Centralized Supplementary Controller to Stabilize an Islanded AC Microgrid ESNRajuP Research Scholar, Electrical Engineering IIT Indore Indore, India Email:pesnraju88@gmail.com Trapti Jain Assistant Professor, Electrical Engineering IIT Indore Indore, India Email:traptij@iiti.ac.in Abstract This paper presents a centralized supplementary controller to stabilize the dynamics caused by an active and passive loads, connected to an islanded AC microgrid with low inertial inverter interfaced DGs. The centralized supplementary controller has been designed on the basis of a linear quadratic regulator with prescribed degree of stability (LQRPDS). Eigenvalue based method has been utilized to analyze the stability of an islanded AC microgrid comprising of four low inertial inverter interfaced DGs having unequal power ratings and supplying active as well as passive loads. To explore this, a full dynamical linearized small signal state space model of the islanded AC microgrid represented in synchronous (DQ) reference frame has been considered. The effectiveness of the proposed controller has been verified by comparing the results with a centralized supplementary controller based on linear quadratic regulator (LQR). Index Terms Centralized supplementary controller, eigenvalue analysis, inverter interfaced DGs based islanded AC microgrid, linear quadratic regulator with prescribed degree of stability, small signal stability analysis. I. INTRODUCTION The concept of microgrids has been introduced to integrate renewable energy sources with the utility grid so that the ever increasing energy demand can be met reliably [1]. Microgrids have the ability to operate either in islanded or grid connected modes of operation. Stability of an AC microgrid (ACMG), containing low inertial converter interfaced DGs such as solar photovoltaic panels and fuel cells, becomes an important concern especially under islanded mode of operation. For reliable and satisfactory operation of an islanded AC microgrid, it should be small signal stable [2]. Small signal instability in an islanded ACMG may occur due to various factors such as feedback controllers, lack of system damping, power limit of DGs and continuous load switching [3], [4] etc. The stability of an islanded ACMG can be improved by providing supplementary control loops, coordinated control of the micro sources, stabilizers and energy management system [3], [4]. Centralized controller with communication link and without communication link has been proposed in the literature [5 7]. Communication link based centralized controller [5] can be adopted in an islanded ACMG system if the converter interfaced DGs are located 978-1-4799-5141-3/14/$31.00 c 2014 IEEE quite close to each other. However, in a practical islanded ACMG system, converter interfaced DGs are located far away to each other requiring high bandwidth links for dynamic sharing of signals. This leads to high cost and less reliable operation of the islanded ACMG system. These limitations have been overcome by using centralized controller without communication link. In this paper, the later method is adopted. Microgrid operation was optimized using microgrid central controller in [6]. A centralized control system that coordinates parallel operation of multiple inverters in a microgrid for both the grid-connected and the islanded operations was proposed in [7]. A central controller was developed based on model predictive control (MPC) algorithm that decomposes the control problem into steady-state and transient sub-problems to reduce the overall time of computation. The stability improvement of microgrids was obtained under different operating conditions by coordinated operation of different DGs [6], [7]. However, instability may be caused in the microgrids due to the dynamics of active and passive loads. Under this condition, a coordinated centralized control of DGs may not be able to stabilize the microgrid system and a centralized supplementary controller may be required to stabilize the system. In this paper, a centralized supplementary controller has been proposed to stabilize an islanded ACMG consisting of four inverter interfaced DGs of unequal power ratings. Passive loads viz. resistive load (R load), impedance load (RL load), constant power load (CPL) and active load viz. rectifier interfaced load active load (RIAL) are assumed to be connected at different nodes as shown in Fig. 1. The network dynamics and load dynamics, which may impact the small signal stability of the islanded ACMG due to small time constant of inverter interfaced DGs, have also been considered. To analyze the stability of the islanded ACMG, a full dynamical linearized small signal state space model of the islanded AC microgrid represented in synchronous (DQ) reference frame has been utilized. The centralized supplementary controller has been designed on the basis of a linear quadratic regulator with prescribed degree of stability (LQRPDS). The proposed controller has been evaluated by comparing the results with a centralized supplementary controller based on linear quadratic regulator (LQR).
