State-space model of quasi-z-source inverter-PV systems for transient dynamics studies and network stability assessment

energies Article State-Space Model of Quasi-Z-Source Inverter-PV Systems for Transient Dynamics Studies and Network Stability Assessment Lluís Monjo 1,* , Luis Sainz 2, Juan JoséMesas 3and Joaquín Pedra 2   Citation: Monjo, L.; Sainz, L.; Mesas, J.J.; Pedra, J. State-Space Model of Quasi-Z-Source Inverter-PV Systems for Transient Dynamics Studies and Network Stability Assessment. Energies 2021,14, 4150. https://doi. org/10.3390/en14144150 Academic Editor: Rui Esteves Araújo Received: 13 May 2021 Accepted: 7 July 2021 Published: 9 July 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Department of System Engineering and Design, Jaume I University, Av. Vicent sos Baynat s/n, 12071 Castellóde la Plana, Spain 2 Department of Electrical Engineering, Barcelona School of Industrial Engineering (ETSEIB-UPC), Polytechnic University of Catalonia, Av. Diagonal 647, 08028 Barcelona, Spain; [email protected] (L.S.); [email protected] (J.P.) 3Department of Electrical Engineering, Barcelona East School of Engineering (EEBE-UPC), Polytechnic University of Catalonia, Av. Eduard Maristany 16, 08019 Barcelona, Spain; [email protected] *Correspondence: [email protected] Abstract: Photovoltaic (PV) power systems are increasingly being used as renewable power generation sources. Quasi-Z-source inverters (qZSI) are a recent, high-potential technology that can be used to integrate PV power systems into AC networks. Simultaneously, concerns regarding the stability of PV power systems are increasing. Converters reduce the damping of grid-connected converter systems, leading to instability. Several studies have analyzed the stability and dynamics of qZSI, although the characterization of qZSI-PV system dynamics in order to study transient interactions and stability has not yet been properly completed. This paper contributes a small-signal, state-space-averaged model of qZSI-PV systems in order to study these issues. The model is also applied to investigate the stability of PV power systems by analyzing the influence of system parameters. Moreover, solutions to mitigate the instabilities are proposed and the stability is verified using PSCAD time domain simulations. Keywords: PV systems; quasi-Z-source inverter; stability 1. Introduction Photovoltaic (PV) systems have become some of the most popular renewable generation sources [ 1 – 3 ]. They can have different configurations based on centralized or multiple inverters with single- or two-stage topologies [ 4 , 5 ]. The flexibility and efficiency of two-stage voltage source inverter (VSI) topologies have led to their widespread use; however, single-stage topologies based on impedance-source networks with a centralized inverter [ 6 – 9 ] or micro-inverters [ 10 , 11 ] are increasing in popularity as simple and economical configurations that can overcome the shortcomings of two-stage VSI topologies. Reviews of single-stage impedance-based converters (including the main topologies, modeling, control and cutting-edge techniques) are presented in [ 6 – 8 ]. An analytical comparison of the passive components and semiconductor stress of the previous inverters and multilevel buck–boost inverters is presented in [ 9 ]. Single-stage impedance-based converters such Z-source and quasi-Z-source inverters (qZSIs) are currently used for the integration of renewables and grids [ 6 , 9 ]. A review of the use of micro-inverters as a rising technology in PV systems is also presented in [ 10 , 11 ]. In particular, qZSIs are promising because buck– boost voltage is efficiently and reliably generated in a single-stage operation [ 6 – 9 , 12 – 17 ]. A traditional qZSI with a semiconductor and impedance network between the DC energy source and the AC grid inverter is investigated in [ 6 – 9 , 12 – 14 ]. Four improved qZSIs are theoretically studied in [ 15 ], while a three-level NPC qZSI, which provides high energy density, short circuit immunity and voltage regulation (step-down and step-up) capability, is examined in [16,17]. Energies 2021,14, 4150. https://doi.org/10.3390/en14144150 https://www.mdpi.com/journal/energies Energies 2021,14, 4150 2 of 15 Grid integration of PV power systems can lead to stability problems because the damping of power conversion systems connected to the grid is reduced by power electronics. Many studies have dealt with this issue in PV power systems based on traditional two-stage converter topologies by means of small-signal state-space (SS) models. Specifically, the impacts of PV power system variables (e.g., solar irradiance and temperature [ 18 , 19 ]) and control parameters [ 3 ] on stability have been investigated. Other works have looked at qZSI-PV system stability, but most only have provided qZSI dynamic models to analyze qZSI stability and derive general conclusions about qZSI-PV system stability [ 6 – 8 , 12 – 14 , 17 ]. The qZSI control methods and their influence on qZSI stability are explored in [ 6 , 14 ], while qZSI design guides are given from these dynamic models in [ 12 , 13 ]. Exhaustive operating range studies on the impacts of converter variables on the transient response of different Z-source inverters such as qZSIs are also performed in [ 7 , 8 ]. Moreover, qZSI modeling for analysis of qZSI controls is applied in [ 17 ]. It is worth noting that very few studies have provided dynamic models of a complete qZSI-PV system [ 20 – 25 ]. Such studies have mainly explored the influence of an AC network on DC-side stability [ 20 , 23 , 24 ] and qZSI dynamics [ 21 , 22 ]. On the other hand, dynamic interactions between AC and DC networks have not been fully studied yet. Recently, two qZSI-PV system simulation models based on PSCAD and Simulink were introduced in [ 25 ], and these dynamic interactions and qZSI-PV system stability were studied from the proposed models; however, these models (and their corresponding studies) are limited by the features of the PSCAD and Simulink tools, and a qZSI-PV system model based on the small-signal state-space equation is required to have more flexibility in the qZSI-PV system dynamic simulations and stability analysis. This paper extends the work in [ 25 ] and contributes a fully developed small-signal state-space averaged (SSA) equation of qZSI-PV systems for implementation in customized codes of time domain simulation and stability studies. The equation is systematically and rigorously obtained by considering the main qZSI-PV system controls, i.e., maximum power point (MPP) tracking (MPPT), PV voltage, grid current and qZSI duty cycle controls. The models in [ 25 ] only allow Simulink and PSCAD dynamic studies of qZSI-PV system behavior to be performed, while our model enables the use of different software programs (e.g., Simulink and MATLAB environments), increasing the possibility to carry out qZSI-PV system dynamics studies, such as the following: •PV system stability studies in the frequency domain; • Participation factor (PF) assessments and analytical studies of the influence of PV system parameters on stability; • AC grid-connected PV system dynamics studies based on a single Simulink model of qZSI-PV systems derived from the proposed model. Here, the contributions to the knowledge of the following stability issues are presented through the application of the proposed equation: •Impacts of qZSI-PV system parameters on stability; • Proposals for stability improvement from PFs and analysis of qZSI-PV system damping parameters. All of these contributions, which are numerically validated by PSCAD simulations, can be used together as a valuable transient modeling tool and in qZSI-PV system stability studies. 