Virtual Conductance Based Cascade Voltage Controller for VSCs in Islanded Operation Mode

JOURNAL OF MODERN POWER SYSTEMS AND CLEAN ENERGY, VOL. 10, NO. 6, November 2022 Virtual Conductance Based Cascade Voltage Controller for VSCs in Islanded Operation Mode Francisco Jesús Matas-Díaz, Manuel Barragán-Villarejo, Juan Carlos Olives-Camps, Juan Manuel Mauricio, and José María Maza-Ortega Abstract——Voltage source converters have become the main enabler for the integration of distributed energy resources in microgrids. In the case of islanded operation, these devices nor‐ mally set the amplitude and frequency of the network voltage by means of a cascade controller composed of an outer voltage control loop and an inner current control loop. Several strate‐ gies to compute the gains of both control loops have been pro‐ posed in the literature in order to obtain a fast and decoupled response of the voltages at the point of common coupling. This paper proposes an alternative and simple methodology based on the introduction of a virtual conductance in the classic cas‐ cade control. This strategy allows to design each control loop in‐ dependently, obtaining a closed-loop response of a first-order system. In this way, the gains of each control loop are easily de‐ rived from the parameters of the LC coupling filter and the de‐ sired closed-loop time constants. Furthermore, a state observer is included in the controller to estimate the inductor current of the LC filter in order to reduce the number of required mea‐ surements. A laboratory testbed is used to validate and com‐ pare the proposed controller. The experimental results demon‐ strate the effectiveness of the proposal both in steady-state and transient regimes. Index Terms——Voltage source converter, microgrid, cascade controller tuning, virtual conductance. I. INTRODUCTION THE success of the integration of distributed renewable energy sources (DRESs), electrical vehicles or energy storage systems is largely linked to the development of pow‐ er electronics [1]. In particular, the voltage source converter (VSC) is of utmost importance nowadays because it pro‐ vides an efficient and flexible interface between the DC re‐ sources with the AC network [2]. VSCs may bring a pletho‐ ra of functionalities for improving the power system opera‐ tion and pushing towards a fully decarbonized scenario in the near future. For this reason, VSC control algorithms have evolved during the last decades according to the de‐ manded requirements of each application. In case of conven‐ tional power systems, where synchronous machines are re‐ sponsible of maintaining the voltage and the frequency, most of the VSCs operate in grid-feeding mode [3]. These VSCs are controlled as current sources which require a synchroni‐ sation with the power grid for injecting the active and reac‐ tive power references. Conversely, in the case of islanded systems, the VSCs are operated as voltage sources because it is required to form an AC grid, i.e., to define the frequency and voltage amplitudes of the power system. Nevertheless, during the last years, the grid-forming concept is widely dis‐ cussed for grid-connected applications because of the need of enhancing the DRES functionalities when the convention‐ al synchronous generation is being displaced by this new technology. An interesting discussion about the evolution of the terminology and VSC characterization can be found in [4]. Grid-forming VSCs are composed of different control lay‐ ers which can be integrated in a hierarchical manner. In the case of grid-connected applications or islanded microgrids with more than one VSC, a higher control layer is required for defining the frequency and voltage amplitudes to achieve a set of functionalities. With this regard, different techniques relying on droop controllers for active and reactive power sharing [5], advanced algorithms mimicking the dynamic per‐ formance of a synchronous generator [6] or even minimizing the subsynchronous oscillations in power systems [7] have been proposed. Then, a voltage controller is applied to track the computed voltage reference. On the contrary, in the case of a single VSC supplying an islanded system, only a volt‐ age controller to set the desired voltage reference (amplitude and frequency) is required without the need to implement the higher control layer. Focusing on the voltage controller, which is the main ob‐ jective of this paper, two possible alternatives have been ap‐ plied for grid-connected and islanded applications: single voltage control loop or cascade voltage control loops. The former methods control the voltage at point of com‐ mon coupling (PCC) directly through a single control loop based on state feedback as presented in [8]-[10]. The main challenge of this type of controllers is to deal with overcur‐ Manuscript received: February 14, 2021; revised: July 28, 2021; accepted: April 7, 2022. Date of CrossCheck: April 7, 2022. Date of online publication: July 15, 2022. This work was supported by the European Union Horizon 2020 under grant agreement 764090 (EASY-RES), Spanish Ministry of Economy under grant ENE2017-84813-R and CERVERA Research Programme of CDTI, the Industri‐ al and Technological Development Centre of Spain, under the research project HySGrid+(CER-20191019), and in part by Universidad de Sevilla in the frame‐ work of VI PPIT-US. This article is distributed under the terms of the Creative Commons Attribu‐ tion 4.0 International License (http://creativecommons.org/licenses/by/4.0/). F. J. Matas-Díaz, M. Barragán-Villarejo, J. C. Olives-Camps, J. M. Mauricio, and J. M. Maza-Ortega (corresponding author) are with the Department of Elec‐ trical Engineering, Universidad de Sevilla, Seville 41092, Spain (e-mail: fma‐ [email protected]; [email protected]; [email protected]; [email protected]; jmma‐ [email protected]). DOI: 10.35833/MPCE.2021.000121 1704 MATAS-DÍAZ et al.: VIRTUAL CONDUCTANCE BASED CASCADE VOLTAGE CONTROLLER FOR VSCS IN ISLANDED OPERATION MODE rent situations as the current control loop is fully removed [11]. To solve this problem, the PCC reference voltage can be modified according to the VSC current using a voltage droop characteristic [8] or by means of a threshold virtual impedance [10]. Despite the good performance offered by these single-loop strategies, even in case of overcurrent situations, cascade control loops are also a popular alternative widely used in se‐ ries-connected and grid-forming VSCs. These algorithms are based on an outer control loop (OCL) and an inner control loop (ICL) in charge of controlling the voltage and current, respectively [12]-[17]. Conventionally, the ICL and the OCL are based on proportional integral (PI) controllers formulated on a rotating synchronous frame and designed with suffi‐ cient separation between their corresponding bandwidths in such a way that their dynamics do not interact with each oth‐ er [18]. To achieve this goal, the ICL is tuned faster than the OCL, allowing an independent controller design. Probably, this is the reason why cascade controllers are widely used. However, despite the apparent ease of this control strategy, an adequate computation of the controller gains to guarantee the time constants of each control loop is not straightforward due to the characteristics of the plants to be controlled. Thus, the use of PI controllers in the ICL with proportional controllers in the OCL is proposed in [12] at the cost of a steady-state error at the voltage of PCC. This error can be eliminated by introducing a proportional resonant (PR) con‐ troller in the outer voltage control loop. However, it is re‐ quired to use a quite small cutoff frequency to avoid intro‐ ducing a phase shift towards the crossover frequency which decreases the phase margin and leads to a slow dynamics. Recently, a sliding-mode control for the ICL and a mixed H2/H¥for the OCL have also been proposed [13]. In this way, a robust controller without a precise model of the VSC and its coupling filter can be obtained. Nevertheless, this complex technique is suitable for VSCs operating with lowswitching frequencies where the possible resonances with the power system must be damped. Other sophisticated con‐ trol techniques such as predictive control [19] or neural net‐ works [20] have been reported. Despite the good perfor‐ mance of these proposals, it has to be considered that the controller tuning relies on some auxiliary processes, i.e., the selection of adequate weighting factors of the cost function for the predictive controllers or training stages in the case of neural networks. Finally, an interesting interpretation of the controllers used in grid-forming VSCs as circuit elements has been recently outlined in [21], giving clues of the role of each controller parameter on the overall system perfor‐ mance. This paper proposes to modify the classic cascade voltage control of a single LC-coupled grid-forming VSC operating in islanded mode by adding a negative feedback through a virtual conductance at the PCC. This allows to design the voltage and current control loops independently so that they respond following a given first-order closed-loop time con‐ stants with guaranteed system stability. The capacitor voltage and the injected PCC current are used as measurements in the control algorithm due to their low content of switching harmonics. Therefore, the inductor-side current of the VSC is unknown, preventing the application of a cascade control‐ ler. In order to apply the proposed cascade control strategy, a state observer to estimate this current is implemented. In this way, the number of current sensors is reduced while the control algorithm benefits from a filtered inductor-side cur‐ rent estimation with reduced harmonic content. The rest of this paper is organized as follows. Section IIA and II-B present the mathematical model of the LC-cou‐ pled VSC and the classic cascade control algorithm, respec‐ tively. After that, the proposed virtual conductance and the tuning of the OCL gains are presented. Section III outlines a sensitivity analysis where the influence of the virtual conduc‐ tance value and controller time constant are evaluated using a frequency domain and stability analyses. Section IV de‐ scribes the experimental results obtained in the laboratory to validate the proposed virtual conductance technique. The controller performance is analyzed for steady-state and tran‐ sient test cases, including a step change in the load and a large perturbation to evaluate the performance of the pro‐ posed controller. This paper closes with the main conclu‐ sions and future research lines. II. THEORETICAL FRAMEWORK This section is devoted to give the details of the proposed control algorithm based on a virtual conductance in parallel with the filter capacitor. For this purpose, the averaged mod‐ el of the LC-coupled VSC is presented first, followed by the definition of the virtual conductance which allows a simpli‐ fied tuning of the cascade controller gains. The section clos‐ es with