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Feasible Range and Optimal Value of the Virtual Impedance for Droop-Based Control of Microgrids Xiangyu Wu, Student Member, IEEE, Chen Shen, Senior Member, IEEE, and Reza Iravani, Fellow, IEEE
Abstract—This paper presents a systematic method to determine the feasible range and optimal value of the virtual impedance of the droop-based control to enhance a microgrid system performance with respect to power decoupling, reactive power sharing, system damping, and node voltage profile. A modified power flow analysis and an augmented small-signal dynamic model of the droop-based controlled microgrid, considering the impact of the virtual impedance, are developed. Subsequently, based on the developed methods, the feasible range of the virtual impedance, which can satisfy all the system performances requirements, is determined and presented. Based on a particle swarm optimization technique, an optimization process is introduced to select a virtual impedance value within the feasible range to achieve the overall optimal microgrid performance. Finally, simulation results in the PSCAD/EMTDC platform are provided to validate the feasibility and effectiveness of the proposed methods. Index Terms—Droop-control, microgrid, optimal virtual impedance, power decoupling, power flow, power sharing, small signal stability.
I. I NTRODUCTION ITH THE increasing degree of utilization of distributed generation (DG) units, the concept of microgrid to facilitate grid integration of various DG units has emerged as a viable technical and economical option. The droop-based control method has been identified as a viable approach for control and operation of DG units within a microgrid [1]–[3]. However, the basic droop-based control exhibits the following limitations: • coupling between active power and reactive power components under low X/R ratios [4]–[8], • improper reactive power sharing among DG units due to the line voltage drop [9], [10],
W
Manuscript received June 15, 2015; revised October 5, 2015; accepted January 12, 2016. Date of publication February 3, 2016; date of current version April 19, 2017. This work was supported in part by the National High Technology Research and Development Program of China (863 Program) under Grant 2012AA050217, and in part by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China under Grant 51321005. Paper no. TSG-00682-2015. X. Wu and C. Shen are with the Department of Electrical Engineering, Tsinghua University, Beijing 100084, China (e-mail:
[email protected];
[email protected]). R. Iravani is with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSG.2016.2519454
•
dynamic interactions among DG units and poor damping of dynamic modes when DG units operate as voltagecontrolled units [11]. One approach to address the above limitations is to embed “virtual impedance” in the control system of each DG unit. The virtual impedance can (i) provide power decoupling by artificially changing the X/R ratio [4]–[8], (ii) impose accurate reactive power sharing through dynamic regulation of voltage drop [5], [9], [10], and (iii) enhance dampings of the oscillatory modes [11]. However, if the virtual impedance is determined based on any of the above performance requirement, e.g., active/reactive power decoupling, the other requirements may not be satisfied or even degraded. Furthermore, due to the voltage drop associated with the virtual impedance, the node voltages may violate their limits. Thus the virtual impedance must be selected from a feasible region that meets all the requirements. Reference [11] introduces a feasible range for the virtual impedance to satisfy all performance criteria. The limitations of [11] are as follows. • The line impedance and the virtual impedance are combined and treated as one entity while they have different impacts on the node voltages, power losses, power decoupling, and system damping. This is specially the case when the real part of the virtual impedance is negative. • Node voltage limits are not taken into account. • Proper/accurate reactive power sharing is not considered. • Only identical DG units is assumed. Although a virtual impedance inside the feasible range can guarantee that the system performance criteria are satisfied, still there is a need to determine the virtual impedance (within the feasible range) that can provide the overall system optimal performance. The notion of optimal droop-based control has been investigated [12]–[14], however, to the best of our knowledge no systematic methodology to design the optimal virtual impedance to achieve the overall optimal system performance has been reported. This paper overcomes the limitation of [11] by proposing a systematic approach to identify the feasible range for the virtual impedances of a droop-based DG controllers to simultaneously satisfy system performance criteria with respect to power decoupling, reactive power sharing, system damping, and node voltage profile. This paper also presents a particle swarm optimization (PSO) approach to identify the optimal virtual impedance from the feasible range to optimally
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WU et al.: FEASIBLE RANGE AND OPTIMAL VALUE OF THE VIRTUAL IMPEDANCE FOR DROOP-BASED CONTROL
Fig. 1.
Fig. 2.
Details of power controller block.
Fig. 3.
Equivalent circuit for virtual impedance implementation.
Fig. 4.
Details of voltage controller block.