2. State-Space Modeling of PV Power Systems In order to evaluate the qZSI-PV system stability, SSA modeling of the circuit in Figure 1is presented based on the following state-space equation: d∆x dt =A∆x+B1∆u1+B2∆u2∆y=C∆x+E1∆u1+E2∆u2, (1) where x , u1 , u2 and y are the state, internal input, external input and output vectors, respectively, while the small-signal variables are denoted by the symbol ∆. Energies 2021,14, 4150 3 of 15 Figure 1. The qZSI-PV power system circuit. Broadly speaking, qZSI-PV systems have a PV installation supplying the qZSI (i.e., the N p× N s PV panel, the capacitor C p and the DC conductor resistance R c ), which allows the output voltage v dc to be boosted at the VSI terminals. The N p× N s PV panel has N p strings in parallel with NsPV cells in series. To fully extract the PV panel’s maximum power, an MPPT algorithm and the voltage control of the PV panel are used. The VSI current control loop fixes the power that the VSI delivers to the grid. The DC peak voltage v dc,p is adjusted using the qZSI duty cycle control. The pulse width modulation (PWM) block generates the trigger signals of the IGBTs (i.e., the shoot-through states of the qZSI) from the grid dq-frame reference voltage v dqr and the duty cycle dby means of the carrier-based sinewave pulse width modulation method, referred to as simple boost control (see Chapter 4 in [ 6 ]). The dynamic behavior of the qZSI-PV system components is dictated by their SS equations, while the averaging approach for these equations used for characterizing the two qZSI states is also applied for qZSI modeling. The development of SSA equations is well documented in [ 25 ], so only a summary is presented in the following subsections. All qZSI-PV system component equations are turned into a single SSA equation that characterizes the qZSI-PV system dynamics. This equation is validated using PSCAD simulations. 2.1. Model of the PV Installation The SS equation for the PV installation is presented in this subsection. A typical I–V plot of a PV panel is shown to highlight the different parameter values in Figure 2a, where the MPPs of the different I–V plots are labelled with dots. The panel has an Np×Ns60 W PV Solarex MSX60 module (see specifications in [ 2 ]). It is noted that for a constant irradiance level Gand temperature T, an increase in the number of PV panels in series N s leads to an increase in the voltage of the PV panel v pv , while the same is true for the number of PV panels in parallel N p and the current of the PV panel i pv . This I–V plot of a PV panel is expressed as [2,4,25]: ipv =NpIpvL −Ipv0expvpv/Ns+Rsipv/Np nVT −1, (2) where I pvL and I pv0 are the photovoltaic and saturation currents of the PV cell, R s is the equivalent series resistance of the PV cell, nis the diode quality factor, V T =k · T/q is the Energies 2021,14, 4150 4 of 15 thermal constant of the PV cell, kis the Boltzmann constant (1.38 × 10 −23 J/K), qis the Coulomb constant (1.602 × 10 −19 C) and Tis the PV cell temperature in Kelvin. The photovoltaic current I pvL is proportional to the irradiance level Gand is usually referred to as the rated irradiation (i.e., G= 1 Sun = 1000 W/m 2 ). The photovoltaic and saturation currents, IpvL and Ipv0, also depend on the temperature T, which is 25 ◦C. The influence of both parameters is illustrated on the right side of Figure 2a. Figure 2. PV installation: ( a ) I-V plot of the PV panel; ( b ) equivalent circuit of the PV panel; ( c ) smallsignal equivalent circuit of the PV installation. The I-V plots of PV panels are modeled with the equivalent circuit in Figure 2b, which is deduced from the linearization of the I-V Equation in (2) around the MPP (V pv =V mpp and Ipv =Impp, see plot on the left in Figure 2a) [4,25]: ipv =Ipvs −1 Rpv vpv, (3) where: Ipvs =Ipv +Vpv Rpv Rpv =−dvpv dipv (Vpv,Ipv) =Ns Np nVT Ipv0expVpv/Ns+RsIpv/Np nVT −1 | {z } Rpv_cell +Rs. (4) The PV panel’s small-signal circuit derived from the linearization of the I-V plots near the PV panel operating point (3), the DC conductor and the shunt capacitor are shown