Figure 1: Islanded AC Microgrid Figure 2: Photovoltaic (PV) DG Model Figure 3: Block Diagram of Complete Small-Signal Linearized State-Space Model II. SMALL SIGNAL LINEARIZED STATE SPACE MODEL OF AN ISLANDED AC MICROGRID Small signal linearized state space model of the islanded AC microgrid (ACMG) can be obtained from the generalized modelling of the microgrid given in [2]. Inverter interfaced photovoltaic (PV) DG contains PV panel, PV side capacitive energy storage, DC/DC boost converter, DC side capacitive energy storage and three-leg voltage source inverter (DC/AC converter), as shown in Fig. 2. AC side dynamics viz. dynamics of inverters, lines and loads are considered in full dynamical model [2]. DC source dynamics such as dynamics of photovoltaic panel, photovoltaic capacitive energy storage and DC/DC buck converter are neglected for simplified analysis. The systematic modelling approach [2], which divides the whole ACMG system into three sub-modules such as inverter sub-module, network sub-module and load submodule, as shown in Fig. 3, is adapted to model the islanded AC microgrid. A. Inverter Sub-module Inverter sub-module involves the modelling of each of four three-leg voltage source inverters (VSI) on its individual local reference frame (dq). Each VSI model contains power sharing controller, voltage controller, current controller, output LC filter and coupling inductor [2], [8 11]. The first inverter reference frame is selected as common reference frame (DQ) and all other inverters are translated to this common reference with the help of equation given in (1) ( ) ( [ ]= ) [ ( ) ( ) ] (1) Small signal linearized state space model of an individual VSI unit in common reference frame can be written as given below. 0 [ ] 13 1 =[ ] 13 13 [ ] 13 1 +[ ] 13 2 [ ] 2 1 +[ ] 13 1 [ ] 1 1 3 1 = [ ] 1 13 [ ] 2 13 3 13 [ ] 13 1 where, = 0 0 where, represents a state matrix of ith inverter comprising of 13 states, is an input matrix of inverter corresponding to the two voltage inputs, is an input matrix of inverter corresponding to the frequency input, and are output matrices of inverter corresponding to the two output currents 0 and frequency, respectively [2], [8 11]. Now, the complete small signal linearized state space model of the inverter sub-module can be obtained from (2) and (3) by combining all the states of the four inverters and can be written in common reference frame as given in (4), (5) and (6) [ ] 52 1 =[ ] 52 52 [ ] 52 1 +[ ] 52 8 [ ] 8 1 (4) +[ ] 52 1 [ ] 1 1 [ 0 ] 8 1 =[ ] 8 52 [ ] 52 1 [ ] 1 1 =[ ] 1 52 [ ] 52 1 (2) (3) (5) (6)
where, represents state matrix of the inverter submodule comprising of 52 states, is an input matrix of the inverter sub-module corresponding to the eight voltage inputs, is an input matrix of the inverter sub-module corresponding to the frequency input, and are the output matrices of the inverter sub-module corresponding to the 8 output currents [ 0 1 0 2 0 3 0 4 ] and frequency, respectively, [11]. B. Network Sub-module Small signal linearized state space model of a network submodule, [2], [8 11], comprising of three lines can be given in common reference frame as shown in (7) [ ] 6 1 =[ ] 6 6 [ ] 6 1 +[ 1 ] 6 8 [ ] 8 1 +[ 2 ] 6 1 [ ] 1 1 where,, state matrix of the network submodule, 1, an input matrix of the network sub-module corresponding to the eight voltage inputs [ 1 2 3 4 ] and, 2 an input matrix of the network sub-module corresponding to frequency input, [11]. C. Load Sub-module Complete small signal linearized state space model of a load sub-module can be obtained by considering passive loads in one sub-module and an active load in another sub-module. 