- Email: [email protected]
Coordination control of multiple micro sources in islanded microgrid based on differential games theory
Electrical Power and Energy Systems 97 (2018) 11–16
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Coordination control of multiple micro sources in islanded microgrid based on differential games theory
MARK
⁎
Jing Zhanga, , Yuan Gaoa, Peijia Yub, Bowen Lia, Yan Yangc, Ya Shia, Liangwei Zhaoa a b c
School of Electrical Engineering, Guizhou University, Guizhou Province, China College of Computer Science and Information, Guizhou University, Guizhou Province, China Guiyang Power Supply Bureau, Guizhou Power Grid Corp., Guizhou Province, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Microgrid Differential games theory Non-cooperative feedback Nash equilibrium Load frequency control
To deal with the deviations of frequency of microgrid system composed of multiple micro sources caused by load disturbance, the load frequency control of islanded microgrid system is studied based on linear quadratic differential games theory in this paper. Considering the operating characteristics of each micro source, different objective function weighting matrix is selected, and the control strategy of non-cooperative feedback Nash equilibrium of each micro source in microgrid system is researched. The simulation results show that the control strategy can accomplish system control targets, make full use of the various of characteristics of each micro source and balance the benefit of them. At the same time, the control strategy has good robustness to external disturbance and parameter variation of the internal unit.
0. Introduction The emergence and development of microgrids have contributed to widespread connections among distributed generation (DG) units. The key technological challenge of islanded operation of the microgrid is to retain power balance, voltage stability, and frequency stability through the coordination and regulation of various micro-DG units [1–3]. Researchers around the world have conducted many studies on control strategies for microgrids [4–6]. From the perspective of the operation control structure, the primary control strategies for microgrid systems are master/slave control, peer-to-peer (P2P) control, and hierarchical control. Master/slave refers to the mode in which the microgrid operates based on one master controller coupled with a number of slave controllers. In this mode, communication between the master controller and slave controllers is essential, and the slave controllers are subject to control by the master controller [7–9]. Moreover, because of the introduction of the static droop characteristics of the power sources in conventional synchronous generators, P2P control strategies for DGs with inverter have attracted significant attention. Currently, the most frequently adopted P2P strategy is droop control [10–12]. In hierarchical control strategies, DG units and loads at the field level are managed using multi-level coordination and regulation [13–16]. Microgrid operation control generally requires cooperation, coordination, and negotiation among multiple participants. However, based on the currently existing operation control strategies, controllers
⁎
functioning at different levels could exhibit mutual independence (and even conflict) between control targets. If coordination is disregarded in the control strategy design, the control actions could interfere with each other, producing control effects that deviate from the initial objectives. Nevertheless, even if some control strategies are equipped with global coordination functionality, it remains difficult for them to provide mathematically clear coordination and control schemes that can satisfy all involved participants, which is undesirable. Specifically, with the continuous growth of DG amount in microgrids, this self-oriented combination of DG systems is likely to cause problems such as system power oscillation and energy efficiency reduction. Game theory has been widely adopted for evaluating how multiple parties make decisions that are beneficial to themselves (or their entire group) according to their own capabilities and available information when there exist connections and conflicts of interests. Game theory has widespread applications in power systems to solve the coordination of participants[17–20]. Differential game derived from game theory and optimal control theory can be used to solve for the dynamic system control strategy evolving multiple participants [21]. The linear quadratic differential game, a significant subgroup of the differential game, is developed based on the linear model of a dynamic system, in which the game is described using quadratic payoff functions for every participant [22]. The differential game is introduced in power system researches recently. In [23] a novel framework based on the principles of differential games is proposed to analyze the smart grid security. A two-
Corresponding author. E-mail address: [email protected] (J. Zhang).
http://dx.doi.org/10.1016/j.ijepes.2017.10.028 Received 1 August 2017; Received in revised form 18 September 2017; Accepted 22 October 2017 0142-0615/ © 2017 Elsevier Ltd. All rights reserved.
Electrical Power and Energy Systems 97 (2018) 11–16
J. Zhang et al.
Fig. 1. Linear model of microgrid system with multiple micro-DGs.
1 R1
u1
+
1
-
sLwind
Pwind Rwind
1 R2
u2
+
Ppv
1
-
sL pv
R pv
+
KP 1 sTP
+
Pbattery
+ -
Pload
1 R3 u3
+
f
1
-
sLbattery Rbattery
level differential game framework is proposed in [24] to study the demand response program and a large population demand response management can be achieved [25]. The differential game is also applied in power systems for studying the distrubance suppression of excitation systems and frequency control [26]. In this paper, a differential game model for a microgrid integrating with multiple DGs is constructed by using the linear quadratic differential game. The coordinated frequency control strategy of the microgrid system in islanded operation mode is investigated. The feedback control strategy for each micro-DG units is determined by treating themselves as game participants. Simulation and computation results indicate that superior control efficiency and robustness in terms of frequency fluctuation suppression can be obtained by applying the proposed approach.
system states, that linear time-invariant feedback control strategies are adopted by all participants, and that the participants have no reasons to cooperate with each other, because of conflicts of interests. Therefore, the control strategy for each participant can be written as ui = Fi x (t ) , Fi ∈ Rmi × n , where Rmi × n is the strategy space of the ith participant, mi is the dimension of the control vector, and n is the dimension of the system state variable vector. Each participant can control x (t ) by adjusting its own control strategy ui (t ) for the purpose of minimizing the payoff function value. The payoff function for each participant does not depend on only its own strategy; it is also strongly influenced by strategies of the other participants. Moreover, minimization of one payoff function usually requires coordination with the other participants, from which the game is generated. For the linear quadratic differential game between Eqs. (1) and (2), the Nash equilibrium F = (F1,…,Fn ) ∈ F can be defined using Eq. (3).