the formulation of the complete control algorithm in‐ cluding the cross-coupling cancellation terms and a state ob‐ server for estimating the filter inductor current. A. Averaged Model of LC-coupled VSC This subsection details the averaged model of an LC-cou‐ pled VSC, as shown in Fig. 1, in abc coordinates and its cor‐ responding counterpart in dq rotating reference frame. It is assumed that the DC side of VSC is connected to a constant DC voltage source. Therefore, the differential equa‐ tions related to this system in the abc coordinates are de‐ fined as [22]: vtabc =Rtitabc +Lt ditabc dt+vmabc (1) itabc =Cdvmabc dt+isabc (2) where vtabc = [ vta vtb vtc ] Tis the VSC terminal voltage; vmabc = [ vma vmb vmc ] Tis the capacitor voltage; itabc = [ ita itb itc ] Tis the inductor current; isabc = [ isa isb isc ] T is the injected PCC current; and the LC coupling filter is rep‐ CLoad vdc vt,abc vm,abc is,abc Ltit,abc RtPCC +  Fig. 1. One-line diagram of an LC-coupled VSC connected to an islanded system. 1705 JOURNAL OF MODERN POWER SYSTEMS AND CLEAN ENERGY, VOL. 10, NO. 6, November 2022 resented by the diagonal matrices Lt=diag [ LtLtLt ] ,Rt= diag [ RtRtRt ] , and C=diag [ CCC ] . These equations can be transformed into the dq coordi‐ nates by using the Park transformation as: vtdq =Rtdqitdq +Ltdq ditdq dt+ωLtdqitqd +vmdq (3) itdq =Cdq dvmdq dt+ωCdqvmqd +isdq (4) where the vectors with subscripts dand qcorrespond to volt‐ ages and currents in dq coordinates; ωis the angular fre‐ quency; Ltdq =diag[-LtLt];Rtdq =diag[RtRt]; and Cdq = diag[-C C]. B. Cascade Control Algorithm The classic cascade voltage controller for an LC-coupled VSC on a rotating synchronous frame is represented in Fig. 2. In order to simplify the addition of the virtual conduc‐ tance and the computation of the controller gains, the crosscoupling cancellation terms of (3) have been omitted in the analysis. Note that this simplification leads to the same dy‐ namic system for dq coordinates. The objective of this strate‐ gy is to control the PCC voltage through the capacitor volt‐ age vm. For this purpose, two cascade PI controllers, i. e., ICL and OCL, are applied. On the one hand, the OCL gener‐ ates the reference current i* tto the ICL. On the other hand, the ICL generates the voltage vtat the VSC terminals to track the reference voltage v* m. Usually, dynamics of the OCL is assumed to be much slower than that of the ICL. This assumption allows to design each control loop indepen‐ dently without considering the mutual dynamic interactions [14]. Nevertheless, it has to be considered that this dynamic separation of the controllers requires a high bandwidth of the inner current control loop and a high-switching frequen‐ cy of the VSC. Taking into account these issues, the ICL PI controller can be designed considering just its action on the inductive filter. This plant, composed of Rtand Ltas shown in Fig. 2, pres‐ ents a first-order dynamics. In this case, the controller gains can be easily computed as [22]: ì í î ï ï ï ï ï ï ï ï ï ï kpi=Lt τi kii=Rt τi (5) where kpiand kiiare the proportional and the integral gains of the PI controller, respectively; and τiis the desired ICL time constant. The use of these controller gains leads to a controlled closed-loop response following a first-order dy‐ namics with τi. The same simplification can be applied to the OCL design which leads to consider that itequals i* t. In this way, the OCL PI controller is applied directly to a plant composed of the capacitor Cas shown in Fig. 3(a). The computation of the control gains in this case is not straightforward because, as stated in the introduction section, the closed-loop system has a second-order dynamics. Its tuning requires to adjust the damping ratio and natural frequency by means of modify‐ ing the control gains, and the system would provide a typi‐ cal second-order response with a certain rise time, overshoot and settle time depending on the damping coefficient and natural frequency. The next subsection presents how the in‐ corporation of a virtual conductance in the classical cascade control solves this shortcoming and facilitates the computa‐ tion of the OCL gains. C. Virtual Conductance and Tuning of OCL Gains The main idea is to modify the original control algorithm shown in Fig. 3(a), where the ICL dynamics has been om‐ mited, by including the negative feedback of Gv. This modi‐ fies the ICL reference current as shown in Fig. 3(b). This negative feedback can be transformed using block diagram operations as shown in Fig. 3(c). In this way, the OCL PI controller is applied to a plant with a first-order dynamics which facilitates the computation of the controller gains: ì í î ï ïï ï ï ï ï ï ï ï kpv=C τv kiv=Gv τv (6) where kpvand kivare the proportional and the integral gains of the OCL PI controller, respectively; and τvis the desired OCL time constant. These gains guarantee a first-order dy‐ namics of the closed-loop system with τvin a similar way than those in the ICL case. vm vmis ++ + +   * * kp,vs+ki,v s it=it Cs 1 (a) vm vmis + + + +  * * kp,vs+ki,v s OCL PI controller OCL PI controller OCL PI controller it=it Cs+Gv 1 (c) vm vmis + ++ + + +  * * kp,vs+ki,v s it=it Cs Gv 1 (b) it′ Fig. 3. Simplified block diagrams of OCL within a cascade voltage con‐ troller for an LC-coupled VSC simplifying ICL dynamics. (a) OLC PI con‐ troller and plant. (b) OCL PI controller and plant including the negative feedback of the virtual conductance Gv. (c) OCL PI controller and plant in a compact form. OCL ICL PIPI vm vm vm vt itit is Cs +++ + Lts+Rt 11 * * + ++ +   Fig. 2. Simplified block diagram of the classic cascade voltage controller for an LC-coupled VSC on the rotating synchronous frame. 