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Virtual impedance implementation structure.
achieve all the performance criteria. The proposed method is independent of the DG parameters and the microgrid topology, which is a salient improvement as compared with those of [11] and [15]. This paper also provides a modified power flow analysis that includes the impact of the virtual impedance and overcomes the shortcomings of the existing power flow approaches [12], [16], [17]. This paper presents analytical and time-domain simulation results to verify the proposed concepts. The rest of the paper is organized as follows. Section II presents a general overview of the subject. Section III describes the proposed power flow analysis and highlights its features. Section IV presents a small-signal dynamic model of a droop-based controlled microgrid which utilizes the virtual impedance. The feasible range of the virtual impedance is illustrated in Section V. Optimal design of the virtual impedance value is presented in Section VI. Case studies and simulation results are provided in Section VII. Conclusions are stated in Section VIII.
II. V IRTUAL I MPEDANCE I MPLEMENTATION The investigated virtual impedance concept for a droopbased control [6] is as shown in Fig. 1 for DGi . DGi is represented by a dc voltage source, a voltage source inverter, filter Lf Cf , and inductance Lc , and connected to the point of common coupling (PCC) through a feeder line. DGi includes an internal dq-current controller through which provides control over its terminal voltage, voi . Fig. 2 shows a block-representation of the power controller of Fig. 1. Based on Fig. 2, the P/f and Q/V droop equations for DGi are ωi = ω0 − mi Pi − P∗i , |Eoi | = V0 − ni Qi ,
(1) (2)
where mi and ni are active and reactive power droop coefficients, respectively, ωi is the frequency of DGi , ω0 is the rated system frequency, V0 is the no-load voltage for reactive power droop control, E0i is the voltage reference from the power controller, Pi and Qi are the output active and reactive power components of DGi at ω0 , respectively, and P∗i is the rated active power. mi and ni are determined by the following
equations based on the concept of [6] ω0 − ωmin mi = , (3-1) Pi max − P∗i Vn − Vmin , (3-2) ni = Qi max where ωmin and Vmin are the minimum allowable operating frequency and voltage magnitude of the microgrid, Vn is the rated voltage magnitude, and Pimax and Qimax are the maximal output active and reactive power components of DGi . Fig. 3 illustrates an equivalent circuit for realization of the virtual impedance Z0i = R0i + jX0i . From Fig. 3 v∗oi = Eoi − ioi Z0i ,
(4)
v∗oi
is the voltage reference for voi . Fig. 4 shows the where details of the voltage control block diagram of Fig. 1. Details of the current controller are given in [18]. III. P OWER F LOW A NALYSIS I NCLUDING V IRTUAL I MPEDANCE References [12] and [16] provide a power flow method for an islanded microgrid based on the droop control approach without including the effect of the virtual impedance on the power flow. This shortcoming is addressed in [17], however, it considers Eoi , Fig. 3, as the voltage reference for voi and not v∗oi in the power flow model. Thus, it does not accurately
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represent the virtual impedance. This paper also overcomes this limitation through taking v∗oi as the voltage reference for voi of DGi and includes the effect of the virtual impedance on the power flow (hereinafter referred to as “virtual impedance based power flow”). Consider the two types of nodes in the virtual impedance power flow, i.e., PQ nodes and DG nodes. The DG nodes utilize the control method of Fig. 1. The loads and the network nodes are represented by PQ nodes. For the PQ node number j, P∗j − Pj = 0,
(5)
− Qj = 0,
(6)
Q∗j
where P∗j and Q∗j are the pre-specified power components, and Pj and Qj are the injected power components and the unknowns are the real and imaginary voltage components (VlD j and VlQ j ) of the node. For DGi , assuming that voi accurately tracks its reference v∗oi in the steady state, from (1), (2), (4), we deduce (7) ω − ω0 + mi Pi − P∗i = 0, |Eoi | − V0 + ni Qi = 0, (8) (9) Eoi − voi − ioi Z0i = 0. Let δi denote the angle of Eoi and node 1 is taken as the reference node by setting δ1 = 0. voi and ioi can be expressed as voDi + jvoQi and ioDi + jioQi . Then, (9) can be rewritten as |Eoi | cos δi − voDi − ioDi R0i + ioQi X0i = 0, |Eoi | sin δi − voQi − ioQi R0i − ioDi X0i = 0.