in Figure 2c, while the SS model is represented by: d dt ∆vpv=−1 CpRpv  | {z } Apv ∆vpv+−1 Cp | {z } B1pv [∆ii]+1 Cp |{z} B2pv ∆Ipvs[∆vi]=[1] |{z} Cpv ∆vpv+[−Rc] |{z} E1pv [∆ii]. (5) Energies 2021,14, 4150 5 of 15 The MPPT control generates the PV panel reference voltage with the maximum instantaneous power p pv =v pv· i pv , i.e., the voltage at the MPP. According to [ 25 ], the SS model of the MPPT control is characterized by: d dt ∆φpvs=[1] |{z} B1mp ∆vpv ∆vpvr=[km ikm] |{z } Cmp ∆φpvs+hkm pkmi | {z } E1mp ∆vpv. (6) where the variable φpvs is the state-space variable characterizing the dynamic behavior of the MPPT PI control [ 25 ]. This variable does not have a particular physical meaning but it facilitates the development of the state-space model and has the same dimensions as the magnetic flux. The SS model of the MPPT control and the PV installation is derived from (5) and (6) and represented by: d dt ∆vpv ∆φpvs =Apv 01x1 B1mp 01x1 | {z } Apvm ∆vpv ∆φpvs +B1pv 01x1 | {z } B1pvm [∆ii]+B2pv 01x1 | {z } B2pvm ∆Ipvs ∆vpvr=Cpv 01x1 E1mp Cmp  | {z } Cpvm ∆vpv ∆φpvs +E1pv 01x1 | {z } E1pvm [∆ii]. (7) 2.2. Model of the PV Panel Control and Grid-Connected VSI The set composed of the voltage control loop of the PV panel, the VSI current control and the grid-connected VSI is presented. The PI control loop of the PV panel voltage generates the grid dreference current i dr of the dq-frame VSI current control from the PV panel reference voltage (Figure 1). This VSI current control is represented as a PI-based control, which outputs the grid dq-frame reference voltage v dqr to the VSI space vector modulation. The grid q-reference current i qr of the dq-frame VSI current control is fixed to zero by assuming that the power factor of the inverter operation is the unity [ 5 , 6 ]. The influence of the inverter PLL on the system dynamics is disregarded. The small-signal relationship of the grid d-reference current is: idr =− kpv p+kpv i s!(vpvr −vpv)⇒∆idr =kpv p(vpv −vpvr) + kpv i∆φpv s∆φpv =∆vpv −∆vpvr, (8) where kpv p and kpv i are the proportional and integral gains, respectively. The variable φpv is the state-space variable characterizing the dynamic behavior of the PV voltage PI control. This variable does not have a particular physical meaning but it facilitates the development of the state-space model and has the same dimensions as the magnetic flux. According to (8), the SS model of the voltage control of the PV panel is represented by: d dt ∆φpv=1−1 | {z } B1pv_c ∆vpv ∆vpvr ∆vpvr=hkpv ii |{z} Cpv_c ∆φpv+kpv p−kpv p | {z } E1pv_c ∆vpv ∆vpvr . (9) The small-signal relationship of the VSC current control output d-voltage is written as: ud=kcc p+kcc i s(idr −id)⇒∆ud=kcc p(∆idr −∆id) + kcc i∆qcc s∆qcc =∆idr −∆id, (10) where kcc p and kcc i are the compensator’s proportional and integral gains, respectively. The variable q cc is the state-space variable characterizing the dynamic behavior of the VSC current PI control. This variable does not have a particular physical meaning but it facilitates the development of the state-space model and has the same dimensions as the Energies 2021,14, 4150 6 of 15 electric charge; therefore, the SS model of the VSC current control output d-voltage (10) is expressed as: d dt [∆qcc]=[1] |{z} B1cc [∆idr]+[−1] |{z} B2cc [∆id] [∆ud]=[kcc i] |{z} Ccc [∆qcc]+hkcc pi |{z} E1cc [∆idr]+h−kcc pi | {z } E2cc [∆id]. (11) The SS model of the grid-connected VSI is derived as [25]: vd=Lfsid−Lfω1iq+ed vdr =ud−Lfω1iq+edvd=vdr ⇒∆id=1 Lf ∆φds s∆φds =∆ud, (12) With L f being the converter filter inductance (the inner resistance of the inductor is neglected). The variable φds is the state-space variable characterizing the dynamic behavior of the SS model of the grid-connected VSI. This variable does not have a particular physical meaning but it facilitates the development of the state-space model and has the same dimensions as the magnetic flux; thus, the SS model of the