1706 MATAS-DÍAZ et al.: VIRTUAL CONDUCTANCE BASED CASCADE VOLTAGE CONTROLLER FOR VSCS IN ISLANDED OPERATION MODE D. Proposed Control Algorithm with Virtual Conductance The complete control algorithm in dq coordinates includ‐ ing the virtual conductance is shown in Fig. 4. The ICL is in charge of computing the VSC voltage reference vtdq as: vtdq =kpieitdq +kiiξitdq +ωLtdqitqd +vmdq (7) where eitdq and ξitdq are the error and the integral error of the estimated inductor current itdq, respectively. Note that it is proposed to estimate the inductor current itdq, given vmdq and the injected PCC current isdq, following the methodology summarised in Appendix A.1 [8]. This strat‐ egy provides some advantages with respect to directly use the measured inductor current itabc. First, the high-frequency harmonic distortion due to the VSC switching is lower in the injected PCC current due to the LC filter action. This prevents the introduction of a noisy signal in the controller. With this regard, it has to be considered that the effective‐ ness of the synchronized sampling is limited when non-ideal conditions are considered [23], e.g., non-linear evolution of the current within the switching period or the delays intro‐ duced by drivers and insulated gate bipolar transistors (IG‐ BTs). Second, the control algorithm is intended to track the capacitor voltage vmabc to a given voltage reference while the injected PCC current isabc is considered as a system dis‐ turbance. For this reason, the measurement of this current is one of the most efficient ways to reject it. Note that the VSC inductor current is affected by the capacitor dynamics. Therefore, when a rapid change in isabc occurs, e.g., due to a short-circuit fault, the inductor current itabc will experience a delay which may affect the controller performance. The ICL reference current i' tdq is computed by the OCL as: i* tdq =kpvevmdq +kivξvmdq +ωCdqvmqd +isdq (8) i' tdq =i* tdq -Gvvmdq (9) where evmdq and ξvmdq are the error and the integral error of the capacitor voltage vmdq, respectively; and i* tdq is the reference current in the classic cascade control while i' tdq modifies it with the proposed virtual conductance. III. SENSITIVITY ANALYSIS The aim of this section is to evaluate the influence of the introduced virtual conductance in the system dynamics in or‐ der to define its adequate value. First, the plant in which the OCL applies its control action resorting to a transfer func‐ tion analysis is considered. Second, a stability analysis is conducted to guarantee that the computed controller gains and the introduced virtual conductance lead to a stable closed-loop operation. For this purpose, the proposed control algorithm is ap‐ plied to a VSC with the characteristics summarized in Table I. Note that the VSC parameters are expressed in per unit to obtain general conclusions irrespective of the converter rated magnitudes. The analysis includes a wide range of virtual conductances and ICL time constants but assumes τv=10τi. A. Transfer Function Analysis The proposed tuning of the OCL gain detailed in Section II-C neglects the ICL dynamics. The aim of this subsection is to validate this assumption by comparing the frequency domain performance of the complete and simplified plants shown in Fig. 2 and Fig. 3(c), respectively. The correspond‐ ing transfer functions GwICL and GwoICL are: ì í î ï ï ï ï ï ï ï ï ï ï GwICL =kpis+kii s(s2+2δωns+ω2 n)+kiiGv GwoICL =1 Cs +Gv (10) dq dq dq Park transform and angle vm,abc vm,dq vm,dq θ is,abc is,dq is,dq θ θ θ abc abc abc ω s 1 State observer xp=Apxp+Bpup+G(yp yp) xp=it,dq · ~ ~ ~ ~~ yp=Cpxp yp=[vm,dq] up=[vt,dq is,dq]T VSC vdc vdc LtRt it,abc is,abc vm,abc vt,dq vm,dq CPCC ηabc ηdq 2 vmd vmd vmd umd umd itd umq umq isd isq vmq vmq  vmq * * PI PI Outer voltage control loop ′ itq ′= + +  Cω vmq vmd Gv Cross-coupling term cancellation Feedforward Virtual conductance itd itd itd udud vtd uq uq vmd vmq itq itq  itq PI PI Inner current control loop vtq = + + Ltω Cross-coupling term cancellation Feedforward ~ ~ ~ ~ ′ ′ \ \\\\\ ~ ~ Fig. 4. Proposed cascade control with virtual conductance in dq coordinates for grid-forming LC-coupled VSCs. TABLE I VSC AND CONTROLLER PARAMETERS USED IN SENSITIVITY ANALYSIS Parameter Lt Rt C Gv τi τv Value 0.2 p.u. 0.02 p.u. 8´10-6p.u. [ 0.040.160.40.84.08.0 ] p.u. [ 0.250.51.05.010.050.0 ] ms τv=10τi 1707 JOURNAL OF MODERN POWER SYSTEMS AND CLEAN ENERGY, VOL. 10, NO. 6, November 2022 ì í î ï ï ï ï ï ï ï ï ï ï ï ï ωn=1+kpiGv+kiiC LtC δ=Rt+kpi 2Ltωn (11) where ωnand δare the resonance frequency and damping factor of GwICL, respectively. The representation of the transfer functions in the frequen‐ cy domain is shown in Fig. 5, where τi=0.25 ms and the ex‐ treme values of the virtual conductance (0.04 p.u. and 8 p.u.) have been used. Note that the control strategy is implement‐ ed in dq coordinates where time-invariant setpoints are DC magnitudes. With this regard, the magnitude and phase of both transfer functions coincide in the low-frequency range (lower than 10 rad/s) for any Gv. This means that the simpli‐ fication of the dynamics used in the tuning of OCL gain is valid, as the dynamic performance of both plants is identical for low frequencies close to DC. The analysis of the highfrequency range reveals the magnitude drops of 20 dB and 40 dB per