1. Linear quadratic differential game and Nash equilibrium
supJi (F1,…,Fi ,…,FN ,x 0) ⩽ supJi (F1,…,Fi,…,FN ,x 0)
Based on the theory of linear quadratic differential game [27], the Nash equilibrium can be determined by using the method demonstrated below Assuming that there exists a n × n real symmetric matrix Pi satisfying the following condition:
A linear dynamic system with N participants can be expressed as N
x ̇ (t ) = Ax (t ) +
∑
Bi ui (t )
(1)
i=1
where x (t ) is the system state variable vector, ui (t ) is the control strategy adopted by the ith participant, A is the system state matrix, and B is the system control matrix. The payoff function for the ith participant is
minJi =
∫t
∞
0
[x (t )T Qi x (t )
+
ui (t )T Rii ui (t )
+
uj (t )T Rij uj (t )] dt ,
(3)
⎛ A− ⎜ ⎝
T
N
∑ j≠i
Sj Pj
⎞ P + Pi (A− ⎟ i ⎠
N
∑
N
Sj Pj )−Pi Si Pi + Qi +
j≠i
∑
Pj Sij Pj = 0
j≠i
(4)
N
A− ∑ Si Pi is stable
j ≠ i
(5)
i=1
Bi Rii−1 BiT ,Sij
(2)
Bi Rii−1 Rji Rii−1 BiT ,
= i ≠ j , then the Nash equiliwhere Si = brium F = (F1,…,Fn ) ∈ F can be defined as
where t0 is the start time of the game, Qi is the weight coefficient matrix corresponding to the system state variables, and Rii,Rij are the weight coefficient matrices corresponding to the system control variables. Here, Qi is a symmetric matrix, and Rii,Rij are positive definite matrices. Qi defines the weights on the system state variables while Rii,Rij defines the weights on the control input in the cost function. The larger these values are, the more penalization on these corresponding signals. Basically, choosing a large value for Qi means trying to stabilize the system with the least possible changes in the system state variables and vice versa. The choosing of Rii,Rij is similarly. We assume that all of the game participants are able to observe the
Fi = −Rii−1 BiT Pi, i = 1,…,N
(6)
2. Model of microgrid integrated with multiple micro-DGs Treating each micro-DG in a microgrid as a game participant, when the microgrid suffers a disturbance (such as a load fluctuation), participants can choose optimal strategies based on their own payoff functions. A typical mathematical model of a microgrid integrated with micro12
Electrical Power and Energy Systems 97 (2018) 11–16
J. Zhang et al.
DG, such as wind power generation, photovoltaic power generation, and storage batteries, is illustrated in Fig. 1. The state equation of the microgrid shown in Fig. 1 can be written as
Rij = [0] (i, j = 1, 2, 3 and i ≠ j), R11 = R22 = R33 = [1], and let the element values of Qi for wind power generation, photovoltaic power generation, and storage batteries be set as follows.
3
∑
x ̇ (t ) = Ax (t ) +
Q1 = diag [20 1 1 1]
Bi ui (t )
(7)
i=1
Q2 = diag [20 1 1 1]
where
Q3 = diag [5 1 1 1] −1
KP TP −Rwind Lwind
⎧ TP ⎪ −1 ⎪RL ⎪ 1 wind A= ⎨ −1 ⎪ R2 Lpv ⎪ −1 ⎪ R3 Lbattery ⎩ B1T B2T
−Rpv Lpv
0
0
KP TP
⎫ ⎪ 0 ⎪ ⎪ 0 ⎬ ⎪ −Rbattery ⎪ Lbattery ⎪ ⎭
Therefore, the payoff functions for wind power generation, photovoltaic power generation, and storage batteries in the microgrid can be expressed as
1 Lpv
(9)
0⎤ ⎦
∫t
J2 =
∫t
J3 =
∫t
(11)
⎦
x (t )T = [ Δf ΔPwind ΔPpv ΔPbattery ]
0
∞
0
∞
0
2 2 2 [20Δf 2 + ΔPwind + ΔPpv + ΔPbattery + u12] dt
2 2 2 [20Δf 2 + ΔPwind + ΔPpv + ΔPbattery + u22] dt
2 2 2 [5Δf 2 + ΔPwind + ΔPpv + ΔPbattery + u32] dt
(12)
where Δf is the frequency deviation of the microgrid system; ΔPwind , ΔPpv , and ΔPbattery are the practical adjustment amounts contributed by wind power generation, photovoltaic power generation, and storage batteries, respectively; Rwind and L wind are the inverter circuit parameters of the wind power generation system; Rpv and Lpv are the inverter circuit parameters of the photovoltaic power generation system; Rbattery and Lbattery are the inverter circuit parameters of the storage batteries; KP is the gain of the microgrid system; TP is a time constant of the microgrid system; and Ri (i = 1,2,3) is the difference coefficient.
