]. Indeed, they are composed of continuous-time dynamics (voltages and currents) and discrete-time dynamics (switching). Some of the main contributions of this work are based on Hybrid Dynamical Systems (HDS) hence in the next chapter this theory is explained in detail. The following part describes the most commonly used linear type controllers: The PI controller and the PR controller: PI controller: The PI controller is used typically in engineering due to its simple structure, easy implementation, and good performance. Its classical transfer function is depicted below: CPI(s) = Kp+Ki s,(1.1) where Kp and Ki are the proportional and integral gain respectively. This controller is suitable for DC signals due to its high gain at low frequencies. For AC magnitudes, it has a limited response, thus it requires translation of the controller signals from sinusoidal to constant values, that is, the dq reference frame (direct-quadrature). This reference frame rotates at the same frequency, as the magnitude considered in AC, and thus the magnitude expressed in these axes appears as a DC component [14]. 12 Chapter 2. Preliminary concepts dom M={x∈Rm:M(x)=∅}. Remark 1: Note that a set-valued mapping M:Rm⇒Rn associates with every point x∈Rm a subset of Rn . The double arrow notation M:Rm⇒Rn distinguishes a setvalued mapping M from a function. Also, we can write M:Rm⇒S , for S⊂Rn , which indicates that M:Rm⇒Rnis a set-valued mapping with M(x)⊂Sfor all x∈Rm. With the previous definition, the integration of continuous and discrete behavior in (2.1) for a hybrid system in Rnis represented by four objects described below: • A set C ⊂ Rn , called the flow set, which defines the region where the systems presents a continuous-time evolution. • A set-valued mapping F(x) : Rn⇒Rn (or function f ) with C ⊂dom F , called the flow map. It indicates how the system evolves or flows. • A set D ⊂ Rn , called the jump set, which defines the region where the systems presents a discrete-time behaviour. • A set-valued mapping G(x) : Rn⇒Rn (or function g ) with D⊂dom G , called the jump map. It indicates how the system instantaneously changes or jumps. Therefore, a hybrid system with the data as above can be represented by the notation H(C,F,D,G), or briefly by H. 2.1.2 Hybrid time domains and hybrid arcs A solution for a hybrid system is parameterized by a hybrid time domain which is composed of continuous-time and discrete-time variables. The elapsed continuous time is parameterized by t∈R≥0 , and the number of jumps that have occurred is parameterized by j∈N. Definition 2: (Hybrid time domain) [18, Definition 2.3] A subset E⊂R≥0×N is a hybrid domain if E= J−1 [ j=0 ([tj,tj+1],j) where jcan be finite or infinite. It is a compact hybrid domain if jis finite. In other words, E is a hybrid time domain if it is a union of a finite or infinite sequence of intervals [tj,tj+1]×{j} , while E is a compact hybrid time domain if this sequence is finite. Definition 3: (Hybrid arc) [18, Definition 2.4] A function ϕ:E→Rnis a hybrid arc if: •Eis a hybrid time domain and, • if for each j⊂N , the function t→ϕ(t,j) is locally absolutely continuous on the interval Ij={t: (t,j)∈E} 2.1 Hybrid dynamical systems 13 Figure 2.1 Hybrid arc. Fig. 2.1 shows a graphical representation of a hybrid arc ϕ with hybrid time domain dom ϕ . Furthermore, a hybrid arc can be classified based on the structure of their domains. To cite some of them, a hybrid arc is: •nontrivial if dom ϕhas at least two points; •continuous if it is nontrivial and dom ϕ⊂R≥0×{0}; •discrete if if it is nontrivial and dom ϕ⊂{0}×N, and •complete if dom is unbounded i.e., if length(E) = ∞. The solutions of a hybrid system H are given by a hybrid arc that satisfies certain conditions determinated by the hybrid time domain and the data of H. Definition 4: (Solution to a hybrid system) [18, Definition 2.6] A function ϕ is a solution to the hybrid system H(C,F,D,G), if: 1. The initial condition ϕ(0,0) ∈C∪D, where Cdenotes the closure of the set C. 2. For all j∈Nsuch that Ij={t: (t,j)∈E}has nonempty interior ϕ(t,j)∈C for all t∈int Ij, ˙ ϕ(t,j)∈F(ϕ(t,j)) for almost all t∈Ij; 3. For all (t,j)∈dom ϕsuch that (t,j +1) ∈dom ϕ, ϕ(t,j)∈D, ϕ(t,j +1) ∈G(ϕ(t,j)). Fig. 2.2 illustrates a solution admitted by the previous definition. Here are represented the flows and jumps of the solution ϕ , where the flows are allowed only on the flow set Cand the jumps must originate from the jump set D. 14 Chapter 2. Preliminary concepts Figure 2.2 Evolution of a solution to a hybrid system. 2.1.3 Basic assumptions In this section, we present the basic assumptions on a hybrid system to ensure that it is well-posed, which is an important property required for the applicability of many results in hybrid systems theory, as shown in [18]. Assumption 1: (Hybrid basic conditions) [18, Section 6.2] 1. Cand Dare closed subsets of Rn; 2. F:Rn⇒Rn is outer semicontinuos and locally bounded relative to C , C ⊂dom F , and F(x)is convex for every x∈C; 3. G:Rn⇒Rn is outer semicontinuos and locally bounded relative to D , and D⊂ dom G. Theorem 1: (Basic conditions and well-posedness) [18, Theorem 6.30] If a hybrid system H(C,F,D,G)satisfies Assumption 1 then it is well-posed. □ The proof of the well-posedness theorem is depicted in [18, Chapter 6]. If F and G are single value maps, that is, functions, the requirement is that f(x) and g(x) must be continuous (outer semicontinuous as a set-valued mapping), hence the differential equation ˙z=f(x) and the difference equation x+=g(x) can be identified with a hybrid system that satisfies the hybrid basic conditions. Definition 5: (Definition 5.9 [18]) A set-valued mapping F:Rn⇒Rn is said to be outer semicontinuous (osc) if each sequence (xi,yi)∈Rn×Rn that satisfies yi∈F(xi) for each i and converges to a point (x,y)∈Rn×Rn , has the property that y∈F(x) . Moreover, another way to see this property is that a set-valued mapping F is outer semicontinuous if and only if the graph of Fis closed (Fig. 2.3). Definition 6: (Definition 5.14 [18]) A set-valued mapping F:Rm⇒Rn is locally bounded at x∈Rm if there exists a neighborhood Ux of x such that F(Ux)⊂Rn is bounded. Given a set S⊂Rm , the mapping F is locally bounded relative to S if the set-valued mapping from Rm to Rn defined by F(x) for x∈S and ∅ for x /∈S is locally bounded at each x∈S. 2.1 Hybrid dynamical systems 15 Figure 2.3 Mappings:(a) not outer semicontinuous and (b) outer semicontinuous. 2.1.4 Lyapunov conditions for hybrid systems Based on section [18, Section 3.2], we present the following theorem addressing the Lyapunov function as a sufficient condition for stability properties of a compact set A related to a hybrid system. This compact set is defined as the attractor of the system. Moreover, the data of the hybrid system must satisfy the basic hybrid conditions depicted in the previous section. Remark 2: An attractor is defined as the set of all states toward which a system tends to evolve, regardless of the starting conditions of the system. Theorem 2: Given a hybrid system H with state x , and a compact set A⊂Rn , if there exist a Lyapunov function candidate Vfor Hsuch that 1. (Lyapunov function candidate) V(x)=0,∀x∈A, V(x)>0,∀x∈C∪D∪G(D)\A, lim |x|→∞V(x) = ∞,∀x∈C∪D∪G(D) 2. (Flow) ˙ V=⟨∇V(x),f(x)⟩<0,∀x∈C\A, f ∈F(x) 3. (Jump) ∆V=V(g(x))−V(x)<0,∀x∈D\A, g ∈G(x) G(A∩D)⊂A then Ais Uniformly Globally Asymptotically Stable (UGAS) for H.□ Remark 3: A set Ais a compact set if and only if Ais closed and bounded. In this theorem, the first point states the condition for a function V to be a Lyapunov function candidate with respect to A for H . Points 2 and 3 are called the Lyapunov conditions for flow and jump, respectively. A Lyapunov function candidate V must be 16 Chapter 2. Preliminary concepts continuous and positive definite with respect to A , continuously differentiable in an open set containing C , and radially unbounded. Furthermore, the Lyapunov conditions state that for flow: there are strict decrements whenever the system flows away from the set A ; and for jump: the difference between the value after the jump and the value before the jump must be negative. A suitable Lyapunov function candidate from Theorem 2 is illustrated in Fig. 2.4. Figure 2.4 Evolution of a lyapunov function candidate V(x)with x∈Afor H. 2.2 Singular perturbation theory In this section, we present the general theory of singular perturbations applied to stability analysis for both continuous-time and hybrid systems. First, some basic concepts of singular perturbation theory are given, and then stability approaches are explained. 2.2.1 Standard singular perturbation model To simplify complex systems, singular perturbation methods are used to split the dynamics of the system into two parts: slow and fast dynamics. Singular perturbations introduce multitime-scale behavior in the system’s response to external stimulus. By introducing an external parameter and then setting it to zero, a step towards "reduced-order modeling" is achieved. This order reduction is transformed into a parameter perturbation called "singular". As is presented in [19, Chapter 11] and [20, Chapter 1], the singular perturbation model of a dynamical system is obtained by multiplying the derivatives of some of the states by a small positive parameter ε, that is, ˙x=f(t,x,z,ε)(2.2) ε˙z=g(t,x,z,ε)(2.3) where x∈Rn represents the slow variables whereas z∈Rm the fast ones, with f and g being continuously differentiable functions in their arguments (t,x,z,e) . When we set ε= 0 in (2.2) and (2.3) the dimension of the state space equation reduces from n+m to 2.2 Singular perturbation theory 17 nbecause the differential equation (2.3) turned into the algebraic equation: 0 = g(t,x,z,0).(2.4) If (2.4) has k≥1distinct ("isolated") real roots: z=hi(t,x)i= 1,2,...,k, (2.5) the model (2.2) – (2.3) is in standard form. Equation (2.5) describes an n -dimensional manifold for the state space. For each root of (2.4) a well-defined reduced model is obtained, substituting the root (2.5) into the slow dynamics equation (2.2), at ε= 0, so, ˙x=f(t,x,h(t,x),0).(2.6) This model is known as the slow model, also referred to as the quasi-steady-state model, because the fast dynamical variable z has been replaced by its equilibrium, or quasi-steadystate,h(t,x)(2.4). However, we have no information about how z evolves toward its equilibrium. We know that the velocity of z can be large, since ˙z=g/ε . And in fact, by setting ε= 0 in (2.3) , the transient behavior of z becomes instantaneous whenever g= 0 . Therefore, we need to analyze whether this transient zescapes to infinity or converges to its equilibrium. To analyze (2.3) , we first note that ε˙z can remain finite even if ε tends to zero and ˙z tends to infinity. To check this, we stretch or scale the time by a factor of 1/ε. We set εdz dt =dz dτ ,hence, dτ dt =1 ε. On the τtimescale, (2.3) is rewritten as: dz dτ =g(t,x,y +h(t,x),0),(2.7) where y=z−h(t,x) . This shifts the equilibrium of z to the origin. By setting ε= 0 , (t,x) are frozen at their initial values and treated as fixed parameters (constants). This autonomous system (2.7) is known as the boundary-layer system. 2.2.2 Continuous-time stability analysis Singular perturbation theory can be applied to nonlinear systems characterized by both slow and fast dynamics to analyze their stability properties [19, Chapter 11.5]. To apply singular perturbation analysis, we first need to write the system in the standard singular perturbation form (2.2)–(2.3), as described in the previous section. Then, the next three steps in singular perturbation analysis should be followed and some basic assumptions are considered: a) Manifold: 18 Chapter 2. Preliminary concepts First, we need to find the quasi-steady-state equilibrium manifold for the fast subsystem (2.3) , which corresponds to the "isolated root", z=h(t,x) , derived from (2.4) with ε= 0. Assumption 2: The equation (2.4) has an isolated root z=h(t,x) , which is the quasi-steady-state manifold and the stationary solution of the fast subsystem. b) Boundary-layer system (fast subsystem): In the second step, we consider the boundary layer system (2.7) , which represents the rapid transient of the fast subsystem in the initial time interval before it converges to its quasi-steady state. Since these dynamics are considered instantaneous as ε tends to 0 , we have to scale the time t by 1/ε in (2.3) . Here, the slow dynamics are treated as constants ( ˙x= 0). Assumption 3: The origin of the boundary layer system defined in (2.7) is asymptotically stable. c) Reduced system (slow subsystem): In the last step, we consider the reduced system obtained by replacing the isolated root in the slow subsystem, (2.2) . Here, the fast dynamics are in steady state and only the slow variables are considered. Assumption 4: The origin of the reduced system defined in (2.6) is asymptotically stable. Now, we present Theorem 3, based on Assumptions 2– 4, to state that the origin of a singular perturbed system in the standard form (2.2) – (2.3) is asymptotically stable. The following assumptions state asymptotic stability requirements on the reduced and boundary-layer systems, expressed by the existence of Lyapunov functions for each system. Theorem 3: [19, Theorem 11.4] Consider the singular perturbed system (2.2) – (2.3) , and Assumptions 2– 4 are satisfied. Then there exists ε∗>0 such that for all ε<ε∗ , the origin of (2.2)–(2.3) is asymptotically stable. □ The proof of Theorem 3 is detailed in [19]. 2.2.3 Stability analysis for hybrid dynamical systems The singular perturbation stability analysis could be adapted and applied to hybrid dynamical systems with slow and fast dynamics, as detailed in [21]. Consider a hybrid system (2.1) , whose state x∈Rn , where n=n1+n2 , is composed of slow variables x1∈Rn1 and fast variables x2∈Rn2 . Then, we can write it in singular perturbation form as follows: 2.2 Singular perturbation theory 19 (diag(In1,εIn2) ˙x∈F(x), x ∈C1×C2 x+∈G(x), x ∈D1×D2 (2.8) where ε > 0 is a small scalar, Ini denotes the ni×ni identity matrix, C1,D1⊂Rn1 and C2,D2⊂Rn2. In [21] the conditions for semi-global practical asymptotic stability of a compact set A=Ar×C2 as ε→0+ associated with the hybrid system are presented. As in Section 2.2.2, these conditions are expressed considering the three key steps in singular perturbation analysis: the quasi-steady-state manifold, the boundary layer system and the reduced system. Furthermore, the well-posedness of the hybrid system is required in this case. The following steps and basic assumptions are considered: a) Well-posed hybrid system: In the first step of this stability analysis, the well-posedness of the hybrid system is considered. Therefore, the regularity of the data of the system must be guaranteed by satisfying the three points of the Assumption 1. b) Manifold: The quasi-steady-state manifold appears in the case of the hybrid approach as a setvalued mapping Ξ : Rn1⇒Rn2 and represents, as in classical singular perturbation theory, the stationary solution of the fast subsystem. Assumption 5: (Regularity of "Manifold"). The set-valued mapping Ξ : Rn1⇒Rn2 is outer semicontinuous and locally bounded, and for each x1∈C1 then Ξ(x1) is nonempty subset of C2. c) Boundary-layer system (fast subsystem): The family of boundary layer systems is given by ˙x∈diag(0,In2)F(x), x ∈(C1∪ρB)×C2(2.9) with ρ > 0 , which makes the flow set compact. The boundary layer is obtained by scaling the time t by 1/ε in (2.8) . Moreover, the boundary layer systems ignore jumps and slow variables remain constants, i.e. ˙x1= 0 during flows. Assumption 6: (Stability of boundary layer). For each ρ > 0 there is a closed ball ρB, such that the compact set M:= {(x1,x2) : x1∈(C1∩ρB),x2∈Ξ(x1)} associated with the boundary layer system (2.9) , is Globally Asymptotically Stable (GAS). 