decade corresponding to the firstand second-or‐ der dynamics of GwICL and GwoICL, respectively. The influence of the virtual conductance value is also evi‐ dent when a comparison of the transfer functions is carried out. Small values of virtual conductance lead to higher mag‐ nitudes in the low-frequency range but with a small drop be‐ low the resonance frequency. This means that the controller will be able to reach zero steady-state errors but some devia‐ tions with respect to the first-order dynamics for frequencies below the resonance frequency are expected. On the con‐ trary, large values of virtual conductance lead to extremely low magnitudes. Therefore, the controller is expected to have large steady-state errors. The increase of the OCL gains could solve this situation but at the cost of reducing τv which may approach τi. Therefore, interactions between ICL and OCL may appear. Furthermore, a high value of Gvin‐ creases the resonance frequency of the system and reduces the damping factor as shown in (11) and Fig. 5. This situa‐ tion must be avoided in order to push the resonance frequen‐ cy far from the switching frequency of the VSC. B. Stability Analysis The aim of this subsection is to evaluate the impact of the introduced virtual conductance and the controller time con‐ stants on the closed-loop system operation. For this purpose, the state-space equations of the system dynamics given by (3) and (4) and the controller formulated in (7) and (8) are derived in the form as: x =Ax +Bu (12) where x=[itdq vmdq ξitdq ξvmdq ]Tis the vector of state vari‐ ables; u=[v* mdq isdq ]Tcorresponds to the system inputs; and matrices Aand Bare detailed in Appendix A.2. First, the influence of the virtual conductance for τi= 0.25 ms is evaluated. The representation of the eigenvalues as a function of the virtual conductance is shown in Fig. 6(a). The system is always stable and the low-frequency eigenval‐ ues maintain a damping ratio equal to 100% for any value of virtual conductance. Conversely, high-frequency eigenval‐ ues reduce their damping ratio and increase their natural fre‐ quency as the virtual conductance increases, which is consis‐ tent with (11). Therefore, large virtual conductance values must be avoided to reduce the amplification of low-order and switching harmonics of the VSC which may reduce the system performance. With this regard, it is considered that acceptable virtual conductance values are those assuring a damping ratio greater than 10% which corresponds to values lower than 0.4 p.u.. 50 0 -50 -100 -150 Magnitude (dB) 50 0 -50 -100 -200 -150 Phase (°) 10-3 10-1 101103105107109 Frequency (rad/s) GwoICL (Gv=0.04) GwoICL (Gv=8) GwICL (Gv=0.04) GwICL (Gv=8) GwoICL (Gv=0.04) GwoICL (Gv=8) GwICL (Gv=0.04) GwICL (Gv=8) Fig. 5. Transfer function analysis of OCL acting on simplified and com‐ plete plants with different values of Gv. Gv Gv τi τi 0 0 -25000 25000 50000 75000 -50000 -75000 -500-1000-1500-2000 Imaginary axis 0 -5000 5000 10000 -10000 Imaginary axis δ=3% fn=10 kHz δ=40% fn=2.35 kHz δ=10% fn=2.36 kHz δ=20% fn=2.66 kHz δ=100% fn=15.9 Hz δ=100% fn=5 Hz δ=100% fn=63 Hz δ=100% fn=16 Hz Real axis (a) 0-500-1000-1500-2000 Real axis (b) Fig. 6. Stability analysis: eigenvalues of system plant and the proposed controller. (a) Influence of Gvwhen τi=0.25 ms. (b) Influence of τiwhen Gv=0.4 p.u.. 1708 MATAS-DÍAZ et al.: VIRTUAL CONDUCTANCE BASED CASCADE VOLTAGE CONTROLLER FOR VSCS IN ISLANDED OPERATION MODE The influence of the ICL time constant has also been eval‐ uated in the stability analysis. In this case, the virtual con‐ ductance is set to be 0.4 p.u., while the ICL time constant is modified according to the values shown in Table I. The repre‐ sentation of the eigenvalues in this case is shown in Fig. 6(b). On the one hand, the low-frequency eigenvalues maintain a damping ratio close to 100% and reduce the value of their natural frequency as the value of the ICL time constant in‐ creases. On the other hand, the high-frequency eigenvalues reduce the value of their natural frequency and their damp‐ ing ratio as the ICL time constant increases. Despite the fact that the system stability is guaranteed for any value of the ICL time constant, it is observed that the higher this value, the closer the eigenvalues to the imaginary axis, i.e., critical stable system. With this regard, and as in the case of the dis‐ cussion on the virtual conductance, it is considered that ac‐ ceptable values of the ICL time constant are those leading to a damping ratio greater than 10%, which corresponds to val‐ ues lower than 5 ms. IV. EXPERIMENTAL VALIDATION A. Description of Testbed The experimental setup used to validate the proposed con‐ trol algorithm is shown in Fig. 7. This testbed consists of a three-phase three-wire VSC with a DC voltage source con‐ nected at its DC side which provides the energy to the sys‐ tem. The VSC at AC side is connected to the PCC through an LC filter where different loads have been connected to evaluate the performance of the control algorithm. The rele‐ vant parameters of the hardware components and the pro‐ posed voltage control strategy are summarized in Tables II and III, respectively. The three-phase capacitor voltage vmabc and the injected PCC current isabc are measured using transducers from LEM, LV-25P, and HAS-50S, respectively. The control algorithm shown in Fig. 4 has been implemented on a TMS320F28335 Delfino DSP from Texas Instruments with a sampling fre‐ quency of 20 kHz. B. Experimental Results The effectiveness