−0.6345ΔPbattery
−0.7843ΔPbattery
T
j≠i
N
∑
(18)
To verify the correctness and effectiveness of the proposed approach, the modeling and simulation on the control strategy of the microgrid system integrated with multiple DGs shown in Fig. 1 are conducted in Matlab/Simulink. The installed capacities of wind power generation, photovoltaic power generation, and storage batteries are each set at 100 kW, which is also the capacity benchmark. The inverter circuit parameters of the wind power generator are set to Rwind = 0.08 Ω and L wind = 0.6 H; the inverter circuit parameters of the photovoltaic power generator are set to Rpv = 0.06 Ωand Lpv = 0.5 H; and the inverter circuit parameters of the storage batteries are set to Rbattery = 0.05Ω and Lbattery = 0.6 H. The gain, time constant, and difference coefficient of the microgrid system are set to KP = 4.8 Hz/m.u., TP = 3.5 s , and R1 = R2 = R3 = 2.4 Hz/p.u., respectively.
(2) Find the solution Pik + 1 (k ≥ 0) to the decoupled algebraic Riccati equation shown below
⎞ k+1 + Pik + 1 (A− P ⎟ i ⎠
(17)
4. Simulation results and analysis
AT Pi0 + Pi0 A−Pi0 Si Pi0 + Qi = 0
Sj P jk
(16)
u3 = F3 x (t ) = −R33 B3T P3 x (t ) = −0.4461Δf−0.1061ΔPwind−0.0719ΔPpv
To solve for the feedback Nash control strategy adopted by each game participant, ui , the matrix Pi in Eq. (4) must be obtained. However, in general, as a set of coupled algebraic Riccati equations, Eq. (4) is difficult to solve. Therefore, the solution is acquired using numerical algorithms using the iteration steps shown below. (1) Find the solution to the standard decoupled algebraic Riccati equation shown below, Pi0 , and use the solution as the matrix of initial values.
N
(15)
u2 = F2 x (t ) = −R22 B2T P2 x (t ) = −2.1323Δf−0.7109ΔPwind−1.7364ΔPpv
−1.1611ΔPbattery
∑
(14)
u1 = F1 x (t ) = −R11 B1T P1 x (t ) = −1.7347Δf−1.609ΔPwind−0.5122ΔPpv
3. Control strategies for micro-DGs based on linear quadratic differential game
⎛ A− ⎜ ⎝
(13)
The matrix Pi can be solved by setting the error ε to 10−9. Substituting Pi into Eq. (5), we can obtain the eigenvalues of this equation, of which all the real parts are less than zero, proving that this equation is stable. Therefore, the feedback Nash equilibrium strategy adopted for the microgrid system with wind power generation, photovoltaic power generation, and storage batteries is below.
(10)
1 Lbattery ⎤
B3T = ⎡ 0 0 0 ⎣
∞
J1 = (8)
0 0⎤ ⎦
Lwind
= ⎡0 0 ⎣
0
0
1
= ⎡0 ⎣
KP TP
Sj P jk )−Pik + 1 Si Pik + 1 + Qi
j≠i
4.1. Efficiency comparison with droop control
N
+
∑
P jk Sij P jk
=0
Based on the parameters of the microgrid system, the droop coefficient of the droop control is set to 0.25. A load disturbance of −20 kW (−0.2 p.u.) and 20 kW (0.2 p.u.) lasting for one second, which are shown in Figs. 2 Fig. 3 respectively, takes place at the sixth second. The linear quadratic differential game control (LQDGC) and the droop control (DC) are adopted in the same microgrid respectively. The deviations of frequency and active power of each micro-DG for both situations are presented in Figs. 4and 5 respectively. It can be seen from the figures that the suppression effect of the linear quadratic differential game control on the frequency deviation of the microgrid system after the disturbance is superior to that of droop control, because the linear quadratic differential game control prove the coordination of all the
j≠i
(3) Substitute Pik + 1 into Eq. (4). If the value of Eq. (4) is larger than the predetermined error ε , return to step 2); otherwise, let Pi = Pik + 1, termination routine.Qi and Rii,Rij reflect the interests of every participant in the game. The larger these values are, the more penalization on these corresponding interests. Basically, choosing a large value for Qi means trying to stabilize the system with the least possible changes in the outputs, frequency and active power, and vice versa. The choosing of Rii and Rij is similarly. To simplify the analysis while preserving the generality of the problem, and because the control decision of participant i is not impacted by the other participants’ control costs, let 13
Electrical Power and Energy Systems 97 (2018) 11–16
J. Zhang et al.
LQDGC
f / Hz
PL / p.u.
DC
t/s
t/s
(a) Deviation of frequency
Pwind / p.u.
Fig. 2. Disturbance of load decrease.
LQDGC DC
t/s
(b) Deviation of wind power output
Ppv / p.u.