grid-connected VSI is: d dt [∆φds]=[1] |{z} B1g [∆ud] [∆id]=h1/Lfi | {z } Cg [∆φds]. (13) It must be noted that the converter filter capacitor C f does not appear in the SS model of the grid-connected VSI, meaning this capacitor does not affect the qZSI-PV system model or dynamics. It is considered as a component of the grid in the stability studies of grid-connected qZSI-PV systems. The SS model of the PV panel voltage, VSI current control and the grid-connected VSI is derived from (9), (11) and (13) and expressed as: d dt   ∆φpv ∆qcc ∆φds  =  01x101x2 B1cc B1gE1cc Cpv_c 01x1B2ccCg B1gCcc B1gE2ccCg  |{z } Apv_vsc   ∆φpv ∆qcc ∆φds  +  B1pv_c B1cc B1gE1cc E1pv_c   | {z } B1pv_vsc ∆vpv ∆vpvr  [∆id]=01x2Cg | {z } Cpv_vsc   ∆φpv ∆qcc ∆φds  . (14) 2.3. Model of the qZSI The SSA model of the qZSI in Figure 1, which considers the parasitic resistances rof the inductors and series resistances Rof the capacitors is outlined in this subsection. The qZSI has two different operational states within one switching cycle, i.e., the shoot-through and the non-shoot-through states during T 0 (the inverter behaves as a short circuit) and T 1 (the inverter behaves as a current source representing VSI consumption), respectively [ 6 , 14 , 25 ]. The former is identified by the duty cycle d=T 0 /Tand the latter by T 1 /T= 1 − d. The qZSI control is also plotted in Figure 1, where the duty cycle is obtained to adjust the DC peak voltage vdc,p [6,14]. Energies 2021,14, 4150 7 of 15 2.3.1. Model of the Power Circuit The SSA model of the qZSI is expressed from the SS equations of the qZSI operational states (i.e., shoot- and non-shoot-through states) according to Figure 1as [6,25]: d dt     ∆iL1 ∆iL2 ∆vC1 ∆vC2    =     −R+r L10d−1 L1 d L1 0−R+r L2 d L2 d−1 L2 1−d C1−d C10 0 −d C2 1−d C20 0          ∆iL1 ∆iL2 ∆vC1 ∆vC2    +    1 L1R(1−d) 0R(1−d) 0d−1 0d−1    ∆vi ∆idc  ∆ii ∆vdc =1 0 0 0 R(1−d)R(1−d)1−d1−d    ∆iL1 ∆iL2 ∆vC1 ∆vC2    . (15) Finally, the SSA model of the qZSI is expressed from (15) as [6,25]: d dt     ∆iL1 ∆iL2 ∆vC1 ∆vC2    =      −(R+r) L10(D−1) L1 D L1 0−(R+r) L2 D L2 (D−1) L2 1−D C1 −D C10 0 −D C2 1−D C20 0       | {z } Az     ∆iL1 ∆iL2 ∆vC1 ∆vC2    +      1 L1 R(1−D) L1 0R(1−D) L2 0D−1 C1 0D−1 C2       | {z } Blz ∆vi ∆idc +      V1 L1 V1 L2 I1 C1 I1 C2       | {z } B2z [∆d] ∆ii ∆vdc =1 0 0 0 R(1−D)R(1−D)1−D1−D |{z} Cz     ∆iL1 ∆iL2 ∆vC1 ∆vC2    +0 0 0−2R(1−D) |{z} Elz ∆vi ∆idc +0 V2 |{z} E2z [∆d], (16) where Dis the steady-state duty cycle, while V 1 =V C1 +V C2− R · I dc ,I 1 =I dc − I L1− I L2 , V 2 = − V 1 +R · I 1 and Xz = [I L1 I L2 V C1 V C2 ] T and U1z = [V i I dc ] T are the state and input vectors in steady state, respectively. The steady-state voltages and currents are determined by imposing dx/dt = 0 in the SSA model of the qZSI, (1) and (15) [6]. The qZSI input vector (i.e., the qZSI input voltage v i and output current i dc ) is obtained from the SS model of the PV installation (2–5) and the power balance in the VSI, respectively. Assuming a lossless VSI, the AC and DC instantaneous power balance in the circuit of Figure 1can be written as [25]: vdcidc =vdid+vqiq⇒Vdc∆idc +Idc∆vdc =Vd∆id+Id∆vd+Vq∆iq+Iq∆vq Iq=0 ⇒ Vd>>Vq ∆idc =md0∆id−1 md0Gdc∆vd+Gdc∆vdc,(17) where m d0 =V d /V dc is the modulation function’s steady-state operation point, G dc = 1/R dc = − P/ V2 dc , with Pbeing the active power delivered from DC to AC (see Figure 1) and V d ,V dc and I dc being the inverter output voltage, the qZSI voltage and the current in steady state, respectively. The power balance’s small-signal relationship in (17) can be rewritten with (5), (10) and (12) as follows: ∆idc =md0∆id−Gdcm(∆ud+∆ed) + Gdc∆vdc =∆idc0+Gdc∆vdc, (18) where Gdcm =Gdc/md0and: ∆idc0= (md0+kcc pGdcm)∆id−Gdcm∆ed−Gdcmkcc pkpv i∆qpv +kcc i∆qcc +kcc pkpv p(∆vpv −∆vpvr). (19) It must be highlighted that the virtual conductance G dc relates the current i dc and the voltage v dc , while the negative value of this conductance for the