of the proposed controller is evaluated experimentally by using the laboratory setup and control pa‐ rameters defined in the previous subsection. For this pur‐ pose, different loads have been connected to the VSC to as‐ sess its performance both in steady-state and transient re‐ gimes. In both cases, two load conditions have been tested: a three-phase delta resistive balanced load of 42 Ωper phase; no-load. The state observer detailed in Appendix A.1 Section A has been implemented for estimating the inductor current from the injected PCC current. This is an alternative to the synchronous sampling of the inductor current, which prevents the errors due to the non-linear evolution of the cur‐ rent within the switching period and the delays introduced by drivers and IGBTs [23]. Finally, it is important to point out that the second test is quite challenging because no damping of the LC-filter resonance is introduced. The results obtained in the steady-state test are shown in Fig. 8, where the voltage references are set as v* md =0V, v* mq = -330 V. It can be observed that an almost perfect sinusoidal waveform is achieved for both load conditions. However, a light superimposed band appears which corresponds to the high-frequency harmonics generated by the VSC switching. In any case, the obtained power quality is excellent because the total harmonic distortion (THD) of the voltage is 1.40% TABLE II HARDWARE COMPONENTS OF EXPERIMENTAL TESTBED Parameter DC bus voltage VSC rated voltage at AC side VSC rated current at AC side Rated frequency Switching frequency Filter inductor Filter inductor quality factor Filter capacitor LC-filter resonant frequency Value 730 V 400 V 40 A 50 Hz 10 kHz 5 mH 100 1μF 2.25 kHz HAS-50S LV-25P Microcontroller VSC Capacitor Inductor Fig. 7. Laboratory experimental testbed. TABLE III PARAMETERS OF PROPOSED CONTROL STRATEGY Parameter τi kpi kii τv kpv kiv Gv Value 0.25 ms 20 V/A 62 V/A 2.5 ms 4´10-4A/V 4.5 A/V 0.02 S 1709 JOURNAL OF MODERN POWER SYSTEMS AND CLEAN ENERGY, VOL. 10, NO. 6, November 2022 and 0.91% in the load and no-load tests, respectively. Regarding the transient performance of the voltage control‐ ler, a step change of the voltage references has been tested. The initial voltage setpoint is modified from v* mq =0 V to v* mq = -330 V with the result shown in Fig. 9. For comparison purposes, it has been included Fig. 9 an exponential function with τv. It is worth noting that the volt‐ ages evolve following a controlled first-order dynamics as a result of the virtual conductance introduced in the OCL. With this regard, it is interesting to analyze the dynamic per‐ formance of the PCC voltages in the dq coordinates as shown in Fig. 10. The analysis of this figure evidences two important characteristics of the proposed controller. First, and regarding the qcomponent, it can be observed that the voltages follow the desired first-order dynamics with τv. Sec‐ ond, a good uncoupling of the dand qcomponents is achieved as the dcomponent is almost constant with the ref‐ erence change of the qcomponent. C. Performance with Large Perturbation Finally, the controller is tested in case of a three-phase fault at the PCC in order to evaluate its performance under a large disturbance. This test has been performed using a hard‐ ware-in-the-loop (HIL) testing approach on the Typhoon HIL 402-01-005 platform. Therefore, the control algorithm can be safely tested in the microcontroller without jeopardizing the VSC. For this purpose, a saturation block between the OCL and the ICL is integrated in the proposed control algorithm shown in Fig. 4. A back-calculation anti-wind-up method has Voltage (100 V/div) Current (5 A/div) t (5 ms/div) (a) Voltage (100 V/div) Current (5 A/div) t (5 ms/div) (b) Phase a; Phase b; Phase c Fig. 8. Steady-state results with v* md = 0 V and v* mq =-330 V. (a) Threephase delta resistive balanced load of 42 Ωper phase. (b) No-load operation. 100 0 -100 -200 -400 -300 Voltage (V) vmd vmd vmq vmq 20 30 40 50 60 Time (ms) (a) 70 80 90 100 100 0 -100 -200 -400 -300 Voltage (V) 20 30 40 50 60 Time (ms) (b) 70 80 90 100 * * vmd vmd vmq vmq * * Fig. 10. Voltages in dq coordinates for transient behaviour. (a) Threephase balanced delta resistive load of 42 Ωper phase. (b) No-load operation. Voltage (100 V/div) Current (5 A/div) t (5 ms/div) (a) Voltage (100 V/div) Current (5 A/div) t (5 ms/div) (b) 330(e t/τv+1) 330(e t/τv+1) Phase a; Phase b; Phase c Fig. 9. Transient behaviour when v* md = 0 V and v* mq = 0 V→ -330 V. (a) Three-phase delta resistive balanced load of 42 Ωper phase. (b) No-load op‐ eration. 1710 MATAS-DÍAZ et al.: VIRTUAL CONDUCTANCE BASED CASCADE VOLTAGE CONTROLLER FOR VSCS IN ISLANDED OPERATION MODE been implemented to prevent the integration wind-up when the control action is saturated [24]. The current limits for each dand qcomponents is set to be ±20 A in order to pro‐ tect the VSC against overcurrents. In any case, these values can be modified depending on the VSC rated power or the corresponding protection criteria. Figure 11 represents the magnitudes vmabc,isabc, and itabc during this short-circuit fault test. The fault starts at t= 0.1 s and ends at t= 0.22 s. During the fault, the voltage is practi‐ cally zero but once the fault is cleared, the voltage recovers its previous reference with a slight overvoltage during a short transient period. With respect to the current, it is impor‐ tant to point out that itabc is maintained within the imposed limits without any unwanted overcurrent which