20 Chapter 2. Preliminary concepts d) Reduced system (slow subsystem): The reduced system is obtained considering that the fast dynamics are in steady state given by Ξ(x1) , therefore only slow variables are considered. The reduced system is given by: (˙x∈Fr(x1), x ∈C1 x+∈Gr(x1), x ∈D1,(2.10) where Fr(x1) := diag(In1,0)F(x1,x2), x2∈Ξ(x1) Gr(x1) := diag(In1,0)G(x1,x2) The jump map of the reduced system is not expressed in terms of x2∈Ξ(x1) since the boundary layer system ignores jumps. In the most straightforward case, Gr does not depend on x2 , and thus the reduced system ignores x2 when determining jumps. Our focus is on the stability of the compact set A1:= {x1∈C :x1=x∗ 1} represents the equilibrium state of x1in the reduced model. Assumption 7: (Stability for reduced system). For the reduced system (2.10) , the compact set A1⊂Rn1is globally asymptotically stable. Theorem 4: [21, Theorem 1] Under Assumption 1 and Assumptions 5– 7, the compact set A1×C2is Semi-globally Practically Asymptotically Stable (SPAS) as ε→0+.□ The proof of Theorem 4 is detailed in [21]. 2.3 Multi-Agent System theory A multi-agent system consists of a team of elements, called agents, that exchange information over a communication network. In distributed control strategies, these agents must respond to unexpected situations or changes in the system. To achieve effective control coordination, the team of agents must reach a consensus on the coordination data through aconsensus algorithm. Consensus algorithm: A consensus algorithm is an interaction rule that establishes how agents share information with their neighboring agents in the network. Through these algorithms, the entire team can reach agreement on relevant quantities that depend on the states of all agents [22]. An example of a consensus algorithm is presented below: 2.3 Multi-Agent System theory 21 Consider a multi-agent system formed by i={1,...,N} agents. The objective is to design a distributed control law which ensures that the output vector of each agent, represented by yi , achieves agreement. Therefore, the consensus algorithm can be written as follows: ui=−KX j∈Vi (yi−yj),(2.11) where K > 0 is a control matrix, and Vi the neighborhood of agent i . This model was used in [23]. Graph theory: MAS uses graph theory to represent the network communication between agents. Graphs are pictures consisting of a set of points, some of which are connected by lines. The points, called vertices or nodes, represent the agents, while the lines between points, called edges, represent the communication links. Graphs in which edges have no direction, allowing bidirectional communication between vertices, are called undirected graphs. In contrast, directed graphs have edges with fixed directions [24]. Hence, the following definitions are given: Definition 7: [22] A direct graph is defined as G(N,E) consisting of a set of N elements called vertices, N={1,2,...,N} and a set of ordered pairs of vertices called edges, represented by E ⊆N ×N. The pair {i,j}denotes an edge from the element ito j. Definition 8: [22] An undirected graph consists of a set of vertices N and a set of edges Esuch that, for all pairs of elements i,j ∈N,{i,j}∈Eand {i,j}∈E. Fig. 2.5 represents an example of a graph with four vertices (1,2,3,4) and five edges ((1,2), (1,3), (2,3), (2,4) and (3,4)). Note that this graph is undirected; hence the edge (1,2) is the same as the edge (2,1). Figure 2.5 Example of a graph [24]. If a graph has an edge between every pair of vertices, it is said to be a complete graph. The number of edges incident on a vertex is the degree of a vertex represented by δi , in other words the number of its neighbors. 28 Chapter 3. Continuous control scheme for discharging rate consensus Figure 3.1 Islanded AC microgrid with BESS units. Figure 3.2 Structure of a BESSiin an AC islanded microgrid . As mentioned above, the main control objective to manage the set of N -BESSs in discharging mode is SOC balancing or consensus. Taking into account this statement and the previous literature overview, the contributions of this chapter focus on the design of a distributed control scheme related to the N -BESSs in discharging mode, using a control approach that achieves the following objectives: 3.2 Proposed control scheme 29 Figure 3.3 Proposed distributed control scheme for discharging rate consensus in an islanded AC microgrid. 1) To ensure that the estimated SOCs of the set of BESS in discharging mode in an islanded AC microgrid converge to a consensus in order to increase the battery lifespan. 2) To design a distributed complete control scheme that considers both secondary and primary control loops. 3) To provide large-signal stability analysis of the complete control system. 3.2 Proposed control scheme As motivated above and shown in Fig. 3.3, we propose a distributed control scheme with: •Secondary control loop: – Consensus algorithm: a controller that reaches a balancing of SOCs considering any communication failure. •Primary control loop: – Droop control: a mechanism that ensures the convergence of the SOC and active powers to their references, and generates voltage and frequency references for the inner control loop. –Inner control: a control law for half-bridge inverters. The objective of the secondary control loop is to achieve a consensus among the estimated SOCs, Ψi , of the interconnected BESSs. This is reached indirectly through a consensus algorithm based on MAS applied to the reference variables of the estimated SOCs, Ψr,i . The neighboring BESSs are considered as agents that share the value of their SOC references and independently each calculates the consensus value Ψ∗ . The consensus reference feeds the primary control composed of a droop control and an inner control loop. The droop control based on conventional P/w and Q/V curves is designed to ensure that the estimated SOCs and active powers converge to their references, respectively. This 30 Chapter 3. Continuous control scheme for discharging rate consensus droop control generates the frequency, wr,i , and voltage, Vr,i , references for the inner control loop of the inverter. These three controllers, namely, consensus algorithm, droop control, and inverter control, exhibit a three-time-scale separation through an appropriate choice of control parameters. The aim is to obtain a large signal stability analysis using singular perturbation theory. 3.3 Control objectives The control objectives considered here are formulated as follows: Design a complete continuous-time dynamic model for the primary and secondary control loops in an islanded AC microgrid with a set of N -BESSs in discharging mode, such that it accomplishes the following control objectives for each i∈N := {1,2,...,N} agent: 1. Convergence of the inverter state xi:= [iL,i,vC,i]⊤ , with iL,i the inductance current and vC,i the capacitor voltage, to a reference xr,i through a continuous-time control law that guarantees GAS of a small neighborhood of xi=xr,i(Vr,i,ωr,i). 2. Convergence of the SOC references, Ψr,i , of the interconnected BESSs to a consensus value Ψ∗. 3. Convergence of Ψiand Pi, to their references Ψr,i and Pr,i, respectively. 4. SPAS property for the complete closed-loop control system, considering a largesignal analysis, that includes power converter control, droop control, and reference consensus, by using a three time-scale separation model and applying singular perturbation theory. 3.4 Dynamical models in the primary and secondary control loops This section describes the three proposed closed-loop dynamic models that achieve the control objectives mentioned above. 3.4.1 Consensus algorithm The SOC references consensus algorithm for the BESSs is designed using MAS theory, where each BESS i is considered as an agent and the energy exchange between two agents are considered as edges. The energy exchange between different agents is covered using graph theory and will be presented later in Section 3.5.2. 3.4 Dynamical models in the primary and secondary control loops 31 The dynamic of the consensus algorithm for SOC references ( Ψr,i ) is proposed here ∀i∈N according to ˙ Ψr,i =−αiKc  N X j=1 αj(Ψr,i −Ψr,j)  −Ke(Ψr,i −Ψi(xi)) (3.3) with Kc and Ke positive parameters. The intuitive idea of this expression is the following. The first term of (3.3) looks for achieving a consensus between all Ψr,i , while the second term collects the deviation between the estimated SOC of BESS i , Ψi , and its reference Ψr,i ∈[0,1] . αi= 1 represents if the BESS i is communicating and αi= 0 a communication failure of BESSi. The estimation method for the SOC is adopted from [1] based on the Coulomb counting estimation technique described in Section 3.1.1. If we take into account that the power loss in the conversion is omitted and it is assumed that the batteries have the same output voltage, Vin,i , the output current could be rewritten as ib,i =Pi Vin,i in (3.2) . Thus, the SOC of the BESSidepends on the active power as follows, dΨi dt =−δiPi∀i∈N (3.4) with δi:= k Cbat,iVin,i , where k is a time scale ratio, and Cbat,i is expressed in ampere hours (Ah). The next droop control is key to guarantee that the estimated SOC converges to its reference, tending this deviation to zero in a finite time, as will be proven later. 3.4.2 Droop control A droop control is used here to guarantee that the estimated SOCs and active powers converge to their references, which are given by the consensus algorithm (3.3) . We consider the P/ω droop relation to be crucial for the control objectives, as it has a direct influence on the active power and thus the SOC of the BESSi. Based on the well-known P/ω and Q/V curves, and adding a proportional-integral (PI) adjustment term about the SOC reference value, the droop control modifies the frequency and amplitude of the inverter voltage reference ∀i∈N, as follows: ωr,i =ωn−Kd,1(Ψi−Ψr,i)−Kd,2Z(Ψi−Ψr,i)dt −Kd,3(Pi−Pr,i)(3.5) Vr,i =Vn−KV,iQi,(3.6) where ωn and Vn are the nominal frequency and voltage, respectively, and Pi and Qi are the active and reactive powers. We will prove later, that Ψi and Pi converge asimptotically 32 Chapter 3. Continuous control scheme for discharging rate consensus to their reference values Ψr,i and Pr,i if the droop control parameters are defined as Kd,1=Kω Kc1, Kd,2=Kω Kc1KΨ+K2 c1 Kω+K2 0 KdP  Kd,3=KωK0 Kc1KdP , (3.7) where Kω,KΨ,KdP ,K0, and Kc1 are positive parameters, which will be used later in the hybrid scheme. The error dynamics obtained from (3.5), ∀i∈N are: ˙ ˜ωi=−Kω˜ωi−Kc1˜ Ψi ˙ ˜ Ψi=−KΨ˜ Ψi+Kc1˜ωi+K0˜ Pi ˙ ˜ Pi=−KdP ˜ Pi−K0˜ Ψi, (3.8) with ˜ Ψi= Ψi−Ψr,i , ˜ Pi=Pi−Pr,i , and ˜ωi=ωr,i −ωn,p,i , where ωn,p,i is the nominal frequency associated with the desired reference xr,i , such that ˜ωn,i =ωn,p,i −ωn [27]. Note that (3.5) is obtained from simple mathematical manipulations of (3.8) , which are based on the cross-coupling of errors among ˜ Ψi , ˜ Pi , and ˜ωi , designed to be asymptotically stable using a Lyapunov function candidate, which will be proved in Section 3.6. 3.4.3 Continuous-time control law for the inverters We consider a half-bridge converter as the inverter for the N -BESSs in discharging mode. As shown in Fig. 3.4, this inverter is fed by a DC source and, by the commutation of two switches ( U1,i,U2,i ), generates a single-phase AC output followed by an LC filter. The dynamical model of this inverter can be rewritten according to [38] as follows: ˙xi=Aixi+Biui,∀i∈N (3.9) where Ai="−RLS,i Li−1 Li 1 Ci−1 RiCi# , Bi=Vin,i Li 0. with Ci and Li , the capacitance and inductance respectively, Ri and RLS,i , the nominal converter load and the parasitic resistance respectively, and Vin,i the input voltage. xi= [iL,i,vC,i]⊤∈R2 represents the continuoustime state vector, such that iL,i is the inductance current and vC,i is the capacitor voltage, while ui=U1,i −U2,i ∈{−1,1} is the the discrete-time control input, representing two operating modes: Mode 1: ui=−1,if U1,i =OFF and U2,i =ON →vC,i =−Vin,i Mode 2: ui= 1,if U1,i =ON and U2,i =OFF →vC,i = +Vin,i Note that capacitors C1,i and C2,i are large enough such that the ripple is negligible. The desired output voltage reference is defined by vCr,i =Vr,i sin(ωr,it) 3.4 Dynamical models in the primary and secondary control loops 33 Figure 3.4 Half-bridge inverter. where ωr,i and Vr,i represent the desired frequency and voltage, respectively. From vCr,i , we can derive the inductance current reference, iLr,i =Ciωr,iVr,i cos(ωr,it) + Vr,i Risin(ωr,it) . Then, the reference signal for the complete state xr,i can be generated from a generic oscillator: ˙zi= Θizi(3.10) with zi(0) = h0 Vr,i iand Θi=h0−ωr,i ωr,i 0i, such that z2,i =vCr,i . Hence, zi∈Φi,Φi={z1,i,z2,i ∈R:z2 1,i +z2 2,i =V2 r,i}, therefore the complete state reference xr,i = [iLr,i ,vCr,i ]⊤is defined as follows xr,i = Πizi,(3.11) with Πi=hωr,iCi1 Ri 0 1 i. The control law used for these inverters are two PIs controllers, as shown in Fig. 3.5, which are described in the time domain as follows: ˙ei=K1˙ ˜x2,i +K2˜x2,i ˙vi=KP˙ ˜x1,i +KI˜x1,i ui=vi+Γizi, (3.12) where K1,K2,KP,KI are parameters, ˜x2,i =vC,i −vCr,i and ˜x1,i =iL,i −iLr,i +ei , while ui∈R is the control input. Now, let us define the tracking error as ˜xi=xi−Πizi+ [1 0]eiwith ˜xi= [˜xi,1,˜xi,2]⊤. Then, considering AiΠi+BiΓi= ΠiΘi, with Γi=hωr,iLi RiVin,i +ωr,iRLS,iCi Vin,i 1 Li−Ciω2 r,i+RLS,i LiRiLi Vin,i i, the tracking error dynamic is governed by ˙ ˜xi= ˙xi−Πi˙zi+[1 0]˙ei=Aixi+Biui−ΠiΘizi+[1 0]˙ei =Ai(˜xi+Πizi−[1 0]ei)+Biui−(AiΠi+BiΓi)zi+[1 0] ˙ei =Ai(˜xi−[1 0]ei)+Bivi+[1 0]˙ei. (3.13) 34 Chapter 3. Continuous control scheme for discharging rate consensus Figure 3.5 Block diagram of the controlled inverter. Finally, the closed loop of the tracking error dynamic (3.13) and the control law (3.12) can be written as follows: ˙χi=Miχi,(3.14) with χi=˙ ˜xi ˜xi,Mi=¯ Ci−1¯ Ai¯ Ci−1¯ Bi I 0 , ¯ Ai=AiI−0K1 0 0 +BiKP0+0K2 0 0 , ¯ Bi=Ai0K2 0 0 +BiKI0and ¯ Ci=I−0K1 0 0 . Then, next lemma provides a stability property of the closed-loop system (3.12). Lemma 1: Consider the closed-loop dynamic (3.14) for an inverter i∈N . For a given K1,K2,KP,KI∈R, if there exists any PL,i,QL,i ≻0∈R4×4, such that M⊤ iPi+PiMi≺−2QL,i.(3.15) Then, χi=0is GAS. Proof: The proof is direct, selecting V(χi) = χ⊤ iPiχias Lyapunov function. ■ Note that this lemma establishes, under some assumptions, that the output voltage reaches the desired reference, ensuring output voltage regulation of the inverter, while the inductance current converges to another equilibrium modified by ei. 3.5 Complete control dynamic In this section, we define a complete control system that collects all dynamics for i∈N for a fixed communication network. This complete control dynamic structure with three time-scale separation allows us to provide a large-signal stability analysis of the complete system applying singular perturbation method [31], which will be presented in the following section. First, we define the complete closed-loop dynamics for a BESS i . Then, complete 3.5 Complete control dynamic 35 control dynamics are presented, followed by three time-scale separation guarantees between the three control dynamics. 