(10) (11)
There are 4 equations for each DG node, and the corresponding 4 unknowns are |Eoi |, δi , voDi and voQi . There are only 3 unknowns for node 1 since δ1 is set to 0. The power flow can be calculated from (5)-(8), (10) and (11). If the number of PQ nodes and DG nodes are NPQ and NDG , then the number of power flow equations is 2NPQ + 4NDG . Considering that system frequency ω is also an unknown, then the total number of unknowns is also 2NPQ + 4NDG . The unknowns are given by x as in the Appendix. Assume Z0 denotes the virtual resistances R0i and virtual reactances X0i of all the DG units, i.e., Z0 = R01 , . . . R0NDG , X01 , . . . X0NDG . (12) Then the power flow equations, including virtual impedances, can be expressed as g(x, Z0 ) = 0.
(13)
IV. S MALL S IGNAL DYNAMIC M ODEL OF A D ROOP -BASED C ONTROLLED M ICROGRID I NCLUDING V IRTUAL I MPEDANCE The small-signal dynamic model presented in this section is based on the model of [18]. However, the effect of the virtual impedance is not included in [18]. In this paper, the small-signal model of a droop-based controlled microgrid is augmented to take into account the effect of the virtual impedance. Due to space limitations, developments of [18] are not repeated here.
Based on (2), in the local dq frame of DGi , we deduce Eodi = V0 − ni Qi , Eoqi = 0.
(14)
Similarly, (4) in the local dq frame can be expressed as ∗ vodi = Eodi − R0i iodi + X0i ioqi , (15) v∗oqi = Eoqi − R0i ioqi − X0i iodi and linearized as
v∗odi v∗oqi
⎤ ⎡ ⎤ δi ildqi = CPvi ⎣ Pi ⎦ + EPvi ⎣ vodqi ⎦, Qi iodqi
where CPvi and EPvi are 0 0 CPvi = 0 0 0 0 EPvi = 0 0
⎡
−ni , 0 0 0
0 0
−R0i −X0i
X0i . −R0i
(16)
(17)
Considering the dynamics of the power controller, voltage controller, current controller and LCL filter, the small signal dynamic model of DGi under the control method of Fig. 1 is X˙ invi = Ainvi [Xinvi ] + Binvi vbDQi + Biωcom [ωcom ], (18) where Xinvi includes the state variables of DGi , and details of Ainvi , Binvi and Biωcom are given in the Appendix. The last column of Ainvi is different from that of [18] due to the introduction of EPvi , which reflects the impact of the virtual impedance on the system small-signal dynamic model. The overall model of the islanded microgrid can be expressed as: ⎤ ⎡ ⎤ ⎡ X˙ inv Xinv ⎣ ˙ilineDQ ⎦ = Amg ⎣ ilineDQ ⎦, (19) iloadDQ ˙iloadDQ where Xinv , ilineDQ and iloadDQ are the state variables of DGs, lines and loads, respectively, and Amg is the system state matrix. The system steady-state information, including the virtual impedance, is embedded in Amg . Therefore, the virtual impedance based power flow should be calculated first to construct Amg and perform eigenvalue analysis. V. F EASIBLE R ANGE OF V IRTUAL I MPEDANCE Based on the virtual impedance based power flow model and the small-signal dynamic model of the previous sections, this section proposes a methodology to determine the feasible range of the virtual impedance which can satisfy all the system performance requirements. The feasible range is determined by a set of constraints which consider node voltage limits, power decoupling, system damping and reactive power sharing as follows. A. Node Voltage Limits All node voltages in a microgrid should be maintained within a permissible range. Therefore, the lowest (highest)
WU et al.: FEASIBLE RANGE AND OPTIMAL VALUE OF THE VIRTUAL IMPEDANCE FOR DROOP-BASED CONTROL
node voltage μLV (μHV ) should be larger (smaller) than the minimum (maximum) value μ0_LV (μ0_HV ), i.e., μLV ≥ μ0_LV , μHV ≤ μ0_HV ,
(20) (21)
where (20) and (21) specify lower and upper voltage limit. B. Power Decoupling Constraint Under low X/R ratios, the P/f and Q/V droop-based controls can exhibit strong coupling between P and Q [5], [6]. This paper provides a solution to this issue by introducing the virtual impedance to the droop control system. By emulating the effect of a physical impedance based on considering the voltage drop on the virtual impedance in the droop control (Fig. 1 and Fig. 3), the virtual impedance introduces a predominantly inductive impedance to increase the X/R ratio and effectively decouple P and Q [11]. To maintain a satisfactory power decoupling performance, the virtual impedance must be confined by a power decoupling constraint. The power decoupling constraint presented in this paper is based on the concept of [11]. Assume φoi is the angle difference between Eoi and U in Fig. 3. The measure of decoupling of voltage and angle for DGi are [11]