could lead to a VSC damage. Note that the injected PCC current isabc pres‐ ents a spike at the beginning of the short-circuit fault, which is due to the discharge of the LC-filter capacitor. After the fault, both currents return to their initial state smoothly. Therefore, it can be stated that the delay introduced by the state observer does not significantly affect the controller per‐ formance even in the case of a large perturbation. V. CONCLUSION This paper has presented a modification of the classic cas‐ cade voltage control which can be used as a part of the con‐ trol algorithm of grid-forming VSCs with LC coupling filter operating in islanded mode. The proposed control algorithm allows to tune the gains of the OCL and ICL independently and easily. This has been achieved by introducing a virtual conductance in parallel with the coupling filter capacitor which turns the OCL plant as a first-order system if the ICL dynamics is neglected. It is worth noting that the simplified OCL plant is the dual circuit of the ICL one. As a conse‐ quence, both control loops can be designed in the same way following a straightforward strategy that, in addition, assures a controlled closed-loop dynamics. This paper has introduced a sensitivity analysis to evi‐ dence the influence of two key parameters required to prop‐ erly adjust the controller gains: the virtual conductance and the ICL time constant. For this purpose, frequency domain and stability analyses have been carried out. These analyses have been formulated following a per unit approach to ob‐ tain general conclusions independently of the rated values of the VSC. The main outcome that can be derived from the frequency domain analysis is that small virtual conductance values are preferred because of the large controller gains ob‐ tained in the low-frequency range where the controller is in‐ tended for. The stability analysis brings the same conclusion because small virtual conductance values lead to higher damping ratios and reduced resonance frequencies. The proposal has been experimentally validated in a labo‐ ratory testbed where steady-state and transient regimes have been evaluated under load and no-load conditions. The re‐ sults show that the proposed control algorithm obtains zero steady-state error and very low THD values with and with‐ out load. Regarding the transient results, the voltage follows a first-order dynamics with the time constant used in the def‐ inition of the controller gains. In addition to these experi‐ mental tests, an HIL testing approach has been applied in or‐ der to evidence that the delay introduced by the state observ‐ er of the inductor current does not significantly affect the controller dynamics even in the case of large perturbations. Future research lines will incorporate to this cascade volt‐ age controller new control layers dealing with key functional‐ ities of grid-former VSCs for islanded systems like power sharing as well as unbalance and harmonic mitigations. In addition, it is expected to adapt the formulation to grid-form‐ ing VSCs operating in grid-connected mode. APPENDIX A A. State Observer for Inductor Current The purpose of the state observer is to estimate the induc‐ tor current itdq from the available measurements isdq and vmdq. The plant dynamics given by (3) and (4) is formulated in a compact form as: x p=Apxp+Bpup(A1) yp=Cpxp(A2) where the subscript pis used to indicate the plant, xp= [vmdq itdq ]Tis the vector of state variables; yp=vT mdq is the system output vector; and up=[vtdq isdq ]Tis the vector of system inputs. By applying the Luenberger state observer definition over (A1) and (A2), (A3) and (A4) can be obtained [25]: x p=Apx p+Bpup+G(yp-y p)(A3) y p=Cpx p(A4) where x pis the estimated vector state; y pis the estimated output; yp-y pis the estimation error; and Gis the additional term called weighting matrix. This matrix, which defines the observer performance, can be computed by solving an LQR problem with the constraints imposed by (A1) and (A2). In this way, it is assured that the estimation error converges to zero. From a practical point of view, the LQR problem to compute Ghas been solved by using the MATLAB function lqrd.m. Once Gis defined, it is possible to calculate the esti‐ 1000 500 0 -500 200 0 -200 -1000 50 25 0 -25 -50 Voltage (V) Grid current (A) VSC current (A) 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Time (s) Phase a; Phase b; Phase c Fig. 11. HIL simulation of three-phase fault at PCC. 1711 JOURNAL OF MODERN POWER SYSTEMS AND CLEAN ENERGY, VOL. 10, NO. 6, November 2022 mated state vector x pusing (A3) and (A4). It is worth noting that the use of the current itdq instead of the measured cur‐ rent itdq improves the performance of the controller since the state observer also acts as a low-pass filter which eliminates the high-frequency harmonic content of the inductor current itdq [8]. Matrices used in the state observer are defined as: Ap= é ë ê ê ê ê ê ê ê ê ê êù û ú ú ú ú ú ú ú ú ú ú 0ω1C0 -ω0 0 1 C -1/Lt0-Rt/Ltω 0-1/Lt-ω-Rt/Lt (A5) Bp= é ë ê ê ê ê ê ê ê êù û ú ú ú ú ú ú ú ú 0 0 -1/C0 0 0 0 -1/C 1/Lt0 0 0 0 1/Lt0 0 (A6) Cp=é ë ê êê êù û ú úú ú 1 0 0 0 0 1 0 0 (A7) B. Stability Analysis Matrices Matrices Aand Bof the stability analysis outlined in Sec‐ tion III-B are presented as: A= é ë ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê êù û ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú a0b c d 0e0 0a-c b 0d0e 1C0 0 ω0 0 0 0 0 1 C-ω0 0 0 0 0 -1 0 f Cω0 0 kiv0 0-1Cωf0 0 0 kiv 