3.5.1 Closed-loop dynamics for a BESSi First, we define the complete closed-loop dynamics for a BESS i and a fixed communication network, which includes the droop control, the proposed continuous-time control law for the inverter, and the SOC reference consensus algorithm. Let us define the dynamic variables of each control loop for a BESSias follows: •Droop control: ξ1,i = [˜ωi,˜ Ψi,˜ Pi]⊤. •Inverter control: ξ2,i = [˜x⊤ i,ei,ui,zi]⊤. •Consensus algorithm: Ψr,i. Then, the complete closed-loop system for agent iis   ˙ ξ1,i ˙ ξ2,i ˙ Ψr,i  =fi(ξi),(3.16) where ξi= [ξ⊤ 1,i,ξ⊤ 2,i,Ψr,i]⊤and fi(ξi)=                −Kω˜ωi−Kc1˜ Ψi −KΨ˜ Ψi+Kc1˜ωi+K0˜ Pi −KdP ˜ Pi−K0˜ Ψi Ai(˜xi−[1 0]ei)+Bivi+[1 0] ˙ei K1˙ ˜x2,i +K2˜x2,i KP˙ ˜x1,i +KI˜x1,i +Γi(ωr,i,Vr,i)Θ(ωr,i)zi Θi(ωr,i)zi −αiKc N P j=1 αj(Ψr,i−Ψr,j) !−Ke(Ψr,i −Ψi(xi))                . The dynamics for ξ1,i (droop control) are defined in (3.8) and the dynamics for Ψr,i (consensus) in (3.3) . Lastly, the dynamics for ξ2,i (inverter) are given in (3.12) . Moreover, let us recall that αi,αj∈{0,1}represents the communication between BESSs. 3.5.2 Complete control dynamic Now, we define a complete control system that collects all dynamics for i∈ N . We first provide a Laplacian matrix definition, which is used to represent the undirected communication graph among BESSs according to MAS theory and we will consider fixed. This graph is formally described through algebraic graph theory and mathematically represented by a Laplacian matrix. Definition 1: Consider a communication network defined by an undirected graph G(N,E) , where N is the set of BESSs in discharging mode, and E ⊆N ×N the edges, where each 36 Chapter 3. Continuous control scheme for discharging rate consensus edge {i,j}∈E represents a bidirectional communication link between two different agents. The Laplacian matrix represents the communication between neighbours and depends on αij =αiαj∈{0,1} and α={αij ∈{0,1},i,j ∈N} . αij = 0 defines an unintentional disconnection between agent i and j , due to a connection failure of agent i to the PCC or communication failure, for instance, and αij = 1 the connection between them. Note that αij is an exogenous variable. Then, the Laplacian matrix is L(α) = ∆(α)−Ad(α) , with ∆(α)the degree matrix and Ad(α)the adjacency(or connectivity) matrix, where ∆(α):= diag([δ1(α),δ2(α),...,δN(α)]),with δi(α):= N X j=1 aij(α) and, Ad(α):= [aij(α)],with aij(α) : αij if i=jand ∀(i,j)∈E 0if i=jor ∀(i,j)/∈E. The Laplacian matrix L(α)is positive semi-definite [39]. From this definition, for a fixed α, the following complete control system is presented              ˙ ξ1 ˙ ξ2 ˙ Ψr              =              −Kω˜ω−Kc1˜ Ψ −KΨ˜ Ψ+Kc1˜ω+K0˜ P −KdP ˜ P−K0˜ Ψ A(˜x−[1 0]e)+Bv +[1 0]˙e K1˙ ˜x2+K2˜x2 KP˙ ˜x1+KI˜x1+Γ(ωr,Vr)Θ(ωr)z Θ(ωr)z −KcL(α)Ψr+Ke˜ Ψ              (3.17) with ξ1:= [˜ω,˜ Ψ,˜ P]⊤, ξ2:= [˜x,z,v,τ]⊤ Ψr:= [Ψr,1,Ψr,2,..,Ψr,N ]⊤ξ:= [ξ⊤ 1,ξ⊤ 2,Ψr]⊤ ˜ω:= [˜ω1,˜ω2,..,˜ωN]⊤˜ Ψ := [˜ Ψ1,˜ Ψ2,..,˜ ΨN]⊤ ˜ P:= [ ˜ P1,˜ P2,.., ˜ PN]⊤,˜x:= [˜x1,˜x2,..,˜xN]⊤ e:= [e1,e2,..,eN], v := [v1,v2,..,vN] z:= [z1,z2,..,zN]⊤, A := diag{A1,A2,..,AN} B:= [B1,B2,..,BN]⊤,Θ := diag{Θ1,Θ2,..,ΘN} Γ := diag{Γ1,Γ2,..,ΓN} The desired equilibrium associated with system (3.17) is {ξe 1,ξe 2,Ψe r}∈Se 1×Se 2 Se 1={ξ1= 0 0 0,˜x= 0 0 0,e,v ∈RN,zi∈Φi∀i∈N} Se 2={L(α)Ψr= 0 0 0}.(3.18) 3.6 Stability analysis of the complete continuous-time system 37 3.5.3 Three time-scale separation As mentioned before, we can identify three control objectives: power converter control, droop control, and reference consensus, which are required to work in different time-scales, as is an intrinsic property in microgrids. Generally, the droop control evolves slowly enough so that voltage and frequency references can be kept approximately ’constant’ by the converters. It is desirable for the consensus reference loop in the secondary control to be faster than the controlled power converters and the droop control. In this context, where a three-timescale separation can be established for complete system (3.17) , singular perturbation theory [19] is appealed to provide nonlinear stability guarantees. Assumption 8: Consider (3.17) . Then, there exist some parameters Kc>> Ke>0 , K0 , Kc1 , Kω , KΨ , KdP >0 , Kdc = min i∈N(min|eig(K)|) with K=−Kω−Kc10 Kc1−KΨK0 0−K0−KdP  , K1,K2,KP,KI>0and Kinv = min i∈N(min|eig(Mi)|)such that condition (3.15) and Kc>> Kinv >> Kdc are satisfied. Note that 1 Kc,1 Kinv ,1 Kdc represent the estimations of the convergence speed of each control loop. Then, this assumption implies that Ψr is faster than ξ2 , because 1 Kc<< 1 Kinv and, that ξ2 is faster than ξ1 , because 1 Kinv << 1 Kdc . Moreover, note that Kinv results from the selected parameters K1,K2,KP,KI>0and the inverter parameters. 3.6 Stability analysis of the complete continuous-time system Inspired by [40], this section is devoted to the stability analysis of the complete continuoustime system given by (3.17) . Following Section 2.2.2, we provide here a large-signal stability analysis for the distributed control scheme proposed for an islanded AC microgrid, using three-time-scale separation and singular perturbation theory. Singular perturbations approaches are used to simplify complex systems by separating the dynamics of the whole system into slow and fast dynamics if Assumption 8 is satisfied, we find that ξ1 is slower than ξ2 , which in turn is slower than Ψr . This results in a three-time-scale separation, making the system suitable for applying the singular perturbation method. 3.6.1 Singular perturbation form Then, introducing the parameters ν1 and ν2 , the system (3.17) is rewritten in the following singular perturbation form: 44 Chapter 3. Continuous control scheme for discharging rate consensus Figure 3.10 Scenario 1: In the top, evolution of the SOC references, Ψr,i , the SOCs, Ψi , and the active powers, Pi for i={1,2,3} , when a load of 22 Ω is connected at T2 and then it is disconnected back at T3 . In T3 , α1,2= 0 . In the bottom a zoom of SOC references at T1. Fig. 3.13. At time T1 , the SOCs, whose initial conditions are different, start to converge to a consensus. During the time period T3−T2 , the SOC and active power of the disconnected BESS 1 remain constant, and the other BESSs empty out faster because they have to absorb the capacity of the disconnected BESS. From T3 BESS 1 recovers its activity, compensating for its storage energy with the other BESSs. Moreover, the evolution of voltage errors, capacitance voltages, and inductance currents are shown in Fig. 3.14, validating the 3.7 Experimental validation 45 0 20 40 60 -100 0 100 0 20 40 60 -100 0 100 0 20 40 60 -100 0 100 0 20 40 60 -100 0 100 0 20 40 60 -100 0 100 0 20 40 60 -100 0 100 0 20 40 -5 0 5 0 20 40 60 -5 0 5 0 20 40 60 -5 0 5 T2 T3 T1 T1 T2 T3 T3 T2 T1 Figure 3.11 Scenario 1: Evolution of voltage errors, ˜x2,i , capacitance voltages, vC,i , and inductance currents, iL,i for i={1,2,3}. Figure 3.12 Scenario 1: Zoom at T1 of SOC references Ψr,i , and voltage errors ˜x2,i , for i={1,2,3}. robustness of the power converter control loop after any connection/disconnect event. Once again, note the three-time-scale separation. The SOC references converge to a consensus with transient times of less than 0.2 ms. This evolution is faster than the SOCs and the active powers convergence to their references. Moreover, as shown in Fig. 3.15, the time response of the DC/AC converters (approximately 2ms) is longer than the convergence speed of the SOC references, but shorter than convergence speed of the SOCs and the 46 Chapter 3. Continuous control scheme for discharging rate consensus Figure 3.13 Scenario 2: In the top, evolution of the SOC references, Ψr,i , the SOCs, Ψi , and the active powers, Pi , for i={1,2,3} , when BESS 1 is disconnected at T2 and then it is connected back at T3 . In the bottom, a zoom of SOC references at: a) T1and b) T2. active powers convergence speed. From these scenarios, we can validate the statement of Theorem 5. 3.7.3 Comparison with [1] This section focuses on the experimental comparison between our control approach and the one proposed in [1], which introduces a discrete-time average MAS consensus algorithm. 3.7 Experimental validation 47 0 50 -200 0 200 0 50 -200 0 200 0 50 -200 0 200 0 50 -200 0 200 0 50 -200 0 200 0 50 -200 0 200 0 50 -5 0 5 0 50 -5 0 5 0 50 -5 0 5 T1 T2 T3 T1 T2 T3 T1 T2 T3 Figure 3.14 Scenario 2: Evolution of voltage errors, ˜x2,i , capacitance voltages, vC,i and inductance currents, iL,i for i={1,2,3}. Figure 3.15 Scenario 2: Zoom at T2 of SOC references, Ψr,i , and voltage errors, ˜x2,i , for i={1,2,3}. As previously described in Section 4.6.1, this algorithm calculated the average value of its neighbors, SOCmean,i , for the SOC i of the set of Ni BESSs units. Furthermore, the 48 Chapter 3. Continuous control scheme for discharging rate consensus Figure 3.16 Comparison of the SOC and active power evolution between a) our control and b) the controller proposed in [1]. droop control modifies the reference of the frequency with respect to the deviation between the SOCiand the average value of its neighbors. We implement the control algorithm given in [1], selecting KP=Kd,3 , using the same control parameters given in Table 3.2, and setting σ= 0.3333 and KSOC = 0.05 according to [1]. The comparison is performed under the conditions of Scenario 1, which mimics a scenario similar to the one in [1]. At instant time T1 , both consensus algorithms are activated. Then, at instant time T2 a second load of 22 Ω is connected, and then, in T3 , it is disconnected. Figure 3.16 shows a comparison of the SOCs and the active power evolution between our control proposal and the controller proposed in [1]. Each SOCmean,i and Ψr,i converge to a consensus with their neighbors, respectively. Likewise, each SOC converges to SOCmean,i and Ψr,i , respectively. Note that the convergence speed of the SOCs and the active powers is faster with our proposed controller. Moreover, our control algorithm ensures a large-signal stability for the complete nonlinear continuous-time system, while [1] provides a small-signal stability analysis for a discrete system. 3.8 Conclusion 49 3.8 Conclusion This chapter provides a distributed control scheme of three-time scales, composed of a primary and secondary control loops for AC-bus islanded microgrids. The secondary loop consists of a consensus algorithm based on MAS theory connected to a droop control, ensuring the SOC convergence to a consensus, thereby increasing battery lifespan and providing robustness with respect to any communication failure. Moreover, the primary loop consists of a droop control and an inverter control, which includes voltage and current continuous-time control loops to enable the use of PWM, commonly required in the industry. Under the satisfaction of Assumption 10, a three-time scale separation is ensured between each control goal (consensus algorithm, droop control, and power converter control). This property allows us to apply singular perturbation theory to ensure large-signal stability of the complete nonlinear system. The proposed distributed control is experimentally validated through an Imperix power test bench under various scenarios. In addition, a comparison with the controller given in [1] is provided. Furthermore, future work envisions extending these results by adding an extra control loop to manage the operation modes of batteries, loads, and sources (charging mode or discharging mode) to improve battery degradation. Furthermore, extending the results to considering discrete-time dynamics is also proposed. In the next chapter, a hybrid model of the proposed control scheme will be discussed. 4 Hybrid dynamical control scheme for discharging rate consensus T he previous chapter exhibited the contribution to discharging rate consensus through a continuous-time control scheme for the primary and secondary control loops for the set of N -BESSs in an AC islanded microgrid. In contrast with the preceding chapter, this chapter presents a hybrid dynamical control approach for the proposed distributed control scheme that addresses the problem formulation mentioned in Section 3.1.1. The objective of this distributed control scheme is to achieve SOC balancing using the aforementioned proposed secondary control loop and droop control. However, in the primary loop, we consider a hybrid dynamical power converter control loop, taking into account the nonlinearities of the power converter models (switches and affine terms) and of the signals (dwell time constraints). A particular interest has emerged to apply hybrid control techniques to power converters with the aim of considering the subtlety of the hybrid nature of these systems. Indeed, they are composed of continuous-time dynamics (voltages and currents) and discrete-time dynamics (switching). Previous works on switched affine systems have considered a hybrid framework [38], [39], [40], [41] based on HDS theory given in [42]. Some of them have also taken into account practical issues, such as minimum dwell time, robustness with respect to parameter variation, absence of sensor signal or even PWM implementation. In this chapter, the control objectives are first presented, and afterwards the hybrid control loop for the power converter with voltage output regulation is presented. Then, a complete hybrid dynamic model for the secondary and primary control loops using HDS theory is discussed, followed by the stability analysis using singular perturbation theory applied to HDS. Finally, the chapter concludes with simulation validation results, providing a comparison between the proposed control scheme and the one presented in [1]. 51 52 Chapter 4. Hybrid dynamical control scheme for discharging rate consensus 4.1 Control objectives The goal here is to design a complete hybrid scheme for the primary and secondary control loops in an islanded AC microgrid with a set of N -BESSs in discharging mode, such that it accomplishes the following control objectives for each i∈N := {1,2,...,N}agent: 1. Convergence of the half-bridge inverter state xi:= [iL,i,vC,i]⊤ , with iL,i the inductance current and vC,i the capacitor voltage, to a reference xr,i , through a hybrid control that considers a switched affine model and a minimum dwell time. Moreover, UGAS of a small neighborhood of xi=xr,i(Vr,i,ωr,i) , as well as robustness output voltage regulation should be ensured. 2. Convergence of the SOC references, Ψr,i , of the interconnected BESSs to a consensus value Ψ∗. 3. Convergence of Ψiand Pi, to their references Ψr,i and Pr,i, respectively. 4. SPAS of the complete hybrid dynamical system, through a large-signal analysis, that includes inverter control, droop control, and consensus algorithm, by using a three time-scale separation model and applying singular perturbation theory in HDS. 4.2 Inverter model and control In this section, we present a hybrid control law for the inverters of the N -BESSs in discharging mode in AC islanded microgrid. As in Section 3.4.3 we consider half-bridge converters as the inverters for the BESSs as well as the dynamical model presented in [38]. Let us to recall from Chapter 3, that zi is the state of a generic oscillator used by the reference generator block with matrices Θi(ωr,i) = h0−ωr,i ωr,i 0i and Πi(ωr,i) := hωr,iCi1 Ri 0 1 i to generate the complete state reference xr,i = [iLr,i ,vCr,i ]⊤ , previously defined in (3.11). Moreover, Ai="−RLS,i Li−1 Li 1 Ci−1 RiCi#and Bi=Vin,i Li 0. The dynamics of the overall system is described by ˙xi=Aixi+Biui ˙zi= Θi(ωr,i)zi ˜xi=xi−Πi(ωr,i)zi, (4.1) where ˜xiis the tracking error with ˜xi= [˜xi,1,˜xi,2]⊤. 4.2 Inverter model and control 53 Then, consider the following algebraic equation: AiΠi+BiΓi= ΠiΘi.