∂Qi ∂Pi
,
(22) dec_Eoi = ∂Eoi ∂Eoi
∂Pi ∂Qi
. (23) dec_φoi =
∂φoi ∂φoi Large dec_Eoi and dec_φoi provide a higher decoupling value of voltage and angle corresponding to Pi and Qi , respectively. The minimum value of voltage and angle decoupling of all the DGs in the system, μdec , is defined as μdec = min dec_Eo1 , dec_φo1 , . . . dec_EoNDG , dec_φoNDG . (24) If μdec is selected above the minimum allowable value of μ0_dec , i.e., μdec ≥ μ0_dec ,
(25)
it can guarantee that all the voltage and angle decoupling values are above μ0_dec . If (25) is satisfied with the implemented virtual impedances, then a satisfactory power decoupling performance is achieved. μ0_dec is usually selected as 1 to ensure that for all the DGs: 1) the coupling between Pi and φoi is stronger than that of Qi and φoi ; 2) the coupling between Qi and Eoi is stronger than that of Pi and Eoi . C. System Damping Constraint The virtual impedance of the droop-based control approach of Fig. 1 effectively increases the electrical distance between DGi and the other DG units and thus enhances the damping of the dynamic interaction modes of the DG units. Let eigk and ξk be the kth mode and its damping ratio respectively. Assume that the number of modes is Ne and only the modes which have real parts larger than σ (least damped modes) are considered. σ is selected based on multiple case
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studies and parameters of the study system. For the study system in Section VII, σ is selected as -100. μdamp1 is the minimum damping ratio of the modes which have real parts larger than σ, i.e., μdamp1 = min(ξ1 , . . . ξi , . . . ξNA ), i ∈ A A = j | Re eigj > σ, j = 1, . . . Ne ,
(26)
where A is the set of the modes with real parts larger than σ, and NA is the number of elements in A. Assume ξ(eigkA ) = μdamp1 , and kA ∈ A. Then μdamp2 is defined as the absolute value of the real part of the eigenvalue with the minimum damping ratio, i.e., μdamp2 = |Re(eigkA )|.
(27)
μdamp1 and μdamp2 should be larger than their respective minimum allowable values μ0_damp1 and μ0_damp2 to ensure that the system has adequate damping, i.e., μdamp1 ≥ μ0_damp1 ,
(28-a)
μdamp2 ≥ μ0_damp2 .
(28-b)
μ0_damp1 and μ0_damp2 are determined based on the engineering judgement and knowledge of the system. Equation (28) constitutes the system damping constraints. D. Reactive Power Sharing Constraint The reason for undesirable reactive power sharing is the different voltage drops on the line impedances of the DG units [6]. Proper selection of the virtual impedance can adjust the effective voltage drop on each line impedance and its corresponding virtual impedance to achieve the desired reactive power sharing [19]. For the study system in Fig. 6, without considering the local load of each DG unit, i.e., Load1 to Load4, the DG unit equivalent impedance is designed in inverse proportion to the DG maximal power to eliminate the reactive power sharing error [19], i.e., Re1 P1 max = Re2 P2 max = Re3 P3 max = Re4 P4 max ,
(29)
Xe1 Q1 max = Xe2 Q2 max = Xe3 Q3 max = Xe4 Q4 max , (30) where Rei (i=1,2,3,4) is the equivalent resistance of DGi , and Xei (i=1,2,3,4) is the equivalent reactance of DGi . The DG equivalent impedance in (29) and (30) consists of three series parts, i.e., Rei + jXei = (RLinei + jXLinei ) + jωLc + (R0i + jX0i ),
(31)
where RLinei and XLinei are the feeder resistance and reactance of the ith line in Fig. 6. From (31), the proper virtual impedance can adjust the DG equivalent impedance to satisfy (29) and (30). Therefore, the reactive power sharing error can be eliminated. Assume that the desired reactive power sharing ratios of the DGs are: Q1 : Q2 : . . . : QNDG = kq1 : kq2 : . . . : kqNDG .