1) Passive Load Submodule: Resistive load can also be modelled as an type of load with an assumption that the value of inductance is very small. Small signal linearized state space model of load or type load [2], may be written in common reference frame as given in (8). [ ] 2 1 =[ ] 2 2 [ ] 2 1 +[ 1 ] 2 2 [ ] 2 1 +[ 2 ] 2 1 [ ] 1 1 where,, state matrix of an load, 1,an input matrix of load corresponding to the two voltage inputs [ ] and, 2 an input matrix of load corresponding to the frequency input, [11]. Small signal linearized state space model of the constant power load ( ) in common reference frame [9], can be produced by replacing and by and, respectively in (8). Here, = and =, where is the delay angle of a controlled rectifier. Now, the complete small signal linearized state space model of the passive load sub-module, [2], [8 11], comprising of three loads, may be written in common reference frame as given in (9). [ ] 6 1 =[ ] 6 6 [ ] 6 1 +[ 1 ] 6 8 [ ] 8 1 +[ 2 ] 6 1 [ ] 1 1 (7) (8) (9) 2) Rectifier Interfaced Active Load ( ): Small signal linearized state space model of an RIAL with 10 states, 6 inputs and 3 outputs in common reference frame [9], is given in (10), (11) and (12) where, and, [ ] 10 1 =[ ] 10 10 [ ] 10 1 +[ 1 ] 10 2 [ ] 2 1 +[ 2 ] 10 3 [ ] 3 1 +[ ] 10 1 [ ] 1 1 [ ] 2 1 =[ ] 2 10 [ ] 10 1 [ ] 1 1 =[ ] 1 10 [ ] 10 1 = = (10) (11) (12) where,, state matrix of an load, 1,an input matrix of load corresponding to the two voltage inputs, 2, an input matrix of load corresponding to the three control inputs,, an input matrix of load corresponding to the frequency input,, an output matrix of the two states and,aretaken from Reference [9]. D. Mapping of Sub-modules Complete small signal linearized state space model of an islanded ACMG system can be obtained by mapping of the sub-modules through the mapping matrices, [2], [11], which connect the output current to a node. Generally, all the components in the microgrid may either import power or current from the node or export the power or current to the node. It has been assumed that a minus sign represents current flowing away from the node and a plus sign represents current flowing towards the node. All the components are assumed with the two states namely D and Q while forming the mapping matrix. Mapping matrix of the inverter sub-module maps the inverter connection points onto the network nodes. Mapping matrix of the inverter sub-module is represented by of order 8 8. Mapping matrix of the network sub-module maps the connecting lines onto the nodes and is represented by of order 8 6. Mapping matrix of the load submodule ( ) maps the connected loads onto the nodes. Mapping matrix of the passive load sub-module and the active load sub-module is of the order of 8 6 and 8 2, respectively. The elements of, and are defined in [2], [11]. To analyse the small signal stability and the response of various components of the system, a disturbance has to be injected into the system. In an islanded ACMG, the most apparent disturbance is a load change. In this case, small change in the load current is considered as a disturbance.