(4.2) From simple calculations, we yield Γi=hωr,iLi RiVin,i +ωr,iRLS,iCi Vin,i 1 Li−Ciω2 r,i+RLS,i LiRiLi Vin,i i, from (4.1) and following (4.2) we can define the tracking error dynamic as follows: ˙ ˜xi= ˙xi−Πi˙zi=Aixi+Biui−ΠiΘizi =Ai(˜xi+Πizi)+Biui−(AiΠi+BiΓi)zi =Ai˜xi+Bivi, (4.3) with vi=ui−Γizi∈R.(4.4) The new control input to be designed, vi , is composed of a continuous-time signal Γizi and a switching signal of two logical modes ui∈K={−1,1} . Nevertheless, we need to keep in mind that the control applied to the half-bridge inverter is ui . This control law for the switching signal ui must ensure suitable convergence properties of the inverter error variable ˜xito 0. To do so, we introduce next assumption. Assumption 9: [38] Consider model (4.1) . For a given matrix QL,i ≻0∈R2×2 , there exists a matrix PL,i ≻0∈R2×2such that 1. matrix Aiverifies AT iPL,i +PL,iAi+2QL,i ≺0(4.5) 2. and, for any periodic signal zi(t)∈Φi of period Tp , there exists a λj,i(t) for each j∈Ksuch that λ−1,i(t)+λ1,i(t)=1such that λ1,i(t)−λ−1,i −Γizi(t)=0.(4.6) The Hurwitz requirement for matrices Ai is a common property in converter models [38,41]. The control law used here for vi(or equivalently ui) is [38] vi= (argmin ui∈K˜x⊤ iPL,i(Ai˜xi+Bi(ui−Γizi)))−Γizi.(4.7) The “argmin” operator finds the value of ui which minimizes the Lyapunov function, V(˜xi) = ˜x⊤ iPi˜xi. The UGAS property is given in [38]. 4.2.1 Voltage output regulation for inverters It is well known that power converters suffer variations in the load, the voltage input, or other components. Therefore, it is necessary to guarantee robustness in the output voltage 60 Chapter 4. Hybrid dynamical control scheme for discharging rate consensus Proposition 4: Under Assumption 10 and for a given α , the quasi-steady-state equilibrium ‘manifold’ of Hsp is regular. Proof: Note that the dynamic of variable Ψr in (4.16) – (4.17) is purely continiuous. The quasi-steady-state equilibrium ’manifold’ of Hsp is computed when ν1= 0 and −L(α)Ψr+ν1Ke˜ Ψ = 0. Then, it is: Ξ(ξ1,ξ2) = (Ψ∗diag(αi)1(ξ1,ξ2)∈C ∅(ξ1,ξ2)/∈C with Ψ∗ a parameter. Then, Ξ is a set-valued mapping, which is empty outside of set C . ■ 4.5.4 Stability of the boundary layer system The next proposition provides a stability analysis of the ‘boundary layer’ between (ξ1,ξ2) and Ψr , which is obtained by scaling ordinary time by 1/ν1 to τ=t/ν1 in the original system (4.20)–(4.21) and then setting ν1= 0. Proposition 5: Under Assumption 10 and for a given α , there exists a closed ball ρiBi of radius ρi>0such that the compact set M={i∈N : (ξ1,i,ξ2,i)∈(Ci∩ρiBi),Ψr∈Ξ(ξ1,ξ2)} associated with the boundary layer   ˙ ξ1 ˙ ξ2 ˙ Ψr  =  0 0 −L(α)Ψr (ξ,Ψr)∈\ i∈N (Ci∩ρiBi)×RN,(4.22) is GAS. Proof: First note that if Assumption 10 is satisfied there is a time-scale separation between between (ξ1,ξ2) and Ψr that allows getting the boundary layer (4.22) as follows, because ν1<< 1/Ke . In other words, we zoom in on the faster variable Ψr , causing the slower variables to become frozen, remaining ξ1 and ξ2 constant during flows. Moreover, we stress that the boundary layer system ignores jumps. The fast subsystem dynamics from (4.20) can be written as: ν1 dΨr dt =−L(α)Ψr+ν1Ke˜ Ψ replacing τ=t/ν1, and setting ν1= 0 ν1 dΨr d(ν1τ)=dΨr dτ =−L(α)Ψr+ν1Ke˜ Ψ =−L(α)Ψr 4.5 Stability analysis of the complete hybrid system 61 Now, consider the Lyapunov function candidate Vc:= 1 2Ψr⊤L(α)Ψr. Then, ⟨▽Vc(Ψr),fbl⟩=−Ψr⊤L(α)L(α)Ψr≤0, with fbl := [0⊤0⊤−Ψ⊤ rL(α)]⊤ Note that ⟨▽Vc(Ψr),fbl⟩ is negative semidefinite. Then by LaSalle’s principle, we can conclude the proof stating that all agents interconnected converge to a consensus. Consequently, (4.22) is GAS. The dynamic defined in (4.22) guarantees that each Ψr,i with associated αi= 1 converges to a neighborhood of Ψ∗ which defines a consensus between the interconnected agents. ■ 4.5.5 Stability of the reduced system Now, we establish the next result for the reduced system associated with Hsp . Note that the reduced system’s jump map is not expressed in terms of Ψr∈Ξ(ξ) since the boundary layer system ignores jumps. Proposition 6: Under Assumption 10 and for a given α , the attractor Ar:= S i∈N Ar,i with Ar,i := {(ξ1,i,ξ2,i)∈Hi:∥˜xi∥< Xi,ξ1,i =0}associated to the reduced system: Hr(ξ):                                                  ˙ ξ1 ν2˙ ξ2           =           −Kω˜ω−Kc1˜ Ψ −KΨ˜ Ψ+Kc1˜ω+K0˜ P −KdP ˜ P−K0˜ Ψ ν2A˜x+ν2Bv ν2Θi(ωr,Vr)z −ν2Γ(ωr,Vr)Θ(ωr)z ν21           ξ∈C       ξ1 + ν2ξ+ 2      ∈      ξ1 ν2˜x ν2z ν2h(ξ) 0       ξ∈D, (4.23) is SPAS as ν2→0+. Proof: Consider the following Lyapunov function for the reduced system Hr, V(ξ1,˜x) := X i∈N Vi(ξ1,i,˜xi) Vi(ξ1,i,˜xi) = 1 2(ξ⊤ 1,iξ1,i + ˜x⊤ iPL,i ˜xi)∀(ξ1,i,ξ2,i)∈Hi\Ar,i. 62 Chapter 4. Hybrid dynamical control scheme for discharging rate consensus Applying [38, Theorem 1] to hybrid system Hr, it is got ⟨∇Vi(ξ1,i,˜xi)⟩=−ξ⊤ 1,iSξ1,i +αi˜xiPL,i(Ai˜xi+Bivi) ≤−ξ⊤ 1,iSξ1,i −αiηi˜x⊤ iQL,i ˜xi<0∀(ξ1,i,ξ2,i)∈Ci\Ar,i (4.24) Vi(ξ+ 1,i,˜x+ i)−Vi(ξ1,i,˜xi)=0 ∀(ξ1,i,ξ2,i)∈Di\Ar,i (4.25) with S:= diag{Kω,KΨ,KP}. Note that the dynamic of ˙xi1is a continuous-time autonomous system. The proof is concluded by applying [38, Theorem 3]. ■ Theorem 6: Under Assumption 9, 10 and for a given α , the set A=Ar×{L(α)Ψr=0} associated with the hybrid system H, (4.16)–(4.19), is SPAS as ν1,ν2→0+.□ Proof: Note that if Assumptions 9 and 10 are satisfied, then Propositions 3–6 are hold. Consequently, the proof is direct from [21, Theorem 1]. ■ 4.6 Simulation results In this section, the hybrid control proposed above is validated in Matlab/Simulink using the Electrical Toolbox for an AC islanded microgrid composed of three BESSs in discharging mode, as shown in Fig. 4.1. A similar scenario was given in [1]. Each half-bridge is emulated as an AC controlled voltage source whose input is the equivalent voltage modulated by the two switches, given by Vm,i =uiVin,i , followed by its corresponding LC filter. The output current, io,i is also measured to calculate the output active power and the line impedances are included. The bus line is Vbus = 120√2sin(2π60t)V , then Vn= 120√2 V and ωn= 2π60rad/s . Tables 4.1 and 4.2 provide the parameters for the secondary and primary control loops respectively, i.e., the parameters for the hybrid model (4.11) – (4.14) . From these parameters, we have Ke/Kc= 0.01 , Kinv = 34.7 and Kdc = 0.51 , then it is easy to see that Assumption 10 is satisfied. Furthermore, matrices PL,i and QL,i are selected providing some LQR (Linear-Quadratic Regulator) performance guarantee in order to reduce, for instance, the energy cost, current peaks, and response time for the voltage or current. To this end, we apply [45, Theorem 2]. Therefore, satisfying Linear Matrix Inequality (LMI) tools with the inverter parameters given in Table 4.2, we obtain: PL,i =28.20 0.12 0.12 0.08and QL,i =1.50 0 0 5.55. 4.6 Simulation results 63 Figure 4.1 Microgrid configuration scheme. Table 4.1 Consensus algorithm and droop control parameters. Parameter Value Parameter Value Kc10000 KdP 50 Ke100 Kc10.5 KΨ0.45 K00.05 Kω5 Table 4.2 Inverter parameters. Parameter ∀iValue Parameter ∀iValue Vin,i 48 V ωn2π60 rad/s Li50 mHVn120√2V Ci140.72 µFZline,i R=0.1 Ω L=1.6 m H RLS,i 1.5 ΩT0.1 ms Ri180 ΩCbat,i 0.1615 Wh Xi0.041 k1/3600 K1 The Laplacian matrix that represents the communication interconnections among the three BESSs is the same as previously given in (3.22), as shown in Fig. 3.9. To validate the effectiveness of our proposed strategy, we performed the SOC balancing verification through a simulation scenario. This scenario will test the performance of the system against communication failures and the capability to ride out plug-and-play events (i.e. intentional or unintentional shutdown of a BESS). A comparison with a similar consensus proposal in [1] is also presented below. 64 Chapter 4. Hybrid dynamical control scheme for discharging rate consensus 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 50 100 150 200 250 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 012 10-4 0.8 0.9 1 1.7 1.7001 1.7002 0.6 0.65 Figure 4.2 Evolution of Ψr,i , Ψi , and the active powers Pi , for i={1,2,3} , when the battery 1 is disconnected at T2 and then connected back at T3 . In T1 , α1,3 turns to 0. We consider that the net power (load power minus DG power) is Pn= 300 W, which is represented by the load connected in the configuration scheme in Fig. 4.1. The initial conditions of the SOCs are different among them, with values of Ψ1(0) = 1 , Ψ2(0) = 0.9 , and Ψ3(0) = 0.8 , respectively. At the instant time T2= 1.5 s the BESS 1 is disconnected (i.e., α1= 0 ), and then it is connected back in T3= 1.7 s. Also, at T1= 0.5 s, a communication failure occurs between BESS1and BESS3, i.e., α1,3=α3,1= 0. Fig. 4.2 shows the convergence of the SOC references, (or Ψr,i ), to a consensus from the initial conditions and after the BESS plug-and-play event. Note that, at the start of the simulation, the SOC references converge to a consensus with transient times of less than 0.1ms approximately. However, the active powers are perturbed on this time scale. In fact, they initially diverge because the BESS with a higher SOC ( Ψ1 ) shares more power, while the unit with lower SOC ( Ψ3 ) shares less power. Hence, on a larger time scale, active powers and estimated SOCs converge to a consensus. Indeed, the SOCs and the active powers converge to their references before to 2.4s approximately (note that 1/Kdc = 2.4 ) and Ψr in 0.1ms approximately (note that 1/Kc= 10−4 ). Moreover, note that the signals evolve robustly after the communication failure. 4.6 Simulation results 65 1.5 1.6 1.7 1.8 -200 0 200 1.5 1.6 1.7 1.8 -10 0 10 1.5 1.6 1.7 1.8 -1 0 1 1.5 1.6 1.7 1.8 -200 0 200 1.5 1.6 1.7 1.8 -10 0 10 1.5 1.6 1.7 1.8 -1 0 1 1.5 1.6 1.7 1.8 -200 0 200 1.5 1.6 1.7 1.8 -10 0 10 1.5 1.6 1.7 1.8 -1 0 1 Figure 4.3 Evolution of the voltages, currents and duty cycles of inverters for i={1,2,3} between 1.45s to 1.8s. 024 -200 0 200 0 2 4 -200 0 200 024 -200 0 200 Figure 4.4 Evolution of the voltage errors. For the interval T3−T2 , the remaining BESSs have to discharge faster as they have to absorb the capacity of the disconnected BESS 1 . At T3 BESS 1 resumes its operation, balancing its stored energy with the other BESSs. In addition, Fig 4.3 shows the evolution of the capacitance voltages, inductance currents, and control inputs, validating the robustness of the power converter control loop after any connection/disconnect event. Furthermore, in Fig. 4.4 the voltage errors are shown. Note that the voltage errors converge to zero, ensuring that the voltage outputs follow their references, validating voltage output regulation. Note that these errors present a transient time less than 30ms (note that 1/Kinv = 34.48 ). Therefore, inverters evolve slower than the SOC references to a consensus and, faster than the droop control, validating the time-scale separation assumption, hence the statement of Theorem 6. 4.6.1 Comparison with [1] This section focuses on a comparison between our control approach and the one proposed in [1], which introduces a discrete-time average MAS consensus algorithm. In [1], the droop control modifies the reference of the frequency with respect to the deviation between 66 Chapter 4. Hybrid dynamical control scheme for discharging rate consensus 01234 0 0.2 0.4 0.6 0.8 1 01234 0 0.2 0.4 0.6 0.8 1 01234 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 (b)(a) (a) (b) Figure 4.5 Comparison of simulation results of the SOC evolutions between the controller proposed in [1] (a) and our hybrid control proposal (b). the SOCiand the average value of its neighbours, SOCmean,i, as follows: ωr,i =ωn+KSOC(Ψi−SoCmean,i)−KPPi,(4.26) SOCmean,i =SOCi+σZt 0X j∈Ni (SOCmean,j(τ)−SOCmean,i(τ))dτ, (4.27) where Ni is the set of neighbours of agent i , KSOC and KP are the droop gains, and σ is a scalar tuning parameter which depends on N . Note that (4.26) is similar to the one given in (3.5) . Both equations have the proportional term related to the error between the estimated SOC with respect to SOCmean,i and Ψr,i , respectively. However, our proposal adds an integral term of the error between the estimated SOC and a term with the difference between the active power and its reference. Additionally, note that (4.27) lacks the term on the left side in (3.3) , which gathers the deviation between the estimated SOC, Ψi , and its reference Ψr,i . To simulate this algorithm, σ= 0.3333 and KSoC = 0.0015 are optimally selected according to [1] and the batteries parameters are given in Table 4.2. Moreover, in T1= 2s a load of 150W is connected and then, in T2= 3s , is disconnected to mimic the scenario given in [1]. Figure 4.5 shows a comparison of the SOC evolutions between the controller proposed in [1] (a) and our control proposal (b). Note that each SOCmean,i and each Ψr,i converges 4.7 Conclusions 67 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 50 100 150 200 250 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 50 100 150 200 250 (a) (b) Figure 4.6 Comparison of simulation results of the active power evolutions between the controller proposed in [1] (a) and our hybrid control proposal (b). to a consensus, respectively. Moreover, the SOCs converge to SOCmean and to their Ψr,i when using the controller given in [1] and our control solution, respectively. The convergence speed of the SOCs is faster in our proposition. Furthermore, in Fig. 4.6 there is a comparison of the active power evolutions. Here, we again note the improvement obtained with our control algorithm with respect to the response time. The oscillations given in the steady state are due to the event-triggered control used in our proposition. Indeed, different to [1], we do not use PWM in the inverter control signal, but a hybrid control which may not be suitable for industrial applications due to high sampling frequency requirements. Note that our proposition also provides a large-signal stability analysis for the complete hybrid system, i.e., considering the continuous-time and discrete-time dynamics. Indeed, in [1], a small-signal stability analysis is given for a discrete system. 4.7 Conclusions This chapter provides a complete hybrid scheme for the primary and secondary control loops in an islanded AC microgrid to achieve SOC consensus among the BESSs in discharging mode. The scheme considers nonlinearities in the inverter model (switching and affine terms) and in the signals (minimum-dwell time constraint). Moreover, the secondary loop is composed of a consensus algorithm to increase the battery lifespan, 68 Chapter 4. Hybrid dynamical control scheme for discharging rate consensus as well as provide scalability of the microgrid with respect to plug-and-play events and communication failures. Furthermore, a large-signal stability analysis for the three control loops in a whole: inverter control, droop control, and consensus algorithm is obtained by applying singular perturbation theory to a time-scale separation hybrid model obtained by selecting the controller gains appropriately. Then, it is concluded that the attractor A is SPAS given a given interconnected graph L(α). 5 Hybrid dynamical control scheme for reactive power sharing T his chapter presents an extended version of the hybrid control scheme introduced in Chapter 4, incorporating reactive power sharing into the control loops for a set of N -BESSs operating in discharging mode within an AC islanded microgrid. The primary objectives of this enhanced hybrid control scheme are to achieve SOC balancing and optimal reactive power sharing, utilizing a modified version of the previously proposed secondary control loop and droop control. This chapter begins by introducing a literature review and describing the reactive power consensus strategy. Then, a complete hybrid dynamical model for the primary and secondary control loops is provided. This model exhibits a three-time-scale separation, enabling the application of the singular perturbation method to give stability guarantees of the complete system, which will be detailed below. The chapter concludes with simulation results that illustrate the model’s performance across various application scenarios. 