VSI inverter operation Energies 2021,14, 4150 8 of 15 (P> 0) causes the VSI non-passive behavior on the DC side. This may lead to system instability at certain resonances. Any factor increasing the VSI input flow of the active power P(e.g., the number of PV or the irradiance level) affects the qZSI-PV power system’s stability (see Section 3). On the other hand, any factor reducing the value of G dc (e.g., a higher steady-state qZSI output voltage V dc ) improves the qZSI-PV system’s stability (see Section 3). Nevertheless, it must be considered that increasing the qZSI output voltage V dc could lead to higher switch stress and a lower voltage utilization ratio. 2.3.2. Model of the Duty Cycle Control The qZSI control in Figure 1adjusts the DC peak voltage v dc,p by using the duty cycle d. The DC voltage PI control loop imposes the DC peak voltage reference V∗ dc,p , while the inductor-L 2 current loop imposes the inductor-L 2 current reference i L2 through a proportional controller to improve the dynamic response of the control [ 6 , 14 ]. A low-pass filter (LPF) with a corner frequency f c = 25 Hz (i.e., a bandwidth ωc = 2 π f c ) smooths the reference duty cycle supplied to the qZSI. According to [25], the SS model of the duty cycle control is characterized by: d dt [∆qdc]=h01 1−Di | {z } B1dc ∆iL2 ∆vC1+Vdc,p 1−D | {z } B2dc [∆d] [∆dr]=h−kL pkdc ii | {z } Cdc [∆qdc]+h−kL p kL pkdc p D−1i | {z } E1dc ∆iL2 ∆vC1+"kL pkdc p D−1Vdc,p# | {z } E2dc [∆d], (20) where kdc p and kdc i are the proportional and integral gains of the DC voltage controller and kL p is the proportional gain of the inductor-L 2 controller. The SS model of the LPF is characterized by: d dt [∆d]=[−ωc] |{z} Af [∆d]+[ωc] |{z} B1f [∆dr] [∆d]=[1] |{z} Cf [∆d]. (21) The SSA model of the duty cycle control is written from (20) and (21) as: d dt ∆qdc ∆d=01x1 B2dc B1fCdc Af+B1fE2dc  | {z } Ad ∆qdc ∆d+01x1 B1dc 01x1 01x1 B1fE1dc 01x1  | {z } B1d     ∆iL1 ∆iL2 ∆vC1 ∆vC2     [∆d] = 0 1  | {z } Cd∆qdc ∆d. (22) 2.3.3. Complete Model of the qZSI The SSA model of the qZSI in Figure 1is derived from (16) and (22) and represented by: d dt         ∆iL1 ∆iL2 ∆vC1 ∆vC2 ∆qdc ∆d         =AzB2zCd B1d Ad | {z } Aqz         ∆iL1 ∆iL2 ∆vC1 ∆vC2 ∆qdc ∆d         +B1z 02x2  | {z } B1qz ∆vi ∆idc   ∆ii ∆vdc =CzE2zCd | {z } Cqz         ∆iL1 ∆iL2 ∆vC1 ∆vC2 ∆qdc ∆d         +[E1z] |{z} Elqz∆vi ∆idc .(23) Energies 2021,14, 4150 9 of 15 2.4. Model of the qZSI-PV System The diagram of the qZSI-PV system with the small-signal models of all the components is presented in Figure 3, according to the previous sections. The SSA models (1) of the qZSI-PV system modules before obtaining the complete qZSI-PV system are shown in this subsection. Figure 3. The qZSI-PV system block diagram (the symbol ∆is omitted). • Module #1 (M_1 in Figure 3): The PV panel control and grid-connected VSI (14), including the VSI power balance (18), are expressed as: d dt   ∆φpv ∆qcc ∆φds  =hApv−vsci | {z } AM−1   ∆φpv ∆qcc ∆φds  +B1pvvvsc 03x1  | {z } B1M−1   ∆vpv ∆vpvr ∆vdc   ∆idc ∆id=CM−1  ∆φpv ∆qcc ∆φds  +E1M−1  ∆vpv ∆vpvr ∆vdc  +−Gdcm 0 | {z } E2M−1 [∆ed] CM_1 ="md0+Gdcmkcc pCpv−vsc Cpv−vsc #−Gdcmkcc pkpv ikcc i0 0 0 0 E1M−1=−Gdcmkcc pkpv pGdcmkcc pkpv pGdc 0 0 0 . (24) • Module #2 (M_2 in Figure 3): The PV installation (4) and the PV panel control, gridconnected VSI and VSI power balance (24) are represented by: d dt       ∆vpv ∆φpvs ∆φpv ∆qcc ∆φds       =Apvm 02x3 B1M_1HAAM_1 | {z } AM_2       ∆vpv ∆φpvs ∆φpv ∆qcc ∆φds       +B1pvm 02x1 B1M_1HBE1pvm B1M_1IT 001  | {z } B1M_2 ∆ii ∆vdc +B2pvm 02x1 03x103x1 |{z } B2M_2 ∆Ipv ∆ed ∆vi ∆idc =I10Cpvm 01x3 I10E1M_1HAI10CM_1 | {z } CM_2       ∆vpv ∆φpvs ∆φpv ∆qcc ∆φds       +I10E1pvm 01x1 I10E1M_1HBE1pvm I10E1M_1IT 001  | {z } E1M_2 ∆ii ∆vdc +01x101x1 01x1I10E2M_1 | {z } E2M_2 ∆Ipv ∆ed, (25) where HA=IT 100I10 +IT 010I01Cpvm =  1 0 0 0 0 0  +  0 0 0 1 0 0  Cpvm HB=IT 010I01 =  0 0 0 1 0 0   I10 =1 0 I01 =0 1 I100 =010I010 =010. (26)