Mapping matrix of the load disturbance of order 8 2, maps the load disturbance onto the node. Now, the complete small signal linearized state space model of the islanded ACMG can be given in common reference frame as. [ ] 74 1 =[ ] 74 74 [ ] 74 1 +[ ] 74 4 [ ] 4 1 (13) [ ] 26 1 =[ ] 26 74 [ ] 74 1 +[ ] 26 4 [ ] 4 1 (14) where,, state matrix of the islanded ACMG,, an input matrix of the islanded ACMG corresponding to the four inputs,, an output matrix and transfer matrix are obtained from the generalized modelling discussed in [2], [8]-[11]. Here, = and = III. CENTRALIZED SUPPLEMENTARY CONTROLLER DESIGN A centralized supplementary controller has been designed on the basis of LQR with prescribed degree of stability (LQRPDS), [12], to stabilize an islanded AC microgrid, as shown in Fig. 4. The state space equation of the islanded ACMG system can be given as in (15). = + (15a) = + (15b) With feedback control law, ½ = for the ; for the leads to (16) and (17) for the and, respectively. =[ ] (16) =[ ] (17) where, and are the controller gain matrices of order 4 74 as given in (18) and (19), respectively. = 1 = 1 (18) (19) where, and are obtained by solving the algebraic riccati equation as given in (20) and (21), respectively. + 1 + =0 (20) Figure 4: Block Diagram of the Centralized Supplementary Controller ( + 74 ) + ( + 74 ) 1 + =0 (21) The controller gain matrices have been obtained such that the closed loop eigenvalues of an islanded ACMG are shifted to the desired left of s-plane by minimizing the performance indices (PI) as given in (22) and (23), respectively. = = Z [( 0 Z 0 ) +( )] [( ) +( )] (22) (23) where, is a positive semidefinite diagonal weighting matrix of order 74 74, is a positive definite diagonal weighting matrix of order 4 4 and is a prescribed degree of stability, which can be selected according to the operating condition of the islanded ACMG. The diagonal elements of the are selected such that the state variables causing instability are given higher priority to shift towards the stable region as given below. ½ 100 ( ) = if the ( ( )) 0; 0 if the ( ( )) 0 The diagonal elements of corresponding to the controllers variables are set to give an equal importance to all of them. IV. SIMULATION RESULTS The simulation results have been performed on a 230 V (per phase RMS), 50Hz islanded AC microgrid (ACMG) consisting of four inverter interfaced DGs having unequal power ratings. These DGs are interconnected through three lines and are considered to be far away from each other. R load, RL load, CPL load and RIAL are connected at bus 4, bus 1, bus 2 and bus 3, respectively, as shown in Fig.
Table I: Steady State Initial Operating Conditions Figure 5: Eigenvalues of the Islanded AC Microgrid without the Centralized supplementary Controller Parameter Initial Conditions for 4 Inverter DGs and RIAL Value 4 Inverter DGs RIAL Internal Parameters 0 [398.5 398.9 398.6 398.6] [397.6] 0 [-0.5351-0.5375-0.5821-0.4976] [-9.581] 0 [2.616 6.478 21.76 46.21] [29.83] 0 [-0.2029 18.28-3.066-7.89] [0.2289] [2.369 6.232 21.52 45.97] [30.66] [6.035 24.53 3.099-1.679] [-1.076] External Parameters [398.4 400.8 398.1 396.4] [397.5] [-0.825-1.807-2.756-5.356] [-0.2641] 0 [50 50 49.99 49.99] [49.47] 0 [0 0.001-0.04-0.058] [-0.073] Load Steady State Parameters (3 Loads in order R, RL and CPL) [62.21 9.34-24.93] [-4.675-6.785 16.45] Line Steady State Parameters (LR Type Line) [-6.725 24.71 16.57] [6.575 8.394 6.049] Figure 6: Eigenvalues of the islanded ACMG with the LQR based Centralized supplementary Controller 1. These four inverter based DGs are controlled to share the active and reactive powers with droop control method. The active droop gains have been obtained considering the maximum and minimum allowable frequencies to be 50.5 Hz and 49.5 