5.1 Introduction Optimal reactive power sharing is one of the main objectives for parallel interconnected DG units in a microgrid, addressing crucial issues such as diminishing voltage deviations and uncertainties in the line impedances. In high-voltage networks, reactive power sharing between generators is usually not a major issue due to capacitive compensation of both the loads and the transmission lines. Therefore, generators voltages are controlled to fixed values. However, the situation is different in small-scale microgrid applications where islanded AC microgrids are involved. In these cases, the low capacities of the DGs, the short electrical distances between the units, and the lack of static compensation require a precise distribution of the reactive power demand between the DGs. This precise sharing is necessary to avoid overloading of the individual units [47]. Droop control methods are extensively used to obtain an acceptable power sharing [48,49]. These network control methods are of particular interest due to their scalability, 69 76 Chapter 5. Hybrid dynamical control scheme for reactive power sharing 5.6.1 Three time-scale separation We rewrite Assumption 10 to include the reactive power consensus parameters, establishing the conditions necessary for achieving a three-time-scale separation among the three control dynamics. The inverter and droop control response speed are estimated from Kinv := min i∈N(min|Re(eig(Ai))|)and Kdc := min i∈N(min|Re(eig(K))|), such that K:      −Kω−Kc10 0 0 Kc1−KΨ0 0 K0 0 0 −KV−K20 0 0 K2−KQ0 0−K00 0 −KP       , respectively. Therefore, the three-time scales among ξ1,ξ2and (Ψr,Qr)are given if the next assumption is satisfied. Assumption 11: Consider (5.8) – (5.11) . Then, select T , Kc>> Ke>0 and matrix K , such that (i) Kc>1 T>> Kinv (ii) Kc>> Kinv >> Kdc are satisfied. Note that 1 Kc , 1 Kinv and 1 Kdc represent the estimations of the convergence speeds of (Ψr,Qr),ξ2and ξ1, respectively. The next parameter design guideline is given to provide a method of tuning the parameters in order to satisfy Assumption 11. Algorithm 2 Parameter design guideline 2 1: Estimate: Kinv. 2: Select: T,Kcand Ke, s.t. Ass. 11, item (i) is satisfied. 3: Select : Kω,KΨ,KV,K0and K1s.t. Kd,1>Kd,2>Kd,3. 4: Select : KQ,KPand K2s.t. Kd,4<Kd,5. 5: Test : Kinv >> Kdc s.t. Ass. 11, item (ii) is satisfied. 5.7 Stability analysis of the complete hybrid system This section focuses on analyzing the stability of H . This analysis is grounded in the theory of singular perturbation applied to hybrid control systems, as described in [21] similar to the development provided in the Section 4.5. Let us consider that the tuned parameters are selected satisfying Assumption 11 such that system H is composed by 5.7 Stability analysis of the complete hybrid system 77 three time-scale separated dynamic loops where ξ1 is slower than ξ2 and these last ones slower than (Ψr,Qr). Then, His rewritten in the following singular perturbation form: Hsp(ξ) :                                ˙ ξ1 ν2˙ ξ2 ν1˙ Ψr ν1˙ Qr     =fsp(ξ,Ψr,Qr),(ξ,Ψr,Qr)∈C×R2N,     ξ1 + ν2ξ+ 2 Ψ+ r Q+ r    ∈Gsp(ξ,Ψr,Qr),(ξ,Ψr,Qr)∈D×R2N, (5.12) such that fsp(ξ) :=                    −Kω˜ω−Kc1˜ Ψ −KΨ˜ Ψ+Kc1˜ω+K0˜ P −KV˜ V−K2˜ Q −KQ˜ Q+K2˜ V −KdP ˜ P−K0˜ Ψ ν2A˜x+ν2Bv ν2Θ(ωr)z −ν2Γ(ωr,Vr)Θ(ωr)z ν21 −L(α)Ψr+ν1Ke˜ Ψ −L(α)Qr+ν1KeQ                    , Gsp(ξ) =           ξ1 ν2˜x ν2z ν2h(ξ) 0 Ψr Qr           (5.13) with ν1:= 1/Kc and ν2= 1/Kinv . Highlighting that the jump set G does not change for dynamics ξ1and (Ψr,Qr)in a singular perturbation form. 5.7.1 Regularity of System’s Data Following [21], we impose basic assumptions on (5.12). Proposition 7: (Assumption 1) For a given αsystem Hsp(C,fsp,D,Gsp)is well-posed. Proof:Hsp(C,fsp,D,Gsp)verifies the following properties •C×R2Nand D×R2Ngiven in (5.10) and (5.11) respectively, are closed sets; •fsp is a continuous function, thus locally bounded and outer semicontinuous. Furthermore, it is convex and non-empty for each ξ∈C×R2N; •Gsp is outer semicontinuous and locally bounded. Moreover, for each ξ∈D×R2N , it is non-empty; Then, Hsp satisfies the hybrid basic conditions and therefore it is well posed (Theorem 1). ■ 78 Chapter 5. Hybrid dynamical control scheme for reactive power sharing 5.7.2 Regularity of the ’manifold’ The next result establishes a property of the quasi steady state manifold. It is a set-valued map from the slow variables to the fast variables, as Ξ : H⇒R2N. Proposition 8: For a given α and under Assumption 11, the quasi-steady-state equilibrium ’manifold’ of Hsp is regular. Proof: Observe that the variables Ψr and Qr in equations (5.8) – (5.9) exhibit purely continuous dynamics. Hence, we calculated the the quasi-steady-state equilibrium ’manifold’ doing ν1= 0 and, −L(α)Ψr+ν1Ke˜ Ψ = 0 −L(α)Qr+ν1Ke˜ Q=0 Therefore, Ξ(ξ1,ξ2) =      Ψ∗diag(αi)1 Q∗diag(αi)1(ξ1,ξ2)∈C ∅(ξ1,ξ2)/∈C where Ψ∗ and Q∗ are parameters. Then, Ξ is a set-valued mapping, which is empty outside of set C. ■ 5.7.3 Stability of the ‘boundary layer’ The following proposition presents a stability analysis of the ‘boundary layer’ between (ξ1,ξ2) and (Ψr,Qr) , which is obtained by scaling the ordinary time by 1/ν1 in the original system (5.12)–(5.13) and then setting ν1= 0. Proposition 9: Consider that Assumption 11 is satisfied then, there exists a closed ball ρiBiof radius ρi>0such that the compact set M={i∈N : (ξ1,i,ξ2,i)∈(Ci∩ρiBi),(Ψr,Qr)∈Ξ(ξ1,ξ2)} associated with the boundary layer between (ξ1,ξ2)and (Ψr,Qr)     ˙ ξ1 ˙ ξ2 ˙ Ψr ˙ Qr     =    0 0 −L(α)Ψr −L(α)Qr     (ξ,Ψr,Qr)∈\ i∈N (Ci∩ρiBi)×R2N,(5.14) is GAS. Proof: Note that if Assumption 11 is satisfied, there is a time-scale separation between between (ξ1,ξ2) and (Ψr,Qr) that allows getting the boundary layer (5.14) , which is obtained re-scaling the ordinary time, t , by 1/ν1 , in (5.12) , and doing ν1= 0 . That is, we zoom into the faster variables (Ψr,Qr) , freezing the slower variables so that ξ1 and ξ2 remain constant during flows. Likewise, note that there is no jump in the boundary layer. 5.7 Stability analysis of the complete hybrid system 79 The fast subsystem dynamics from (5.12) can be written as: ν1 dΨr dt =−L(α)Ψr+ν1Ke˜ Ψ ν1 dQr dt =−L(α)Qr+ν1Ke˜ Q replacing τ=t/ν1, and setting ν1= 0 ν1 dΨr d(ν1τ)=dΨr dτ =−L(α)Ψr+ν1Ke˜ Ψ =−L(α)Ψr ν1 dQr d(ν1τ)=dQr dτ =−L(α)Qr+ν1Ke˜ Q =−L(α)Qr Now, let us consider the following Lyapunov function candidate Vc:= 1 2Ψr⊤L(α)Ψr+1 2Qr⊤L(α)Qr≥0. Then, ⟨▽Vc(Ψr,Qr),fbl⟩=−Ψr⊤L(α)L(α)Ψr−Qr⊤L(α)L(α)Qr≤0, being fbl := [0 0 −L(α)Ψr−L(α)Qr]⊤ . Notice that ⟨▽Vc(Ψr,Qr),fbl⟩ is negative semidefinite. Consequently, from the structure of L(α) and standard consensus results [39], (5.14) is GAS. The dynamic defined in (5.14) guarantees that each Ψr,i and Qr,i , for agents with associated αi= 1 , converges to a neighborhood of Ψ∗ and Q∗ , respectively, which defines a consensus between the interconnected agents. ■ 5.7.4 Stability of the reduced system For the reduced system associated with Hsp, we now obtain the next result. Proposition 10: Under Assumption 11 and for a given α , the attractor Ar:= S i∈N Ar,i with Ar,i := {(ξ1,i,ξ2,i)∈Hi:∥˜xi∥< Xi,ξ1,i =0}associated to the reduced system: 80 Chapter 5. Hybrid dynamical control scheme for reactive power sharing Hr(ξ1,ξ2):                                                                ˙ ξ1 ν2˙ ξ2               =               −Kω˜ω−Kc1˜ Ψ −KΨ˜ Ψ+Kc1˜ω+K0˜ P −KV˜ V−K2˜ Q −KQ˜ Q+K2˜ V −KdP ˜ P−K0˜ Ψ ν2A˜x+ν2Bv ν2Θi(ωr,Vr)z −ν2Γ(ωr,Vr)Θ(ωr)z ν21               (ξ1,ξ2)∈C       ξ1 + ν2ξ+ 2      ∈      ξ1 ν2˜x ν2z ν2h(ξ) 0       (ξ1,ξ2)∈D, (5.15) is SPAS as ν2→0+. Proof: The proof for Proposition 10 follows the same approach presented in Section 4.5.5, with S:= diag{Kω,KΨ,KV,KQ,KP}.■ Theorem 7: Under Assumption 9, 11 and for a given α , the set A=Ar×{L(α)Ψr= 0,L(α)Qr=0} associated with the hybrid system H , (5.8) – (5.11) , is SPAS as ν1,ν2→0+ . □ Proof: Note that Ar∈A and that if Assumptions 9 and 11 are satisfied, then Propositions 7– 10 are hold. As a result, the proof is direct following [21, Theorem 1]. ■ 5.8 Simulation results The hybrid control scheme proposed here is validated for an islanded AC-bus microgrid composed of 6 BESSs simulated in Matlab/Simulink using the Electrical Toolbox, as shown in Fig. 5.2. Each half-bridge is emulated as an AC controlled voltage source whose input is the equivalent modulated voltage, followed by its corresponding LC filter and line impedance. The line impedances are heterogeneous and unknown to the controller; hence, the output current, io,i , is also measured to calculate the output active and reactive powers. The nominal voltage of the bus line is Vbus = 120√2sin(2π60t)V , then Vn= 120√2 V and ωn= 2π60rad/s . Table 5.1 and 5.2 provide the parameters for the secondary and primary control loops respectively, i.e., the parameters for the hybrid model (5.6) . From these parameters and PL,i =28.20 0.12 0.12 0.08 , QL,i =1.50 0 0 5.55 , it is easy to see that Assumption 9 is satisfied. Moreover, note that Kinv = 34.48 and Kdc = 0.42 , then Assumption 11 is also satisfied. Finally, from Table 5.1, the droop control (3.5) and (5.2) parameters are: Kd,1:= 10,Kd,2:= 5,Kd,3= 0.01,Kd,4= 10−4and Kd,5= 0.1. 5.8 Simulation results 81 Figure 5.2 Microgrid configuration scheme. Table 5.1 Microgrid and droop control parameters. Parameter Value Parameter Value Kc10000 KV0.0049 Ke100 KdP 50 KΨ0.45 Kc10.5 Kω5K20.49 KQ1K00.05 82 Chapter 5. Hybrid dynamical control scheme for reactive power sharing BESS1 BESS2 BESS3 BESS4 α1,2=α2,1 α6,1=α1,6 α2,3=α3,2 BESS6 α3,4=α4,3 α4,5=α5,4 α5,6=α6,5 BESS5 Figure 5.3 Communication graph among the BESSs. Table 5.2 Inverter parameters. Parameter ∀iValue Parameter ∀iValue Vin,i 48 V Zline1R=0.1ΩL=1.6mH Li50 mHZline2R=0.4ΩL=1.8mH Ci140.72 µFZline3R=0.7ΩL=1.9mH RLS,i 1.5 ΩZline4R=0.6ΩL=1.7mH Ri180 ΩZline5R=0.9ΩL=2mH ωn2π60 rad/s Zline6R=1ΩL=2.2mH Vn120√2VK1 Xi0.041 k1/3600 T0.1 ms Cbat,i 0.1615 Wh The Laplacian matrix that represents the communication interconnections among the 6 BESSs as shown in Fig. 5.3 is L=         α1,2+α1,6−α1,20 0 0 −α1,6 −α2,1α2,1+α2,3−α2,30 0 0 0−α3,2α3,2+α3,4−α3,40 0 0 0 −α4,3α4,3+α4,5−α4,10 0 0 0 −α5,4α5,4+α5,6−α5,6 −α6,10 0 0 −α6,5α6,5+α6,1         . To validate the effectiveness of our proposed strategy, we verify the SOC balancing and reactive power consensus through two simulation scenarios. The first scenario tests performance under load changes and communication failures, while the second evaluates the performance against plug-and-play events. 5.8.1 Scenario 1 In this scenario, the proposed control hybrid scheme is validated considering that a load change and a connection failure occur. Initially, the total load is 810W and 150VAr. Then, 5.8 Simulation results 83 one load of 300W and 60 VAr is disconnected at T1= 1.5 s and connected back at T2= 2.5 s. Moreover, at T3= 3s , a connection failure occurs between the BESS 1 and the BESS 2 , i.e., α1,2=α2,1= 0. Fig. 5.4 shows the SOC, their references, and the active powers of each BESS in discharging mode. Note that they converge to a consensus, even when the before-mentioned perturbations occur. Note that in this figure, the SOC references (or Ψr,i ) converge faster to a consensus than the SOCs and the active powers to their references. Indeed, the SOCs and the active powers converge to their references before to 2.4s approximately (note that 1/Kdc = 2.4) and Ψrin 0.1ms approximately (note that 1/Kc= 10−4). 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 0 100 200 300 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 0 1 2 10-4 0.8 0.9 1 Figure 5.4 Evolution of the SOC references Ψr,i, the SOCs Ψiand the active powers Pi, for i={1,2,3,4,5,6} , when a load of 300W and 60VAr is disconnected at T1 and then it is connected back at T2. In T3α1,2= 0. Fig. 5.5 shows the reactive powers converging to their references and their references converging to a consensus, again noting the different time scales between both dynamics. Finally, Fig. 5.6 shows the error variables ˜ Ψi , ˜ Pi , ˜ Qi , noting that they converge to zero. It can be seen that the connected and disconnected load modifies the slope of Ψr,i and the constant consensus value of Q∗ . However, the disconnection failure does not change any evolution. Now, focusing on the inverters, Fig. 5.7 shows a zoom of the variables of each DC-AC converter. Note that the voltage errors converge to zero, ensuring that the voltage outputs follow their references. Moreover, we highlight that these dynamics ( ξ2,i ) are slower than Ψr,i and Qr,i and, faster than SOCs, reactive and active powers ( ξ1,i ). Indeed, the voltage 84 Chapter 5. Hybrid dynamical control scheme for reactive power sharing response speed is 30ms approximately (note that 1/Kinv = 0.029 ). Then, we can see as the three time-scale separation model is validated. 0 0.5 1 1.5 2 2.5 3 3.5 -20 0 20 40 60 80 0 0.5 1 1.5 2 2.5 3 3.5 4 -50 0 50 100 150 Figure 5.5 Evolution of the reactive power references Qr,i and the reactive powers Qi for i={1,2,3,4,5,6} , when a load of 300W and 60VAr is disconnected at T1 and then it is connected back at T2. In T3α1,2= 0. 0 0.5 1 1.5 2 2.5 3 3.5 4 -0.2 -0.1 0 0.1 0.2 0.3 0 0.5 1 1.5 2 2.5 3 3.5 4 -100 0 100 200 0 0.5 1 1.5 2 2.5 3 3.5 4 -100 -50 0 50 100 150 Figure 5.6 Evolution ˜ Ψi , ˜ Pi , ˜ Qi in Scenario 1 for i={1,2,3,4,5,6} , when a load of 300W and 60VAr is disconnected at T1 and then it is connected back at T2 . In T3 α1,2= 0. 5.8 Simulation results 85 Figure 5.7 Evolution of voltage errors, voltages, currents and duty cycles of converters for the first 0.2s in Scenario 1. 5.8.2 Scenario 2 In this scenario, the microgrid suffers the connection/disconnection of a BESS 1 , which can be related to a plug-and-play event or a failure of this BESS. We consider a load of 400W and 80 VAr. The disconnection occurs at the instant time T1= 1.5 s and then, it is connected back at T2= 1.7s. Fig. 5.8 shows the convergence of the SOC references to a consensus from initial conditions and after connection/disconnection of the BESS 1 , with transient times less than 0.1 ms. These evolutions are faster than the SOCs and the active powers convergence to a consensus, as seen in the Scenario 1. Fig. 5.9 shows the evolution of the reactive power references, Qr,i , and the reactive powers, Qi during the BESS 1 connection/disconnection. Moreover, in Fig. 5.10, the evolution to zero of the error variables ˜ Ψi , ˜ Pi , ˜ Qi is shown, even after the perturbations. Note that the response speed of all Qr,i to a consensus is faster than the evolution of each ˜ Qito zero. Finally, in Fig. 5.11, a zoom in on the voltage errors is shown, noting the robustness of the output voltage regulation after any connection/disconnection event. It can be seen that the current equilibrium is changing with respect to the nominal reference to absorb the parameter variation, guaranteeing this voltage regulation. In addition, this figure shows the evolution of the voltage, current, and duty cycle. Again, we can appreciate the three time-scale separation. Indeed, the time response of the inverter (less than 30ms) is larger than the SOC and the reactive power references (less than 0.1m s), but smaller than the response speed of ˜ Ψi,˜ Pi,˜ Qi(2.4s approximately). From these scenarios, we can validate the statement of Theorem 7 and, therefore, all the objectives of this chapter. 