Fig. 3. Disturbance of load increase (a) Deviation of frequency (b) Deviation of wind power output (c) Deviation of photovoltaic power output (d) Deviation of battery power output.
micro-DGs. 4.2. Impact of system variable change on control effect
LQDGC
In a practical microgrid system, the operation state variations of the micro-DGs can potentially cause system variable changes, affecting the efficiency of the control strategy and even the operational instability of the system. To examine the robustness of the linear quadratic differential game control strategy proposed in this paper, the inverter circuit parameters (i.e., resistance and inductance) of every micro-DG unit in the microgrid system vary by 10% and 20% increasing, respectively, without changing their control strategies. The simulation and comparison results are shown in Fig. 6. It can be seen from Fig. 6 that the microgrid system control strategy based on the theory of differential games results in superior frequency control performance, even when the model parameters are not accurate or different from the real situations, which is indicative of the excellent robustness of the system control strategy against microgrid system parameter variation. Because it is a decentralized control method, this system control strategy is able to better match the operation of the microgrid system integrated with micro-DGs distributed geographically.
DC
t/s
Pbattery / p.u.
(c) Deviation of photovoltaic power output
LQDGC DC
t/s
5. Conclusions
(d) Deviation of battery power output
In this paper, which considered the coordination and regulation of multiple DGs in a microgrid system, the linear quadratic differential game theory is adopted to the frequency control of the Microgrid integrated with multiple DGs. The control performance is compared to that of the droop control. In differential game theory, the optimal control strategy for all participants can be obtained by solving the Nash equilibrium based on the payoff functions of every participant. In this
Fig. 4. System response for the Disturbance of load decrease.
way, the tion and realized. strategy 14
interests of all participants can be considered, and coordinaregulation of the micro-DGs in the microgrid system can be Simulation results demonstrate that the proposed control to multiple DGs coordination and control for islanded
Electrical Power and Energy Systems 97 (2018) 11–16
J. Zhang et al.
DC (20% Parameters increasing) LQDGC (20% Parameters increasing) DC (10% Parameters increasing) LQDGC (10% Parameters increasing) LQDGC (0% Parameters Error)
f / Hz
f / Hz
DC (0% Parameters Error)
LQDGC DC
t/s
(a) Deviation of frequency t/s Fig. 6. Frequency deviation under parameters variance.
LQDGC DC
Pwind / p.u.
microgrid systems based on differential games results in superior control efficiency and robustness. Acknowledgement This research was supported by the National Natural Science Foundation of China (51007009), Science and Technology Foundation of Guizhou Province ([2016]1036), Guizhou Province Reform Foundation forPostgraduate Education ([2016]02).
t/s
(b) Deviation of wind power output
References [1] Golpîra H, Seifi H, Messina AR, et al. Maximum penetration level of microgrids in large-scale power systems: frequency stability viewpoint. IEEE Trans Power Syst 2016;31(6):5163–71. [2] Mohanty A, Viswavandya M, Mohanty S, et al. Modelling, simulation and optimisation of robust PV based micro grid for mitigation of reactive power and voltage instability. Int J Electr Power Energy Syst 2016;81:444–58. [3] Shen J, Jiang C, Li B. Controllable load management approaches in smart grids. Energies 2015;8(10):11187–202. [4] Zhang W, Xu Y, Dong Z Y, et al. Robust security-constrained optimal power flow using multiple microgrids for corrective control under uncertainty. IEEE Trans Informatics; 2016, PP(99):1–1. [5] Meng K, Dong ZY, Xu Z, et al. Cooperation-driven distributed model predictive control for energy storage systems. IEEE Trans Smart Grid 2015;6(6):2583–5. [6] Wang Z, Liu F, Low SH. Distributed frequency control with operational constraints part I: per-node power balance. IEEE Trans Smart Grid 2017. PP(99):1-1. [7] Dou C, Li N, Yue D, et al. A hierarchical hybrid control strategy for microgrid switching stabilization during operating mode conversion. IET Gener Transm Distrib 2016;10(12):2880–90. [8] Javaid S, Kurose Y, Kato T, et al. Cooperative distributed control implementation of the power flow coloring over a nano-grid with fluctuating power loads. IEEE Trans Smart Grid 2016:1–11. [9] Borrega M, Marroyo L, Gonzalez R, et al. Modeling and control of a master-slave PV inverter with N-paralleled inverters and three-phase three-limb inductors. IEEE Trans Power Electron 2013;28(6):2842–55. [10] Wang Z, Wu W, Zhang B. A distributed control method with minimum generation cost for DC microgrids. IEEE Trans Energy Convers 2016;31(4):1462–70. [11] Simpson-Porco JW, Dorfler F, Bullo F. Voltage stabilization in microgrids via quadratic droop control. IEEE Trans Autom Control 2017;62(3):1239–53. [12] Gao F, Bozhko S, Costabeber A, et al. Comparative stability analysis of droop control approaches in voltage source converters-based DC microgrids. IEEE Trans Power Electron 2017;32(3):2395–415. [13] Xu Z, Yang P, Zhang Y, et al. Control devices development of multi-microgrids based on hierarchical structure. IET Gener Transm Distrib 2016;10(16):4249–56. [14] Xu Z, Callaway DS, Hu Z, et al. Hierarchical coordination of heterogeneous flexible loads. IEEE Trans Power Syst 2016;31(6):4206–16. [15] Li D, Jayaweera SK. Distributed smart-home decision-making in a hierarchical interactive smart grid architecture. IEEE Trans Parallel Distrib Syst 2015;26(1):75–84. [16] Jiang B, Fei Y. Smart home in smart microgrid: a cost-effective energy ecosystem with intelligent hierarchical agents. IEEE Trans Smart Grid 2015;6(1):3–13. [17] Mei S, Wei W, Liu F. On engineering game theory with its application in power systems. Control Theory Technol 2017;15(1):1–12. [18] Banaei M, Buygi MO, Zareipour H. Impacts of strategic bidding of wind power producers on electricity markets. IEEE Trans Power Syst 2016;31(6):4544–53. [19] Wei W, Wang J, Mei S. Convexification of the Nash bargain based environmentaleconomic dispatch. IEEE Trans Power Syst 2016;31(6):5208–9.