Hz, respectively. Whereas, the reactive droop gains have been obtained assuming the minimum and maximum allowable voltages to be 225 V and 235 V, respectively. Test parameters of the rectifier interfaced active load and inverter interfaced DGs, are assumed to be same with the realization that inverter ratings are closer to each other, and are given in Appendix. The initial operating conditions have been obtained by performing time step simulations in MAT- LAB/SIMULINKandareshowninTableI. Small signal stability of the islanded ACMG has been analysed using eigenvalue based method. The eigenvalues obtained from the state matrix of the islanded ACMG,, are shown in Fig. 5. Out of 74 eigenvalues, 52 eigenvalues correspond to inverter sub-module, 6 eigenvalues belong to network sub-module, 6 eigenvalues stand for load sub-module and 10 eigenvalues represent rectifier interfaced active load. These 74 modes can be categorized into group of low damping modes, medium damping modes and high damping modes. It can be seen from Fig. 5 that the six eigenvalues corresponding to 65 and 66, 69 and 70, and 73 and 74 state variables lie in Table II: Comparison of the Damping Factors of modes obtained without and with the Centralized Supplementary Controller S.No State Variables Number Eigenvalue Damping Factor Frequency of Oscillation (rad/sec) Without Centralized Controller 1 65 and 66 61.76±j49.89-777.89e-3 49.89 2 69 and 70 33.87 ±j21.17-848.01e-3 21.17 3 73 and 74 14.01± j19.81-577.39e-3 19.81 With LQR based Centralized Controller 4 65-80.93+j0.00 1.00 0.00 5 66-60.13+j0.00 1.00 0.00 6 69 and 70-27.97 ±j 20.99 799.83e-3 20.99 7 73 and 74-40.62 ± j23.57 864.93e-3 23.57 With LQRPDS based Centralized Controller 8 65 and 66-235.29 ±j 22.11 995.62e-3 22.11 9 69 and 70-216.72 ±j 20.64 995.49e-3 20.64 10 73 and 74-199.18 ±j 26.28 995.41e-3 26.28 unstable region. In order to stabilize the islanded ACMG system, a centralized supplementary controller has been designed on the basis of the LQRPDS such that the eigenvalues of the islanded ACMG are shifted from the unstable region to the stable region. The prescribed degree of stability is dependent upon the operating condition of the islanded ACMG. In this paper, prescribed degree of stability ( ) has been chosen as 100. The
Figure 7: Eigenvalues of the islanded ACMG with the LQRPDS based Centralized supplementary Controller effectiveness of the proposed controller has been evaluated by comparing the results with a centralized supplementary controller based on LQR. The eigenvalues of the islanded ACMG system with the centralized supplementary controller on the basis of LQR and LQRPDS are shown in Fig. 6 and Fig. 7, respectively. It can be observed that the eigenvalues in the unstable region are moved to the stable region. Table II compares the damping factors of modes obtained without and with the centralized supplementary controller. It can be seen that both the controllers are able to stabilize the islanded ACMG, however, LQRPDS is more effective as compared to LQR in improving the damping. V. CONCLUSIONS A centralized supplementary controller on the basis of the LQR with prescribed degree of stability (LQRPDS) has been developed to stabilize the low inertial inverter interfaced DGs based islanded AC microgrid. The test system comprises of 230 V(RMS phase), 50Hz, islanded ACMG containing four low inertial inverter interfaced DGs having unequal power ratings connected to each other through three lines. The islanded ACMG has been assumed to supply passive loads as well as an active load simultaneously, but connected at different nodes. A full dynamical small signal linearized state space model represented in synchronous (DQ) reference frame including interaction between the inverters, network dynamics and