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 (A8) B= é ë ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê êù û ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú kpvkpi Lt 0kpi Lt 0 0kpvkpi Lt 0kpi Lt 0 0 -1/C0 0 0 0 -1/C kpv0 1 0 0kpv0 1 1 0 0 0 0 1 0 0 (A9) where terms a,b,c,d,e, and fare defined as: a= -(kpi+Rt) Lt ;b= -kpi(Gv+kpv) Lt ;c= -Ckpiω Lt ;d=kii Lt ;e= kivkpi Lt ; and f= -(Gv+kpv). REFERENCES [1] J. M. Maza-Ortega, E. Acha, S. García et al., “Overview of power electronics technology and applications in power generation transmis‐ sion and distribution,”Journal of Modern Power Systems and Clean Energy, vol. 5, no. 4, pp. 499-514, Jul. 2017. [2] S. H. Ko, S. R. Lee, H. Dehbonei et al., “Application of voltageand current-controlled voltage source inverters for distributed generation systems,”IEEE Transactions on Energy Conversion, vol. 21, no. 3, pp. 782-792, Sept. 2006. [3] J. T. Bialasiewicz, “Renewable energy systems with photovoltaic pow‐ er generators: operation and modeling,”IEEE Transactions on Indus‐ trial Electronics, vol. 55, no. 7, pp. 2752-2758, Jun. 2008. [4] T. Qoria, T. Prevost, G. Denis et al., “Power converters classification and characterization in power transmission systems,”in Proceedings of 2019 21st European Conference on Power Electronics and Applica‐ tions, Genova, Italy, Sept. 2019, pp. 1-6. [5] J. Rocabert, A. Luna, F. Blaabjerg et al., “Control of power converters in AC microgrids,”IEEE Transactions on Power Electronics, vol. 27, no. 11, pp. 4734-4749, Nov. 2012. [6] P. Rodriguez, C. Citro, J. I. Candela et al., “Flexible grid connection and islanding of SPC-based PV power converters,”IEEE Transactions on Industry Applications, vol. 54, no. 3, pp. 2690-2702, May 2018. [7] G. N. Baltas, N. B. Lai, L. Marin et al., “Grid-forming power convert‐ ers tuned through artificial intelligence to damp subsynchronous inter‐ actions in electrical grids,”IEEE Access, vol. 8, pp. 93369-93379, May 2020. [8] J. C. Olives-Camps, J. M. Mauricio, M. Barragan-Villarejo et al., “Voltage control of four-leg VSC for power system applications with nonlinear and unbalanced loads,”IEEE Transactions on Energy Con‐ version, vol. 35, no. 2, pp. 640-50, Jun. 2020. [9] R. Perez-Ibacache, G. A. Pimentel, C. A. Silva et al., “State feedback control with full disturbance compensation for an LC filtered gridforming converter,”in Proceedings of 2019 IEEE PES Innovative Smart Grid Technologies Conference –Latin America (ISGT Latin America), Gramado, Brazil, Sept. 2019, pp. 1-6. [10] T. Qoria, C. Li, K. Oue et al., “Direct AC voltage control for gridforming inverters,”Journal of Power Electronics, vol. 20, no. 1, pp. 198-211, Dec. 2019. [11] P. C. Loh and D. G. Holmes, “Analysis of multiloop control strategies for LC/CL/LCL-filtered voltage-source and current-source inverters,” IEEE Transactions on Industry Applications, vol. 41, no. 2, pp. 644654, Mar. 2005. [12] Q. Lei, F. Peng, and S. Yang, “Multiloop control method for high-per‐ formance microgrid inverter through load voltage and current decou‐ pling with only output voltage feedback,”IEEE Transactions on Pow‐ er Electronics, vol. 26, no. 3, pp. 953-960, Dec. 2011. [13] Z. Li, C. Zang, P. Zeng et al., “Control of a grid-forming inverter based on sliding-mode and mixed H2/H¥control,”IEEE Transactions on Industrial Electronics, vol. 64, no. 5, pp. 3862-3872, May 2017. [14] T. Qoria, F. Gruson, F. Colas et al., “Tuning of cascaded controllers for robust grid-forming voltage source converter,”in Proceedings of 20th Power Systems Computation Conference, Dublin, Ireland, Aug. 2018, pp. 1-7. [15] Z. Li, C. Zang, P. Zeng et al., “Analysis of multi-agent-based adaptive droop-controlled AC microgrids with PSCAD: modeling and simula‐ tion,”Journal of Power Electronics, vol. 15, no. 2, pp. 455-468, Mar. 2015. [16] T. Qoria, F. Gruson, F. Colas et al., “Critical clearing time determina‐ tion and enhancement of grid-forming converters embedding virtual impedance as current limitation algorithm,”IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 8, no. 2, pp. 10501061, Dec. 2019. [17] D. Pan, X. Wang, F. Liu et al., “Transient stability of voltage-source converters with grid-forming control: design-oriented study,”IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 8, no. 2, pp. 1019-1033, Jun. 2020. [18] Y. Y. Tzou, “DSP-based fully digital control of a PWM DC-AC con‐ verter for AC voltage regulation,”in Proceedings of PESC Record – IEEE Annual Power Electronics Specialists Conference, Atlanta, USA, Jun. 1995, pp. 138-144. [19] V. Yaramasu, M. Rivera, M. Narimani et al., “Model predictive ap‐ proach for a simple and effective load voltage control of four-leg in‐ verter with an output LC filter,”IEEE Transactions on Industrial Elec‐ tronics, vol. 61, no. 10, pp. 5259-5270, Oct. 2014. [20] J. W. Jung, V. Q. Leu, D. Dang et al., “Intelligent voltage control strat‐ egy for three-phase UPS inverters with output LC filter,”International Journal of Electronics, vol. 102, no. 8, pp. 1267-1288, Aug. 2015. [21] Y. Li, Y. Gu, Y. Zhu et al., “Impedance circuit model of grid-forming inverter: visualizing control algorithms as circuit elements,”IEEE Transactions on Power Electronics, vol. 36, no. 3, pp. 3377-3395, Mar. 2021. [22] A. Yazdani and R. Iravani, Voltage-sourced Converters in Power Sys‐ tems: Modeling, Control, and Applications. New York: John Wiley, Jan. 2010. [23] F. Briz, D. Díaz-Reigosa, M. W. Degner et al., “Current sampling and measurement in PWM operated AC drives and power converters,”in 1712