LQDGC
Ppv / p.u.
DC
t/s
(c) Deviation of photovoltaic power output
LQDGC
Pbattery / p.u.
DC
t/s
(d) Deviation of battery power output Fig. 5. System response for the Disturbance of load increase.
15
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2014;6(1):291–300. [25] Zhu Q, Başar T. Multi-resolution large population stochastic differential games and their application to demand response management in the smart grid. Dynamic Games Appl 2013;3(1):68–88. [26] Chen H, Ye R, Wang X, et al. Cooperative control of power system load and frequency by using differential games. IEEE Trans Control Syst Technol 2015;23(3):882–97. [27] Jacob Engwerda LQ. Dynamic optimization and differential games J. Wiley & Sons; 2005. p. 371–82.
[20] Mei S, Wei W. Hierarchal game and its applications in the smart grid. J Syst Sci Math Sci 2014;34(11):1331–44. [in Chinese]. [21] Basar T, Olsder GJ. Dynamic noncooperative game theory. New York, NY, USA: Academic Press; 1999. p. 139–52. [22] Engwerda J. Linear quadratic differential games: an overview. In: Proceedings of the 12th international symposium on dynamic games and applications; 2009. [23] Srikantha P, Kundur D. A DER attack-mitigation differential game for smart grid security analysis. IEEE Trans Smart Grid 2016;7(3):1476–85. [24] Forouzandehmehr N, Esmalifalak M, Mohsenian-Rad H, et al. Autonomous demand response using stochastic differential games. IEEE Trans Smart Grid
16
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Coordination control of multiple micro sources in islanded microgrid based on differential games theory
MARK
⁎
Jing Zhanga, , Yuan Gaoa, Peijia Yub, Bowen Lia, Yan Yangc, Ya Shia, Liangwei Zhaoa a b c
School of Electrical Engineering, Guizhou University, Guizhou Province, China College of Computer Science and Information, Guizhou University, Guizhou Province, China Guiyang Power Supply Bureau, Guizhou Power Grid Corp., Guizhou Province, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Microgrid Differential games theory Non-cooperative feedback Nash equilibrium Load frequency control
To deal with the deviations of frequency of microgrid system composed of multiple micro sources caused by load disturbance, the load frequency control of islanded microgrid system is studied based on linear quadratic differential games theory in this paper. Considering the operating characteristics of each micro source, different objective function weighting matrix is selected, and the control strategy of non-cooperative feedback Nash equilibrium of each micro source in microgrid system is researched. The simulation results show that the control strategy can accomplish system control targets, make full use of the various of characteristics of each micro source and balance the benefit of them. At the same time, the control strategy has good robustness to external disturbance and parameter variation of the internal unit.
0. Introduction The emergence and development of microgrids have contributed to widespread connections among distributed generation (DG) units. The key technological challenge of islanded operation of the microgrid is to retain power balance, voltage stability, and frequency stability through the coordination and regulation of various micro-DG units [1–3]. Researchers around the world have conducted many studies on control strategies for microgrids [4–6]. From the perspective of the operation control structure, the primary control strategies for microgrid systems are master/slave control, peer-to-peer (P2P) control, and hierarchical control. Master/slave refers to the mode in which the microgrid operates based on one master controller coupled with a number of slave controllers. In this mode, communication between the master controller and slave controllers is essential, and the slave controllers are subject to control by the master controller [7–9]. Moreover, because of the introduction of the static droop characteristics of the power sources in conventional synchronous generators, P2P control strategies for DGs with inverter have attracted significant attention. Currently, the most frequently adopted P2P strategy is droop control [10–12]. In hierarchical control strategies, DG units and loads at the field level are managed using multi-level coordination and regulation [13–16]. Microgrid operation control generally requires cooperation, coordination, and negotiation among multiple participants. However, based on the currently existing operation control strategies, controllers
⁎
functioning at different levels could exhibit mutual independence (and even conflict) between control targets. If coordination is disregarded in the control strategy design, the control actions could interfere with each other, producing control effects that deviate from the initial objectives. Nevertheless, even if some control strategies are equipped with global coordination functionality, it remains difficult for them to provide mathematically clear coordination and control schemes that can satisfy all involved participants, which is undesirable. Specifically, with the continuous growth of DG amount in microgrids, this self-oriented combination of DG systems is likely to cause problems such as system power oscillation and energy efficiency reduction. Game theory has been widely adopted for evaluating how multiple parties make decisions that are beneficial to themselves (or their entire group) according to their own capabilities and available information when there exist connections and conflicts of interests. Game theory has widespread applications in power systems to solve the coordination of participants[17–20]. Differential game derived from game theory and optimal control theory can be used to solve for the dynamic system control strategy evolving multiple participants [21]. The linear quadratic differential game, a significant subgroup of the differential game, is developed based on the linear model of a dynamic system, in which the game is described using quadratic payoff functions for every participant [22]. The differential game is introduced in power system researches recently. In [23] a novel framework based on the principles of differential games is proposed to analyze the smart grid security. A two-
Corresponding author. E-mail address: [email protected] (J. Zhang).