92 Chapter 6. Bifurcation analysis of islanded AC microgrids with Constant Power Loads microgrids, is to apply bifurcation theory to their nonlinear models [63]. By applying bifurcation analysis techniques, it is possible to identify a safe operating region for microgrids, and this analysis can then be used to design appropriate protection mechanisms, improving the overall reliability and stability of the system. One of the most critical elements that affect stability is CPLs, so in this chapter, we will focus on a microgrid with a load of this type. In the literature, various approaches to large signal stability analysis applied to power systems and microgrids with CPL have emerged, most of them using bifurcation techniques [61,64 – 67]. In [68], the stability analysis of a three-phase AC microgrid with a CPL is presented. Assuming that there is no reactive power consumption, the bifurcation analysis for the three-phase CPL is reduced to the analysis for the model of the DC microgrid system. Also, the bifurcations are computed with a direct numerical method using the Newton-Raphson procedure. The authors in [64] proposed a stability study of a Dual Active Bridge converter using bifurcation analysis with Poincaré map theory and Jacobian matrix, considering the effect of input filter and a CPL using a discrete-time model. In [65] and [66], CPL models considering delays are presented. Specifically, [65] presented the CPL model with Delay-Differential Equations (DDEs); the model is presented for a single-phase AC microgrid but could be applied to balanced three-phase systems. These studies have examined different microgrid structures and multiple operating modes, and most of them proposed different CPL models for their studies. However, despite achieving greater model precision, these methods result in an excessive number of equations even for basic microgrid systems, complicating the analysis process, and almost none of them presents experimental validations. In [61], a numerical bifurcation analysis using XPP-AUTO is presented for an reduced model of an three-phase AC microgrid with a CPL, where is proposed a reduced model for AC-DC converters as loads. Furthermore, large signal stability analysis of AC microgrids with CPL also considers other study techniques. In [67], a large signal stability analysis of AC microgrids based on mixed potential theory is presented, also considering the influences of the storage system. 6.2 Preliminary concepts 6.2.1 Some notions of bifurcation theory As mentioned above, bifurcations theory is the study of changes in the qualitative structure of the flow of a nonlinear dynamical as parameters are varied. The parameter values at which these bifurcations occur are called the bifurcation values or points [69]. Some bifurcations, known as local bifurcations, can be identified simply by analyzing the equilibrium equations ˙x=f(x) = 0 and the Jacobian matrix ∂f(x)/∂x . However, other more complex bifurcations require numerical methods to detect them. In particular, one of the most important bifurcations in power electronic systems is the Hopf bifurcation, as it often leads to the instability of the system. Poincaré-Andronov-Hopf bifurcation This bifurcation occurs when an equilibrium point switches its stability as a parameter varies, accompanied by the occurrence of a periodic solution (limit cycle) around this point 6.2 Preliminary concepts 93 [70]. A typical example occurs when increasing the load power (a system parameter) leads to instability. From a local perspective, this bifurcation is identified by two eigenvalues crossing the imaginary axis. Hopf bifurcations can be classified into two categories: supercritical and subcritical. Supercritical Hopf bifurcation: To study this bifurcation, we consider the following R2system [71] : ˙x1=x2+x1λ−x2 1−x2 2 ˙x2=−x1+x2λ−x2 1−x2 2. where λ is a parameter. When λ≤0 , the system solutions are a clockwise spiral converging on the origin, which is a stable focus, as t increases. However, when λ > 0 , the origin becomes an unstable focus and a periodic orbit or limit cycle appears in such a way that all solutions converge to it as t increases. Fig.6.1 and Fig.6.2 show the phase portrait (or plane) and the evolution of the state variables for λ≤0and λ > 0, respectively. Figure 6.1 Phase portrait of the supercritical hopf bifurcation for parameter λ. Figure 6.2 Evolution of the system in the stable region when λ≤0 and in the unstable region when λ > 0. 94 Chapter 6. Bifurcation analysis of islanded AC microgrids with Constant Power Loads Figure 6.3 Bifurcation diagram of a supercritical Hopf bifurcation. When the parameter λ changes from negative to positive values, the eigenvalues of the Jacobian matrix, which are complex conjugates, move from the left half-plane to the right half-plane of the complex plane. For λ= 0 the real part of the eigenvalues is zero (imaginary eigenvalues), and at this point the Hopf bifurcation appears. For negative values, the system is locally stable, and for positive values it is locally unstable. The supercritical Hopf bifurcation diagram for the plane {x1,λ}is shown in Fig. 6.3. Notice that the parameter is assigned to the horizontal axis; the stable equilibrium point is drawn in solid lines, and the unstable equilibrium point is drawn in dashed lines, as well as for limit cycles. We will follow these conventions in bifurcation diagrams. Subcritical Hopf bifurcation: To study this bifurcation, we consider the following system [63]: ˙x1=λx1−x2+x1x2 1+x2 2 ˙x2=x1+λx2+x2x2 1+x2 2 where λis a parameter. when λ > 0 , the system has an unstable equilibrium; while, for values of λ < 0 , an unstable limit cycle appears, which surrounds a stable equilibrium. Thus, in this case, the system evolves following spirals away from the unstable limit cycle as t increases, tending towards the equilibrium at the origin if x(0) is in the interior of the unstable orbit or towards infinity if it is in the exterior. Fig.6.4 and Fig.6.5 show the phase portrait and the evolution of the state variables for λ < 0and λ > 0, respectively. 6.2 Preliminary concepts 95 Figure 6.4 Phase portrait of the subcritical hopf bifurcation for parameter λ. Figure 6.5 Evolution of the system in the stable region when λ≤0 and in the unstable region when λ > 0. Figure 6.6 Bifurcation diagram of a subcritical Hopf bifurcation. The subcritical Hopf bifurcation diagram for the plane {x1,λ} is shown in Fig. 6.6. The subcritical Hopf bifurcation indicates that around the stable equilibrium point there is an 96 Chapter 6. Bifurcation analysis of islanded AC microgrids with Constant Power Loads unstable limit cycle. This unstable limit cycle delimits the region (or basin) of attraction of the stable equilibrium, and if the system crosses this region, it becomes unstable, even though the Jacobian matrix has all stable eigenvalues. In this situation the non-linear terms of the model are making the system unstable, but note that close to equilibrium the system remains stable. The supercritical Hopf bifurcation indicates that around the unstable equilibrium point there is a stable limit cycle, and the system oscillates in this region. The stable limit cycle provides the maximum and minimum values of the oscillations to which the system is subject when the equilibrium becomes unstable. Although the Jacobian matrix does not explain the origin of the ripples, in order to guarantee stability it is necessary to know only the linear part of the model for this type of Hopf bifurcation. Note that the subcritical type is worse since local stability is not sufficient when the system is far from equilibrium [63]. Typically, CPLs instabilize an electrical system through a subcritical Hopf, as will be discussed in detail in the next sections. 6.3 AC microgrid model In this section, we present the mathematical model of the selected AC microgrid topology. First, we present the model of a three-phase converter based on the graphical dq - transformation of power switching converters depicted in [72]. We obtained the mathematical model of a three-phase converter with an LC filter and resistive load. Then, the complete model with two three-phase converters with line impedances and the addition of a CPL is presented. 6.3.1 Modelling of a three phase inverter Based on the graphical dq transformation method for power switching converters [72], we model an islanded three-phase converter with an LC filter and resistive load. The three-phase converter is represented in a reduced form as a controlled three-phase voltage source equivalent to the modulated voltage by switching, as shown in Fig. 6.7. The graphical modeling method reduces the effort to obtain the model. The three-phase inverter is modeled as a linear time-invariant circuit obtained by changing balanced AC reactors into equivalent DC reactors combined by gyrators [72]. The circuit is decomposed into four subcircuits: three-phase voltage source, inductor, capacitor, and resistor sets. Borrowing from [72], the graphical dq transformation of these subcircuits is presented, assuming identical parameters for each phase. 6.3 AC microgrid model 97 Figure 6.7 Electrical circuit of a three-phase inverter with an LC filter. a) Three-phase voltage source: The dq transformed circuit for a three-phase voltage source set is shown in Fig. 6.8. From Fig. 6.8 (b) we have: vdq ref =Kabc→dqvabc ref , where vabc ref = [va ref ,vb ref ,vc ref ]Tand vdq ref = [vd ref ,vq ref ]T, and Kabc→dq =r2 3  cos(θ) cosθ−2π 3cosθ+2π 3 −sin(θ)−sinθ−2π 3−sinθ+2π 3 √2 2 √2 2 √2 2   Figure 6.8 dq transformation of a voltage source set. (a) Original circuit. (b) dq transformed circuit. b) Three-phase inductor set: The dq transformed circuit for a three-phase inductor set is shown in Fig. 6.9. The equivalent dynamics from Fig. 6.9 (b) are: L˙ iq L=−wLid L+vq L˙ id L=wLiq L+vd L˙ i0 L=v0 L= 0; balanced condition. (6.1) 98 Chapter 6. Bifurcation analysis of islanded AC microgrids with Constant Power Loads The proof is presented in [72]. Figure 6.9 dq transformation of a inductor set. (a) Original circuit. (b) dq transformed circuit. As the sources are balanced, the 0-axis variable can be excluded from our discussion. c) Three-phase capacitor set: The dq transformed circuit for a three-phase capacitor set is shown in Fig. 6.10. The equivalent dynamics from Fig. 6.10 (b) are: C˙vq C=−wCvd C+iq C˙vd C=wCvq C+id C˙v0 C=i0 C= 0; balanced condition. (6.2) The proof is presented in [72]. Figure 6.10 dq transformation of a capacitor set. (a) Original circuit. (b) dq transformed circuit. d) Three-phase resistor set: The dq transformed circuit for a three-phase resistor set is shown in Fig. 6.11. The equivalent dynamics from Fig. 6.11 (b) are: vdq =Kabc→dqvabc =Kabc→dqRiabc =Ridq (6.3) The 0 axis variables are excluded for both the balanced sources and the balanced initial conditions. 6.3 AC microgrid model 99 Figure 6.11 dq transformation of a resistor set. (a) Original circuit. (b) dq transformed circuit. Now, from the equivalent equations obtained from the graphical dq transformation of each subcircuit we can obtained the complete model of the three-phase inverter of Fig. 6.7 for a dq equivalent circuit given by Ldid L dt =vd ref −RLid L+wLiq L−vd c Ldiq L dt =vq ref −RLiq L−wLid L−vq c Cdvd c dt =wCvq c+id L−vd c R Cdvq c dt =−wCvd c+iq L−vq c R (6.4) In the next section, we introduced the CPL selected model to be used in the microgrid model. 6.3.2 Model of a CPL In AC microgrids, most components connected to the bus are voltage-controlled rectifiers supplying a load, such as a battery in charging mode, drawing constant power from the bus. In balanced three-phase AC networks, as well as in DC microgrids, this combination of a voltage-controlled converter and a load can be modeled as a CPL. Constant power loads usually present a negative impedance characteristic, which can destabilize the power system [66]. As explained in [63,73], the basic diagram of a constant power load, such as a DC/DC converter connected to a linear load, is shown in Fig. 6.12. Figure 6.12 CPL block diagram. (a) Converter + load. (b) Equivalent circuit. 100 Chapter 6. Bifurcation analysis of islanded AC microgrids with Constant Power Loads As this converter regulates its output voltage, any disturbance in the bus voltage will be compensated by the control loop, ensuring output voltage regulation. This adjustment requires the current to compensate for changes in the bus voltage to maintain constant power. Figure 6.13 Negative impedance behavior of CPLs. The static voltage-current characteristic of a CPL load is depicted in Fig. 6.13. Since Pload is constant, the plot forms a hyperbolic curve. The input resistance, Rload , is given by the ratio of small-signal voltage changes over small-signal current changes, ∆v ∆i , and this value depends on the converter’s operating point. As shown in the graph, the slope is negative, indicating that the input resistance of the converter exhibits a negative characteristic. This means that any decrease in voltage will cause an increase in current, and vice versa. Assuming that the converter has no losses, the negative input resistance can be calculated by differentiating the voltage with respect to the current in v=Pload i , as follows. dv di =−i−2Pload =−| v2 Pload |=−Rload (6.5) This negative resistance is also nonlinear depending upon the current and voltage, and would causes instability in the system as depicted in [73]. In the literature, most CPL models for DC microgrids are based on their negative impedance characteristics. The works in [74,75] analyze the current-voltage characteristics of DC/DC converters as CPLs, considering limits in duty cycle and maximum regulated voltage. In fact, the converter behaves as a CPL when operating under closed-loop control, and as a positive resistance in regions where it is in open-loop operation or the control loop is saturated, as shown in Fig. 6.14. Figure 6.14 Current-voltage characteristic of a load converter as a CPL. 6.3 AC microgrid model 101 Following Fig. 6.14, the CPL is given by: i=   Pload v,if v≥Vth v Rth ,if v < Vth where Rth =V2 th Pload is the resistance of the linear region. For AC microgrids, models exist for single-phase and three-phase systems. The works in [65,66] propose CPL models for single-phase systems based on the concept of average power, which involves integrating the product of voltage and current signals over a finite moving window. In [65], a model based on DDEs is also proposed. For balanced three-phase systems, the same model based on negative resistance characteristics is applicable since, after rotating the coordinates to the synchronous frame, the system can be treated as DC [65]. In [63], a three-phase AC/DC rectifier controlled with power factor correction (PFC) is modeled as a constant three-phase active power load. This modeling is based on PQ theory, which assumes zero instantaneous reactive power. However, modeling CPLs for single-phase (or unbalanced multi-phase) AC systems remains largely unexplored due to the complexity of defining active power P(t) in these systems [65]. Selected CPL model In this work, we are going to consider the model proposed in [61]. Here, a CPL model is presented as an alternative representation of multiple identical three-phase controlled rectifiers, modeled as a single time-invariant nonlinear dynamic load. This proposed model is expressed in function of the total demanding load current given by did load dt =−λid load +λPo vd bus (6.6) diq load dt =−λiq load (6.7) where λ is the inverse of the load time-constant, Po the extracted constant active power from the common bus, and vd bus the direct component of the bus voltage. Equations (6.6) and (6.7) represent the idea of using a low-pass filter to model a power converter at the equilibrium point. It is assumed that there is ideal power factor correction (PFC) on the common bus, so the reactive power is zero. Furthermore, this model considers saturation of the current to a maximum drawn current Imax , so the equivalent load dynamic changes to did Load dt =−λid Load +λImax (6.8) diq Load dt =−λiq Load (6.9) 108 Chapter 6. Bifurcation analysis of islanded AC microgrids with Constant Power Loads power reference to Po≈2940 W, where id load is saturated to Imax = 40 A, leading the system back to a stable operating point. The effect of the supercritical Hopf bifurcation can also be observed on bus voltage vbus and in the current iL1 , as depicted in Fig. 6.22 and Fig. 6.21, respectively. Here, we observe that id L1 is saturated to ≈20.8 A because the two inverters are also drawing power to the resistors connected to the common bus. 