load dynamics has been considered to analyze the stability. The stability analysis showed that the presence of RIAL dynamics causes instability in the islanded AC microgrid. The effectiveness of the proposed controller has been evaluated by comparing the results with the centralized supplementary controller based on linear quadratic regulator (LQR). APPENDIX Microgrid test system parameters and operating conditions: DGs Ratings: DGs Ratings:DG1-(10+j6) kva; DG2-(15+j9) kva; DG3-(20+j12) kva; DG4-(25+j15) kva. Static active power droop gains: m 1 =6.2832e-4 rad/sec/watt, m 2 =4.188e-4 rad/sec/watt, m 3 =3.142e-4 rad/sec/watt, m 4 =2.5133e-4 rad/sec/watt. Static reactive power droop gains: n 1 =1.6e-3 Volt/VAR, n 2 =1.111e-3 Volt/VAR, n 3 =8.3e-4 Volt/VAR, n 4 =6.666e-4 Volt/VAR. DGs Parameters: L =1.35 mh, C =50 F, R =0.1 Ω, f =8 khz, w =31.41 rad/sec, K =0.05, K =390, K =10.5, K =16e3, F=0.75, f =50.5 Hz, R =0.03 Ω, L =0.35mH. RIAL Parameters: L =2.3mH, C =8.8 F, R =0.1 Ω, f =10 khz, w =31.41 rad/sec, K =0.5, K =150, K =7, K =25e3, R =0.03 Ω, L =0.93mH. Network Parameters: Line 1: (0.23+j0.35) Ω, Line 2: (0.35+j1.85) Ω, Line 3: (0.30+j1.50) Ω. Load parameters: RL Load: 6 kva, Z =(28+j20.21) Ω; CPL: 12 kva r =13.224 Ω and cos =0.85; RIAL: 12 kw and 40.833 Ω; R Load: 25 kw and 6.347 Ω. REFERENCES [1] S. Chowdhury, S. P. Chowdhury, and P. Crossley, Microgrids and Active Distribution Networks, London, U.K.: Institution of Engineering and Technology, 2009. [2] N. Pogaku, M. Prodanovic, and T. C. Green, Modelling, analysis and testing of autonomous operation of an inverter-based microgrid, IEEE Transactions on Power Electronics, vol.22,no.2,mar.2007,pp.613-625. [3] Ritwik Majumder, Some Aspects of Stability in Microgrids, IEEE Transactions on Power Systems, Vol.28,No.3,August,2013,pp.3243-3252. [4] E S N Raju P and Trapti Jain, Hybrid AC/DC Micro Grid: An Overview, Fifth International Conference on Power and Energy Systems, October, 2013. [5] J. Guerrero, L. de Vicuna, J. Matas,M. Castilla, and J. Miret, A wireless controller to enhance dynamic performance of parallel inverters in distributed generation system, IEEE Transactions on Power Electronics, vol. 19, no. 5, Sep. 2004, pp. 1205-1213. [6] Antonis G. Tsikalakis and Nikos D. Hatziargyriou, Centralized Control for Optimizing Microgrids Operation, IEEE Transactions on Energy Conversion, Vol. 23, No. 1, March 2008, pp. 241-248. [7] K. T. Tan, X.Y. Peng, and P.L. So, Centralized Control for Parallel Operation of Distributed Generation Inverters in Microgrids, IEEE Transactions on Smart Grid, Vol. 3, No. 4, December 2012, pp. 1977-1987. [8] Y. Mohamed and E. F. El-Saadany, Adaptive decentralized droop controller to preserve power sharing stability of paralleled inverters in distributed generation microgrids, IEEE Transactions on Power Electronics, vol. 23, no. 6, Nov. 2008, pp. 2806-2816. [9] Nathaniel Bottrel, Milan Prodanovic, andtimothyc. Green, Dynamic Stability of a Microgrid with an Active Load, IEEE Transactions on Power Electronics, Vol. 28, No. 11, November, 2013, pp. 5107-5119. [10] A. Kahrobaeian and Y. A.-R. I. Mohamed, Analysis and Mitigation of Low-Frequency Instabilities in Autonomous Medium-Voltage Converter- Based Micro-Grids With Dynamic Load, IEEE Transactions on Industrial Electronics, vol.61, no.4, April 2014, pp. 1643-1658. [11] E S N Raju P and Trapti Jain, Small Signal Modelling and Stability Analysis of an Islanded AC Microgrid with Inverter Interfaced DGs, International Conference on Smart Electric Grid, September, 2014. [12] B. Kalyan Kumar, S.N. Singh and S.C Srivastava, A decentralized nonlinear feedback controller with prescribed degree of stability for damping power system oscillations, Electric Power Systems Research, vol.77, no.3, 2007, pp. 204-211.