http://dx.doi.org/10.1016/j.ijepes.2017.10.028 Received 1 August 2017; Received in revised form 18 September 2017; Accepted 22 October 2017 0142-0615/ © 2017 Elsevier Ltd. All rights reserved.
Electrical Power and Energy Systems 97 (2018) 11–16
J. Zhang et al.
Fig. 1. Linear model of microgrid system with multiple micro-DGs.
1 R1
u1
+
1
-
sLwind
Pwind Rwind
1 R2
u2
+
Ppv
1
-
sL pv
R pv
+
KP 1 sTP
+
Pbattery
+ -
Pload
1 R3 u3
+
f
1
-
sLbattery Rbattery
level differential game framework is proposed in [24] to study the demand response program and a large population demand response management can be achieved [25]. The differential game is also applied in power systems for studying the distrubance suppression of excitation systems and frequency control [26]. In this paper, a differential game model for a microgrid integrating with multiple DGs is constructed by using the linear quadratic differential game. The coordinated frequency control strategy of the microgrid system in islanded operation mode is investigated. The feedback control strategy for each micro-DG units is determined by treating themselves as game participants. Simulation and computation results indicate that superior control efficiency and robustness in terms of frequency fluctuation suppression can be obtained by applying the proposed approach.
system states, that linear time-invariant feedback control strategies are adopted by all participants, and that the participants have no reasons to cooperate with each other, because of conflicts of interests. Therefore, the control strategy for each participant can be written as ui = Fi x (t ) , Fi ∈ Rmi × n , where Rmi × n is the strategy space of the ith participant, mi is the dimension of the control vector, and n is the dimension of the system state variable vector. Each participant can control x (t ) by adjusting its own control strategy ui (t ) for the purpose of minimizing the payoff function value. The payoff function for each participant does not depend on only its own strategy; it is also strongly influenced by strategies of the other participants. Moreover, minimization of one payoff function usually requires coordination with the other participants, from which the game is generated. For the linear quadratic differential game between Eqs. (1) and (2), the Nash equilibrium F = (F1,…,Fn ) ∈ F can be defined using Eq. (3).
1. Linear quadratic differential game and Nash equilibrium
supJi (F1,…,Fi ,…,FN ,x 0) ⩽ supJi (F1,…,Fi,…,FN ,x 0)
Based on the theory of linear quadratic differential game [27], the Nash equilibrium can be determined by using the method demonstrated below Assuming that there exists a n × n real symmetric matrix Pi satisfying the following condition:
A linear dynamic system with N participants can be expressed as N
x ̇ (t ) = Ax (t ) +
∑
Bi ui (t )
(1)
i=1
where x (t ) is the system state variable vector, ui (t ) is the control strategy adopted by the ith participant, A is the system state matrix, and B is the system control matrix. The payoff function for the ith participant is
minJi =
∫t
∞
0
[x (t )T Qi x (t )
+
ui (t )T Rii ui (t )
+
uj (t )T Rij uj (t )] dt ,
(3)
⎛ A− ⎜ ⎝
T
N
∑ j≠i
Sj Pj
⎞ P + Pi (A− ⎟ i ⎠
N
∑
N
Sj Pj )−Pi Si Pi + Qi +
j≠i
∑
Pj Sij Pj = 0
j≠i
(4)
N
A− ∑ Si Pi is stable
j ≠ i
(5)
i=1
Bi Rii−1 BiT ,Sij
(2)
Bi Rii−1 Rji Rii−1 BiT ,
= i ≠ j , then the Nash equiliwhere Si = brium F = (F1,…,Fn ) ∈ F can be defined as
where t0 is the start time of the game, Qi is the weight coefficient matrix corresponding to the system state variables, and Rii,Rij are the weight coefficient matrices corresponding to the system control variables. Here, Qi is a symmetric matrix, and Rii,Rij are positive definite matrices. Qi defines the weights on the system state variables while Rii,Rij defines the weights on the control input in the cost function. The larger these values are, the more penalization on these corresponding signals. Basically, choosing a large value for Qi means trying to stabilize the system with the least possible changes in the system state variables and vice versa. The choosing of Rii,Rij is similarly. We assume that all of the game participants are able to observe the
Fi = −Rii−1 BiT Pi, i = 1,…,N
(6)
2. Model of microgrid integrated with multiple micro-DGs Treating each micro-DG in a microgrid as a game participant, when the microgrid suffers a disturbance (such as a load fluctuation), participants can choose optimal strategies based on their own payoff functions. A typical mathematical model of a microgrid integrated with micro12
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DG, such as wind power generation, photovoltaic power generation, and storage batteries, is illustrated in Fig. 1. The state equation of the microgrid shown in Fig. 1 can be written as
Rij = [0] (i, j = 1, 2, 3 and i ≠ j), R11 = R22 = R33 = [1], and let the element values of Qi for wind power generation, photovoltaic power generation, and storage batteries be set as follows.