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0 500 1000 1500 2000 2500 3000 3500 4000 Figure 6.20 CPL power evolutions for power reference steps around bifurcation point H2 . 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 -30 -20 -10 0 10 20 30 Figure 6.21 Current iL1 evolutions in abc and dq coordinates around bifurcation point H2. 6.6 Experimental results 109 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 -200 -150 -100 -50 0 50 100 150 200 250 Figure 6.22 Bus voltage evolutions in abc and dq coordinates around bifurcation point H2 . 6.6 Experimental results This section provides experimental results to validate the bifurcation analysis described above. First, we present experimental tests to obtain the time constant of an electronic CPL. Then, we describe the experimental setup of the AC microgrid in Fig. 6.15 and present experimental results for selected operating points that validate the bifurcation analysis. 6.6.1 CPL time constant test First, we carried out tests to determine the time constant of the electronic CPL used in the experimental validation. We used the Cinergia EL+ vAC AC electronic load, configured in programmable power mode, which can be controlled by power reference steps. We obtained the time constant measurement through tests in CPL mode by connecting it directly to a grid generator (NHR 9410-12). We then applied grid voltage steps to observe the current variations, as shown in Fig.6.23(c). Here, the interface of the CPL control software is shown, where the steps in grid voltage, their effect on CPL current, and the constant power value in the load can be observed. Moreover, Fig.6.23(a) shows the schematic connection for the test, and Fig. 6.23(b) shows the Imperix software interface for current measurements. Fig. 6.24 shows the CPL current iLoad measurements obtained using Imperix external ModuLink ±50 A isolated current sensors for a voltage step resulting in a current step (from iref,1 to iref,2 ). Moreover, a zoom in on the id Load transient is highlighted, showing a time constant ∆t≈0.0625 ms, which is the time interval for id Load , from iref,1 , to reach approximately 63.2% of iref,2, leading λ≈1/0.0625 ≈1600. 110 Chapter 6. Bifurcation analysis of islanded AC microgrids with Constant Power Loads Figure 6.23 CPL time constant experimental test. (a) Connection scheme. (b) Imperix software interface. (c) Interface of the CPL control software.. 0.88 0.9 0.92 0.94 0.96 0.98 1 -6 -4 -2 0 2 4 6 0.9 0.91 0.92 0.93 0.94 0.95 0.96 -6 -4 -2 0 2 4 X 0.937015 Y 2.91008 X 0.936389 Y 2.15 Figure 6.24 CPL current measurements for time constant calculation. 6.6.2 Experimental results of Bifurcation analysis We validate the regions around the bifurcation point H1 where a subcritical Hopf bifurcation occurs, however, the supercritical Hopf bifurcation cannot be validated, as it is a consequence of current saturation in the CPL model, and we use a fixed electronic CPL. 6.6 Experimental results 111 The experimental setup shown in Fig. 6.25 represents an islanded AC microgrid composed of 2 three-phase inverters with LC output filters implemented with Imperix power test bench, 2 DC sources, 2 three-phase inductors, 3 resistors and an electronic CPL, as shown in Fig. 3.7. The microgrid parameters are the same as those presented in Table 6.1. Each DC/AC converter is configured from 3 PEB8038 half-bridge modules, followed by an LC output filter, which is composed of 3 power inductors Li , and three film capacitors Ci , embedded in a passive filter box, as well as line impedances, which are 3 power inductors Lf,i . For DC sources, two PSB 9360-40 units are used. The inductance currents, iL,i and the capacitor voltages, vC,i , are measured using galvanically isolated sensors onboard the PEB8038 module, specifically LEM CKSR 50-P and Avago ACP-C87B, respectively. In addition, external ModuLink ±50A isolated current sensors are used to measure the output current iLoad of the CPL. Figure 6.25 Experimental setup of an islanded AC microgrid with a CPL. Furthermore, each DC/AC converter operates as a voltage source inverter (VSI), controlled by a dual control loop in αβ coordinates, similar to Fig. 1.3. The voltage loop uses a PR controller, while the current loop uses a proportional-integral PI controller. These control algorithms are programmed using the Simulink blockset of the Imperix ACG SDK and then assembled in the B-BOX RCP. Recall that the internal dynamics of the control loop are neglected in the microgrid model as they are assumed to be faster than the rest of the system dynamics and in a stable state. Therefore, to ensure the stability of the two inverters, some tests are carried out without the CPL, using only resistive loads at different operating points. Fig. 6.26 shows the AC microgrid voltages ( vabc C1 and vabc C2 ), currents ( iabc L1 and iabc L2 ) and power measurements for both inverters under different power steps generated by the electronic load operating as a constant resistor. To validate the bifurcation analysis obtained above, we implement some power steps in the electronic load operating in CPL mode to reach the first bifurcation point H1 , as shown 112 Chapter 6. Bifurcation analysis of islanded AC microgrids with Constant Power Loads Figure 6.26 Evolution of microgrid voltages ( vabc C1 and vabc C2 ), currents ( iabc L1 and iabc L2 ) and powers P1and P2under steps of constant resistors. Figure 6.27 Evolution of microgrid variables around bifurcation point H1 . (a) CPL power and current id L1. (b) voltages vabc C1and vabc C2, currents iabc L1,iabc L2and iabc Load. in Fig.6.27. At time T1 , Po≈1310 W, and the system starts to oscillate until T2 , where poor grid quality causes the CPL to disconnect, leaving only the power consumed by the resistors (sometimes Imperix protections are triggered as well). Therefore, we confirm that the experimental operating point is very close to the bifurcation point H1 obtained 6.6 Experimental results 113 in the numerical and simulation analyses, where a subcritical Hopf bifurcation occurs. Furthermore, Fig. 6.27(b) shows the capacitor voltages and inductor currents for both inverters, as well as the CPL current around the bifurcation H1. Additionally, we have carried out tests to obtain experimental validation of the effect on the basin of attraction of the stable equilibrium in region 2 in Fig.6.16(b). To this end, when the system is at the equilibrium point, we introduce a perturbation to move away from it. The perturbation consists of varying the reference value of the inverter output voltage ( Vn ) and, after a few moments, returning to the initial value. We have done this for different values of Po as depicted in Fig. 6.28, which shows the evolution of the CPL power ( Po , top window) and the d component of the filter inductor current of inverter 1 ( id L1 , bottom window). At the three points, the value of Vn changes from 80 V to 65 V at time T1and then returns to 80V at T2. At point 1, which corresponds to Po≈900 W, the system remains stable after returning from the change. At the second point, corresponding to Po≈1100 W, it starts to become unstable when the change occurs, but stabilizes when it returns. However, at point 3, at Po≈1250 W, a power close to the Hopf bifurcation, when it returns, it starts to become unstable and then the system shuts down (Imperix protections are triggered). Figure 6.28 Evolution of Po (top) and id L1 (bottom) for 3 different operation points under Vnchanges. We compared these results with the MATCONT bifurcation diagram as shown in Fig. 6.29. Both points 1 and 3 coincide well with the behavior of the limit cycle around the subcritical Hopf bifurcation ( H1 ). At the first point with a power of Po≈900 W, the current id L1 goes from 6.25 A to 7.25 A, which is within the unstable limit cycle. However, at point 3, at Po≈1250 W, the current goes from 9 A to 10.5 A, which is already on the 114 Chapter 6. Bifurcation analysis of islanded AC microgrids with Constant Power Loads edge of the unstable limit cycle and makes the trajectory not return to equilibrium. This reveals that the domain of attraction of each equilibrium becomes very thin as we approach the Hopf bifurcation. Figure 6.29 Comparison of Po and current id L1 values for 2 operation points between experimental results and bifurcation diagram of Fig 6.16(b). Furthermore, Fig.6.30 and Fig. 6.31 show a zoom in on the capacitor voltages and inductor currents for both inverters, as well as the CPL current around for operating points 1 and 3, respectively. 6.6 Experimental results 115 Figure 6.30 Zoom in on the capacitor voltages, inductor currents and CPL current around for operation point 1. Figure 6.31 Zoom in on the capacitor voltages, inductor currents and CPL current around for operation point 3. 116 Chapter 6. Bifurcation analysis of islanded AC microgrids with Constant Power Loads 6.7 Conclusions This chapter provides the bifurcation analysis and experimental validation of a three-phase AC microgrid in islanded mode with a CPL considering line impedances and resistive loads connected to the bus. Although this development was inspired by [61], the bifurcation diagrams are quite different, being validated by simulations and several experimental results through an Imperix power test bench under various scenarios. The experimental results focused on the subcritical Hopf bifurcation, as it is the most important practical operating point of the system. The results obtained provide information on the safe operating point of the microgrid and where the instability region is, by estimating the region of attraction of the stable operating point in region 2 of the bifurcation diagram (Fig. 6.16). Based on this analysis, we can establish safety limits for general operation and allowable disturbance ranges. Moreover, reactive power analysis and variations in droop control parameters are expected to be studied in extended future versions. 7 Conclusions and future work 7.1 Conclusions and contribution summary This PhD manuscript contributes to the design of control laws and to large-signal stability analysis for AC islanded microgrids. Specifically, it addresses the design of distributed control loops for state-of-charge consensus and reactive power sharing among a set of batteries in discharging mode. Therefore, continuous-time and hybrid dynamic three-time scale control schemes for secondary and primary control loops are proposed with large signal stability properties using singular perturbation theory. Furthermore, a qualitative large-signal stability analysis using bifurcation theory for AC islanded microgrids with constant power loads was presented and validated through simulation and experimental results. Due to the main results not being directly correlated, this thesis was divided into two parts: The first part focuses on the design of control laws for consensus algorithms and optimal active and reactive power sharing. And the second part is devoted to bifurcation analyses of AC microgrids with nonlinear loads. Next, we highlight the main important contributions of this thesis. For the first part, the main contributions are listed below: • A distributed control scheme with secondary and primary control loops has been developed to achieve SOC consensus for a set of BESS in discharging mode. BESS are crucial elements for reliability and grid-forming generation in islanded microgrids, and SOC is an important factor for batteries’ lifespan. The secondary control loop corresponds to a SOC reference consensus algorithm designed according to MAS theory, providing robustness against communication failures and plug-and-play capability. In addition, a droop control method has been designed in the primary control, which uses cross-coupling errors of SOC, active power and frequency to achieve asymptotic stability in the Lyapunov sense. This primary control is essential 117 4238 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 71, NO. 9, SEPTEMBER 2024 point since power converters can operate in a limited set of operating points. Hence, we include the following assumption. Assumption 2: For a given desired operating voltage vCe, there exists a corresponding state xeand weights λe=[λ1λ2] such that λ1in [0,1] (and λ2=1−λ1), and 2  j=1 λj¯ Bj:= 2  j=1 λj(Aj−I)xe+Bj=0.(4) This assumption is based on the computation of the solutions Ajand Bjto the least squares optimization problem. The value of λ1in this assumption gives an information about the duty cycle that corresponds to the desired voltage vCe. To translate this information into a cycle ν, we propose the following procedure to build the corresponding cycle. For a given Nν∈N, define the cycle νgiven by ∀∈[1,Nν],ν ()=1if≤Round(Nνλ1), 2if>Round(Nνλ1).