3
∑
x ̇ (t ) = Ax (t ) +
Q1 = diag [20 1 1 1]
Bi ui (t )
(7)
i=1
Q2 = diag [20 1 1 1]
where
Q3 = diag [5 1 1 1] −1
KP TP −Rwind Lwind
⎧ TP ⎪ −1 ⎪RL ⎪ 1 wind A= ⎨ −1 ⎪ R2 Lpv ⎪ −1 ⎪ R3 Lbattery ⎩ B1T B2T
−Rpv Lpv
0
0
KP TP
⎫ ⎪ 0 ⎪ ⎪ 0 ⎬ ⎪ −Rbattery ⎪ Lbattery ⎪ ⎭
Therefore, the payoff functions for wind power generation, photovoltaic power generation, and storage batteries in the microgrid can be expressed as
1 Lpv
(9)
0⎤ ⎦
∫t
J2 =
∫t
J3 =
∫t
(11)
⎦
x (t )T = [ Δf ΔPwind ΔPpv ΔPbattery ]
0
∞
0
∞
0
2 2 2 [20Δf 2 + ΔPwind + ΔPpv + ΔPbattery + u12] dt
2 2 2 [20Δf 2 + ΔPwind + ΔPpv + ΔPbattery + u22] dt
2 2 2 [5Δf 2 + ΔPwind + ΔPpv + ΔPbattery + u32] dt
(12)
where Δf is the frequency deviation of the microgrid system; ΔPwind , ΔPpv , and ΔPbattery are the practical adjustment amounts contributed by wind power generation, photovoltaic power generation, and storage batteries, respectively; Rwind and L wind are the inverter circuit parameters of the wind power generation system; Rpv and Lpv are the inverter circuit parameters of the photovoltaic power generation system; Rbattery and Lbattery are the inverter circuit parameters of the storage batteries; KP is the gain of the microgrid system; TP is a time constant of the microgrid system; and Ri (i = 1,2,3) is the difference coefficient.
−0.6345ΔPbattery
−0.7843ΔPbattery
T
j≠i
N
∑
(18)
To verify the correctness and effectiveness of the proposed approach, the modeling and simulation on the control strategy of the microgrid system integrated with multiple DGs shown in Fig. 1 are conducted in Matlab/Simulink. The installed capacities of wind power generation, photovoltaic power generation, and storage batteries are each set at 100 kW, which is also the capacity benchmark. The inverter circuit parameters of the wind power generator are set to Rwind = 0.08 Ω and L wind = 0.6 H; the inverter circuit parameters of the photovoltaic power generator are set to Rpv = 0.06 Ωand Lpv = 0.5 H; and the inverter circuit parameters of the storage batteries are set to Rbattery = 0.05Ω and Lbattery = 0.6 H. The gain, time constant, and difference coefficient of the microgrid system are set to KP = 4.8 Hz/m.u., TP = 3.5 s , and R1 = R2 = R3 = 2.4 Hz/p.u., respectively.
(2) Find the solution Pik + 1 (k ≥ 0) to the decoupled algebraic Riccati equation shown below
⎞ k+1 + Pik + 1 (A− P ⎟ i ⎠
(17)
4. Simulation results and analysis
AT Pi0 + Pi0 A−Pi0 Si Pi0 + Qi = 0
Sj P jk
(16)
u3 = F3 x (t ) = −R33 B3T P3 x (t ) = −0.4461Δf−0.1061ΔPwind−0.0719ΔPpv
To solve for the feedback Nash control strategy adopted by each game participant, ui , the matrix Pi in Eq. (4) must be obtained. However, in general, as a set of coupled algebraic Riccati equations, Eq. (4) is difficult to solve. Therefore, the solution is acquired using numerical algorithms using the iteration steps shown below. (1) Find the solution to the standard decoupled algebraic Riccati equation shown below, Pi0 , and use the solution as the matrix of initial values.
N
(15)
u2 = F2 x (t ) = −R22 B2T P2 x (t ) = −2.1323Δf−0.7109ΔPwind−1.7364ΔPpv
−1.1611ΔPbattery
∑
(14)
u1 = F1 x (t ) = −R11 B1T P1 x (t ) = −1.7347Δf−1.609ΔPwind−0.5122ΔPpv
3. Control strategies for micro-DGs based on linear quadratic differential game
⎛ A− ⎜ ⎝
(13)
The matrix Pi can be solved by setting the error ε to 10−9. Substituting Pi into Eq. (5), we can obtain the eigenvalues of this equation, of which all the real parts are less than zero, proving that this equation is stable. Therefore, the feedback Nash equilibrium strategy adopted for the microgrid system with wind power generation, photovoltaic power generation, and storage batteries is below.
(10)
1 Lbattery ⎤
B3T = ⎡ 0 0 0 ⎣
∞
J1 = (8)
0 0⎤ ⎦
Lwind
= ⎡0 0 ⎣
0
0
1
= ⎡0 ⎣
KP TP
Sj P jk )−Pik + 1 Si Pik + 1 + Qi
j≠i
4.1. Efficiency comparison with droop control
N
+
∑
P jk Sij P jk
=0
Based on the parameters of the microgrid system, the droop coefficient of the droop control is set to 0.25. A load disturbance of −20 kW (−0.2 p.u.) and 20 kW (0.2 p.u.) lasting for one second, which are shown in Figs. 2 Fig. 3 respectively, takes place at the sixth second. The linear quadratic differential game control (LQDGC) and the droop control (DC) are adopted in the same microgrid respectively. The deviations of frequency and active power of each micro-DG for both situations are presented in Figs. 4and 5 respectively. It can be seen from the figures that the suppression effect of the linear quadratic differential game control on the frequency deviation of the microgrid system after the disturbance is superior to that of droop control, because the linear quadratic differential game control prove the coordination of all the
j≠i
(3) Substitute Pik + 1 into Eq. (4). If the value of Eq. (4) is larger than the predetermined error ε , return to step 2); otherwise, let Pi = Pik + 1, termination routine.Qi and Rii,Rij reflect the interests of every participant in the game. The larger these values are, the more penalization on these corresponding interests. Basically, choosing a large value for Qi means trying to stabilize the system with the least possible changes in the outputs, frequency and active power, and vice versa. The choosing of Rii and Rij is similarly. To simplify the analysis while preserving the generality of the problem, and because the control decision of participant i is not impacted by the other participants’ control costs, let 13
Electrical Power and Energy Systems 97 (2018) 11–16
J. Zhang et al.
LQDGC
f / Hz
PL / p.u.
DC
t/s
t/s
(a) Deviation of frequency
Pwind / p.u.
Fig. 2. Disturbance of load decrease.
LQDGC DC
t/s
(b) Deviation of wind power output
Ppv / p.u.
Fig. 3. Disturbance of load increase (a) Deviation of frequency (b) Deviation of wind power output (c) Deviation of photovoltaic power output (d) Deviation of battery power output.
micro-DGs. 4.2. Impact of system variable change on control effect
LQDGC
In a practical microgrid system, the operation state variations of the micro-DGs can potentially cause system variable changes, affecting the efficiency of the control strategy and even the operational instability of the system. To examine the robustness of the linear quadratic differential game control strategy proposed in this paper, the inverter circuit parameters (i.e., resistance and inductance) of every micro-DG unit in the microgrid system vary by 10% and 20% increasing, respectively, without changing their control strategies. The simulation and comparison results are shown in Fig. 6. It can be seen from Fig. 6 that the microgrid system control strategy based on the theory of differential games results in superior frequency control performance, even when the model parameters are not accurate or different from the real situations, which is indicative of the excellent robustness of the system control strategy against microgrid system parameter variation. Because it is a decentralized control method, this system control strategy is able to better match the operation of the microgrid system integrated with micro-DGs distributed geographically.
DC
t/s
Pbattery / p.u.
(c) Deviation of photovoltaic power output
LQDGC DC
t/s
5. Conclusions
(d) Deviation of battery power output
In this paper, which considered the coordination and regulation of multiple DGs in a microgrid system, the linear quadratic differential game theory is adopted to the frequency control of the Microgrid integrated with multiple DGs. The control performance is compared to that of the droop control. In differential game theory, the optimal control strategy for all participants can be obtained by solving the Nash equilibrium based on the payoff functions of every participant. In this
Fig. 4. System response for the Disturbance of load decrease.
way, the tion and realized. strategy 14
interests of all participants can be considered, and coordinaregulation of the micro-DGs in the microgrid system can be Simulation results demonstrate that the proposed control to multiple DGs coordination and control for islanded
Electrical Power and Energy Systems 97 (2018) 11–16
J. Zhang et al.
DC (20% Parameters increasing) LQDGC (20% Parameters increasing) DC (10% Parameters increasing) LQDGC (10% Parameters increasing) LQDGC (0% Parameters Error)
f / Hz
f / Hz
DC (0% Parameters Error)
LQDGC DC
t/s
(a) Deviation of frequency t/s Fig. 6. Frequency deviation under parameters variance.
LQDGC DC
Pwind / p.u.
microgrid systems based on differential games results in superior control efficiency and robustness. Acknowledgement This research was supported by the National Natural Science Foundation of China (51007009), Science and Technology Foundation of Guizhou Province ([2016]1036), Guizhou Province Reform Foundation forPostgraduate Education ([2016]02).
t/s
(b) Deviation of wind power output
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