(5) In practice, one has to select Nνsuch that Round(Nνλ1)is sufficiently close to Nνλ1. D. Data-Driven Control Design We are now in position to state our main result. Theorem 1: Given data experiments Xj,X+ j, a desired operating voltage vCe, associated with corresponding operating vector xeand cycle ν(i.e., λ1)such that Assumptions 1and 2 is satisfied, and for a given a parameter μ∈(0,1), consider {(Wi,ρ i,η i)}i∈Dνin S2×R2×(R>0)2as the solution to the following problem. min {(Wi,ρi,ηi)}i∈Dν ε s.t. iX+ ν(i),Xν(i),xe0,η i>0, εIWi0,∀i∈Dν,(6) where matrices iare given by iX+ ν(i),Xν(i),xe:= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ (1−μ)Wi0WiA ν(i)Wi0 ∗μ (23) i(ρi−xe)1 ∗∗(33) i00 ∗∗∗ηiXν(i)X ν(i)ηXν(i)1ν(i) ∗∗∗ ∗ ηipν(i) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , (7) where (23) i=ρ iA ν(i)+¯ B ν(i)−ρ i+1ν,(33) i=Wi+1ν− ηiαν(i)ν(i). Then, under Assumption 1,theset Sν:= i∈Dν EW−1 i,ρ i+xe(8) is robustly globally exponentially stable for system (2) with ˜x=x−xeand the switching control law u(x):=ν(θ),θ∈argmin i∈Dν (˜x−ρi)W−1 i(˜x−ρi)⊂K. (9) Proof: Following [17,Th.2], the conclusion of the previous theorem holds if the following conditions 0≺iAν(i),Bν(i),xe:= ⎡ ⎢ ⎣ (1−μ)Wi0WiA ν(i) ∗μ(ρi−xe)A ν(i)+B ν(i)−ρi+1ν−xe ∗∗ Wi+1ν ⎤ ⎥ ⎦. are ensured for all uncertain matrices Aν(i),Bν(i). Then, replacing Aν(i)and Bν(i)by Aj−(Aν(i)−Aν(i))and Bν(i)−(Bν(i)− Bν(i)), respectively, the previous LMI can be rewritten as iAν(i),¯ Bν(i)+HeWi0 (ρi−xe)1 003×2A ν(i)−A ν(i) B ν(i)−B ν(i)0, (10) where we recall that ¯ Bν(i)=(Aj−I)xe+Bj. Then, Young’s inequality ensures that the previous inequality is equivalent to the existence of ηi>0 such that iAν(i),¯ Bν(i) −η−1 iWi0 (ρi−xe)1 0Xj 1 jXj 1 j−1Wi0 (ρi−xe)1 0 −ηi⎡ ⎣ 00 0A ν(i)−A ν(i) B ν(i)−B ν(i)Xj 1 jXj 1 jA ν(i)−A ν(i) B ν(i)−B ν(i)⎤ ⎦0. The proof is concluded by using Assumption 1, stating that matrices Ajand Bjare in j, so that the term in αjjappears naturally and by application of the Schur Complement to the second term of the previous inequality. Remark 1: If matrices Ajand Bjare constant, the least squares approximation Ajand ¯ Bjare equal to the corresponding matrices Ajand Bj. In this situation, [17, Proposition 2] guarantees that the solution to the LMI problem is equivalent to conditions A ν(i)W−1 i+1νAν(i)−W−1 i≺0, which according to [4], is equivalent to having a Schur stable monodromy matrix ν:=ARound(Nνλ1) 1ANν−Round(Nνλ1) 2. In other words, the eigenvalues of νare strictly inside the unit circle. A consequence of the previous statement is that, in the case of time-varying matrices Ajand Bj, a necessary condition for the solvability of the LMI is that νis Schur. Remark 2: Problem (6) is convex if the scalar parameter μ∈(0,1)is fixed, and therefore has a unique global minima. A grid search on μcan be performed to optimize the result. E. Summary of the Data-Driven Method To get a better understanding of the previous developments, we propose in this section a summary of the procedure for the data-driven control design, formulated as Algorithm 1. In this algorithm, we specify the mechanism of choosing controller parameters ρiand Wi,∀i∈Dν, from a selecting output voltage, vCe.First,wehaveawhile loop to select the appropriate amount of data such that rank(Xj 1 j)= 3, to identify the matrices from the data acquisition. This ensures that a desired operating point xeexists, along with its associated λe=λ1, which satisfies Assumption 2. Then, a cycle that approximates λeis selected according to (5), Authorized licensed use limited to: Univ degli Studi di Palermo - Univ of Palermo. Downloaded on October 31,2024 at 11:12:23 UTC from IEEE Xplore. Restrictions apply. MERCHÁN-RIVEROS et al.: DATA-DRIVEN CONTROL DESIGN 4239 Algorithm 1 Control Matrices and Parameter repeat Take: Extract Xj,X+ j∈Rn×pjfrom Xand . Identify: Ajand Bjusing Xj,X+ j,∀j∈{1,2}. Test: for a desired vCe,∃any xeand λe. until Assumption 2is satisfied Select: ν() ∀∈N, according to (5) and μ∈(0,1),αj. Compute: ρi,ηiand Wi,∀i∈Dν, s.t. (6) are satisfied. i.e., ensuring that Round(Nνλ1)≈1. Finally, for a given μ, the optimal solution to (6) provides control matrices and parameters. III. EXPERIMENTAL VALIDATION This section is devoted to the experimental validation of the data-driven design on a boost converter, considering that it can be approximated by a switched affine model (2) with two functioning modes. The power converter is implemented in an Imperix power test bench, as shown in Fig. 1and its general architecture of the experimental setup in Fig. 2. More precisely, the boost converter is configured from a PEB8038 half-bridge power module with silicon carbide (SiC) MOSFETs rated for 800V and 38A. Moreover, a power inductanceLand a film capacitor C0are embedded in a passive filter box. Furthermore, a DC source Elektro-Automtik PSB 936040 is employed to provide the converter input voltage, vin. The control algorithm is implemented in a B-Box rapid prototyping controller (RCP), which sends gate drive signals via optical channels and acquires sensors signals through analog RJ45 connectors, running with an internal sampling frequency of 50 kHz. The inductance current, iL, and capacitor voltage, vC, are measured using galvanically isolated onboard sensors in the PEB8038 module, which are LEM CKSR 50P and Avago ACP-C87B, respectively. The B-BOX RCP is programmed using the Simulink blockset of the Imperix ACG software development kit (SDK). Control tests, debugging, data logging, and signal monitoring are acquired from the Imperix Cockpit real-time monitoring software. The electrical block diagram illustrates two different control programs configured, namely, data acquisition and closed-loop control. Both programs use an ADC and a DO-PWM (Direct output PMW) modules, where it is necessary to convert σinto 1 and 0 for the on/off state of the MOSFET, respectively. A. Data Generation Some data were collected by feeding the system by a random input signal, σc∈{1,2}. The selected data set is depicted in Fig. 3. The selected frequency of the random signal σcis 50KHz, and the data sample has a frequency also of 50KHz. B. Closed-Loop Control Validation The objective is to drive the output voltage to a neighbourhood of vCe=66V. We highlight that the identified system and proposed control law depend on the sampling frequency of 50 KHz. Following the Algorithm 1, the experimental Fig. 1. Configured Imperix unit. Fig. 2. Block diagram of the experimental setup. Fig. 3. Experimental data of iLand vCobtained with a random input σ. data delivers the following desired operating point xe= [1.265.9]with associated λ1=0.13 and identified matrices A1=1.08 0 01.01 ,A2=0.91 0 −0.01 0.97 ,¯ B1=0.38 1.81 ,¯ B2=−0.44 −2.13 , which satisfy Assumption 2. Then, we select the cycle ν=[1222222 ], which is consistent with the required λ1, i.e., it has been built so that Round(Nνλ1)=0.91 ≈1. The robust limit cycle associated to this cycle, which is solution to the optimization Authorized licensed use limited to: Univ degli Studi di Palermo - Univ of Palermo. Downloaded on October 31,2024 at 11:12:23 UTC from IEEE Xplore. Restrictions apply. 4240 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 71, NO. 9, SEPTEMBER 2024 Fig. 4. Imperix Cockpit real-time monitoring software. problem (6) with α1=α2=1, is obtained by the composition of the centers ρithat are stacked in the following matrix ρ=−0.09 0.07 0.03 0.00 −0.03 −0.06 −0.07 0.06 0.94 0.77 0.61 0.45 0.30 0.15 , and the ellipsoids obtained with the following symmetric positive definite matrices W1=210.45 −3.22 −3.22 171.53 ,W2=170.1−4.9 −4.9 158.9,W3=173.58 −4.79 −4.79 161.69 , W4=176.53 −4.70 −4.70 163.92 ,W5=178.39 −4.56 −4.56 165.69 ,W6=180.67 −4.37 −4.37 167.37 , W7=189.74 −4.1 −4.1 169.29 . The computational time to solve the LMIs by using CVXSedumi in MATLAB on a computer with a 2 GHz Intel Core i5 processor and 4 cores is 1.19s. We have implemented an external loop through a PI control such that the output voltage is regulated in vCe. Fig. 4shows the voltage and current evolution of the boost converter configured in the Imperix unit controlled with the presented data-driven approach, including the predictor and the voltage output regulation. The load in the boost converter has been changed from 69to 34.5. The figure also shows the voltage error. The experiment validates the data-driven approach. IV. CONCLUSION In this brief, we present a theoretical contribution addressing the data-driven control design for power converters modelled as a switched affine system. The distinctive feature of this control design lies in the unknown dynamics of the system. However, this lack of knowledge is mitigated by leveraging prior experimental data. Building upon several non-restrictive assumptions, we validate the theoretical results using an experimental setup, specifically an Imperix boost converter. Furthermore, we incorporate an external control loop that regulates the output voltage, which is validated through experiments. This contribution marks a significant milestone for future research. We have already identified several crucial issues that merit further investigation. For example, the method relies on the satisfaction of Assumption 1. It would be of interest to devise a methodology for evaluating the values of parameters αjdirectly from the data. Another important concern lies in mode identification, accounting for additional power converter features such as the discontinuous conduction operation mode. Thorough investigation is needed to extend this result to a broader class of power converters with Nfunctioning modes. Finally, a data-driven algorithm that adapts the control law and detects any false or corrupted data from a cyberattack is another investigation line. REFERENCES [1] C. Albea, G. Garcia, S. Hadjeras, W. P. M. H. Heemels, and L. Zaccarian, “Practical stabilization of switched affine systems with dwell-time guarantees,” IEEE Trans. Autom. 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List of Figures 1.1 Block diagram of an AC microgrid 2 1.2 Primary control loop 4 1.3 Inner control loop 5 1.4 Droop control characteristics: (a) P/ω curve and (b) Q/V curve 7 2.1 Hybrid arc 13 2.2 Evolution of a solution to a hybrid system 14 2.3 Mappings:(a) not outer semicontinuous and (b) outer semicontinuous 15 2.4 Evolution of a lyapunov function candidate V(x)with x∈Afor H16 2.5 Example of a graph [24] 21 3.1 Islanded AC microgrid with BESS units 28 3.2 Structure of a BESSiin an AC islanded microgrid 28 3.3 Proposed distributed control scheme for discharging rate consensus in an islanded AC microgrid 29 3.4 Half-bridge inverter 33 3.5 Block diagram of the controlled inverter 34 3.6 Experimental setup 40 3.7 Microgrid configuration scheme 40 3.8 Block structure of a BESSiunit 41 3.9 Communication graph among the BESSs 42 3.10 Scenario 1: In the top, evolution of the SOC references, Ψr,i, the SOCs, Ψi, and the active powers, Pifor i={1,2,3}, when a load of 22Ωis connected at T2and then it is disconnected back at T3. In T3,α1,2= 0. In the bottom a zoom of SOC references at T144 3.11 Scenario 1: Evolution of voltage errors, ˜x2,i, capacitance voltages, vC,i, and inductance currents, iL,i for i={1,2,3}45 3.12 Scenario 1: Zoom at T1 of SOC references Ψr,i, and voltage errors ˜x2,i, for i={1,2,3}45 127 128 List of Figures 3.13 Scenario 2: In the top, evolution of the SOC references, Ψr,i, the SOCs, Ψi, and the active powers, Pi, for i={1,2,3}, when BESS1is disconnected at T2 and then it is connected back at T3. In the bottom, a zoom of SOC references at: a) T1and b) T246 3.14 Scenario 2: Evolution of voltage errors, ˜x2,i, capacitance voltages, vC,i and inductance currents, iL,i for i={1,2,3}47 3.15 Scenario 2: Zoom at T2 of SOC references, Ψr,i, and voltage errors, ˜x2,i, for i={1,2,3}47 3.16 Comparison of the SOC and active power evolution between a) our control and b) the controller proposed in [1] 48 4.1 Microgrid configuration scheme 63 4.2 Evolution of Ψr,i,Ψi, and the active powers Pi, for i={1,2,3}, when the battery 1 is disconnected at T2and then connected back at T3. In T1,α1,3 turns to 0 64 4.3 Evolution of the voltages, currents and duty cycles of inverters for i={1,2,3} between 1.45s to 1.8s 65 4.4 Evolution of the voltage errors 65 4.5 Comparison of simulation results of the SOC evolutions between the controller proposed in [1] (a) and our hybrid control proposal (b) 66 4.6 Comparison of simulation results of the active power evolutions between the controller proposed in [1] (a) and our hybrid control proposal (b) 67 5.1 Distributed control scheme for discharging rate and reactive power consensus in an islanded AC microgrid 71 5.2 Microgrid configuration scheme 81 5.3 Communication graph among the BESSs 82 5.4 Evolution of the SOC references Ψr,i, the SOCs Ψiand the active powers Pi, for i={1,2,3,4,5,6}, when a load of 300W and 60VAr is disconnected at T1 and then it is connected back at T2. In T3α1,2= 0 83 5.5 Evolution of the reactive power references Qr,i and the reactive powers Qi for i={1,2,3,4,5,6}, when a load of 300W and 60VAr is disconnected at T1 and then it is connected back at T2. In T3α1,2= 0 84 5.6 Evolution ˜ Ψi,˜ Pi,˜ Qiin Scenario 1 for i={1,2,3,4,5,6}, when a load of 300W and 60VAr is disconnected at T1and then it is connected back at T2. In T3α1,2= 0 84 5.7 Evolution of voltage errors, voltages, currents and duty cycles of converters for the first 0.2s in Scenario 1 85 5.8 Evolution of the SOC references Ψr,i, the SOCs Ψiand the active powers Pi, for i={1,2,3,4,5,6}, when the BESS1is disconnected at T1and then connected back at T286 5.9 Evolution of the reactive power references Qr,i and the reactive powers Qi for i={1,2,3,4,5,6}, when the BESS1is disconnected at T1and then connected back at T286 5.10 Evolution ˜ Ψi,˜ Pi,˜ Qifor Scenario 2 87 List of Figures 129 5.11 Evolution of voltage, current and duty cycle of converters for 1.45s to 1.95s of Scenario 2 87 6.1 Phase portrait of the supercritical hopf bifurcation for parameter λ93 6.2 Evolution of the system in the stable region when λ≤0and in the unstable region when λ > 093 6.3 Bifurcation diagram of a supercritical Hopf bifurcation 94 6.4 Phase portrait of the subcritical hopf bifurcation for parameter λ95 6.5 Evolution of the system in the stable region when λ≤0and in the unstable region when λ > 095 6.6 Bifurcation diagram of a subcritical Hopf bifurcation 95 6.7 Electrical circuit of a three-phase inverter with an LC filter 97 6.8 dq transformation of a voltage source set. (a) Original circuit. (b) dq transformed circuit 97 6.9 dq transformation of a inductor set. (a) Original circuit. (b) dq transformed circuit 98 6.10 dq transformation of a capacitor set. (a) Original circuit. (b) dq transformed circuit 98 6.11 dq transformation of a resistor set. (a) Original circuit. (b) dq transformed circuit 99 6.12 CPL block diagram. (a) Converter + load. (b) Equivalent circuit 99 6.13 Negative impedance behavior of CPLs 100 6.14 Current-voltage characteristic of a load converter as a CPL 100 6.15 Electrical circuit of a islanded AC microgrid with a CPL 102 6.16 Bifurcation diagram in the {id L1,Po}plane. (a) Bifurcation points and equilibrium stability. (b) Unstable limit cycle and region of attraction of the stable equilibrium 105 6.17 CPL power evolutions for power reference steps around bifurcation point H1106 6.18 Current iL1evolutions in abc and dq coordinates around bifurcation point H1107 6.19 Bus voltage evolutions in abc and dq coordinates around bifurcation point H1107 6.20 CPL power evolutions for power reference steps around bifurcation point H2108 6.21 Current iL1evolutions in abc and dq coordinates around bifurcation point H2108 6.22 Bus voltage evolutions in abc and dq coordinates around bifurcation point H2109 6.23 CPL time constant experimental test. (a) Connection scheme. (b) Imperix software interface. (c) Interface of the CPL control software. 110 6.24 CPL current measurements for time constant calculation 110 6.25 Experimental setup of an islanded AC microgrid with a CPL 111 6.26 Evolution of microgrid voltages (vabc C1and vabc C2), currents (iabc L1and iabc L2) and powers P1and P2under steps of constant resistors 112 6.27 Evolution of microgrid variables around bifurcation point H1. (a) CPL power and current id L1. (b) voltages vabc C1and vabc C2, currents iabc L1,iabc L2and iabc Load 112 6.28 Evolution of Po(top) and id L1(bottom) for 3 different operation points under Vn changes 113 6.29 Comparison of Poand current id L1values for 2 operation points between experimental results and bifurcation diagram of Fig 6.16(b) 114 6.30 Zoom in on the capacitor voltages, inductor currents and CPL current around for operation point 1 115 130 List of Figures 6.31 Zoom in on the capacitor voltages, inductor currents and CPL current around for operation point 3 115 List of Tables 3.1 Microgrid and droop control parameters 42 3.2 Inverter parameters 42 4.1 Consensus algorithm and droop control parameters 63 4.2 Inverter parameters 63 5.1 Microgrid and droop control parameters 81 5.2 Inverter parameters 82 6.1 Microgrid parameters 104 131 Bibliography [1] C. Li, E. A. Coelho, T. Dragicevic, J. Guerrero, and J. 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