(32)
Then we define the reactive power sharing error μQS as
N DG
kq1
(33) μQS =
Q1 − kq Qi . i i=1
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μQS is the sum of the reactive power sharing error between DG1 and the other DG units. A smaller μQS indicates that reactive power sharing is more accurate. Therefore, the reactive power sharing constraint should impose μQS to be smaller than a maximum permissible value μ0_QS , i.e., μQS ≤ μ0_QS .
(34)
μ0_QS can be approximated based on the empirical expression: μ0_QS = 0.25Q1 max (NDG − 1),
(35)
where Q1max is the maximum output reactive power of DG1 . Equation (35) indicates that the average reactive power sharing error of DG1 and DGi is limited to 25% of Q1max . If (34) is satisfied with the implemented virtual impedances, then a satisfactory reactive power sharing performance is achieved. Equations (20), (21), (25), (28) and (34) describe the virtual impedance constraints with respect to the defined performance criteria. The identification of the feasible range is based on a point-by-point approach [20].
Fig. 5.
The two kinds of normalization function.
Two normalization functions fx1 and fx2 are introduced as 1
(40)
A. System Performance Evaluation To assess the overall system performance, the comprehensive assessment index J is defined as (36)
where fvol , fdec , fdamp and fQS are the assessment indices with respect to the node voltage, power decoupling, system damping, and reactive power sharing performances, respectively, and ci (i=1. . . 4) is the corresponding weighting factor subject 4 to ci = 1. In addition, fvol and fdamp are each divided into i=1
μ0 − μs μs − μ0 , fx2 : τ = . ln 0.05 ln 0.05
(37) (38)
where c11 , c12 , c31 and c32 are the sub-weighting factors, and c11 + c12 = 1 and c31 + c32 = 1. Variables μLV , μHV , μdec , μdamp1 , μdamp2 and μQS , presented in Section V, can be used to evaluate the corresponding system performances. However, their dimensions are not the same, and thus index J of (36) cannot be readily used. Therefore, fLV , fHV , fdec , fdamp1 , fdamp2 and fQS are used to normalize μLV , μHV , μdec , μdamp1 , μdamp2 and μQS within the ranges of 0 to 1, respectively. After normalization, these assessment indices can be added, including appropriate weighing factors, to calculate J. Note that J also varies within 0 to 1.
(41)
Higher μLV , μdec , μdamp1 and μdamp2 provide higher system performance. Fig. 5 (a) also shows that a higher μ provides a higher fx1 . Therefore, fx1 is appropriate for normalizing μLV , μdec , μdamp1 and μdamp2 . Similarly, lower μHV and μQS provide higher system performance. Fig. 5 (b) shows that a lower μ provides a higher fx2 . Therefore, fx2 is appropriate for normalizing μHV and μQS . By substituting μLV , μ0_LV and τLV into fx1 , fLV can be obtained from (42-a) and similarly other indices are deduced as (μLV −μ0_LV ) , μ ≥ μ LV 0_LV , 1 μ −μ ( ) = 1 − e τHV HV 0_HV , μHV ≤ μ0_HV , − 1 (μ −μ ) = 1 − e τdec dec 0_dec , μdec ≥ μ0_dec , − 1 (μdamp1 −μ0_damp1 ) , = 1 − e τdamp1 −τ1
fLV = 1 − e fHV fdec fdamp1
LV
μdamp1 ≥ μ0_damp1 , −τ
two sub-indices, i.e.,
, μ ≤ μ0 ,
where μ0 and τ determine each function. Fig. 5 graphically shows fx1 and fx2 . A pre-specified point (μs , 0.95) on the plots of Fig. 5 is used to determine τ for (39) and (40), i.e., fx1 : τ =
The feasible range of the virtual impedances that meet the imposed performance constraints was specified in Section V. This section provides an approach to select feasible virtual impedances that also ensure the overall optimal performance of the system.
fvol = c11 fLV + c12 fHV , fdamp = c31 fdamp1 + c32 fdamp2 ,
(39)
1 τ (μ−μ0 )
fx2 = 1 − e
VI. O PTIMAL D ESIGN OF V IRTUAL I MPEDANCE
J = c1 fvol + c2 fdec + c3 fdamp + c4 fQS ,
fx1 = 1 − e− τ (μ−μ0 ) , μ ≥ μ0 ,
1 damp2
(42-a) (42-b) (42-c)
(42-d) (μdamp2 −μ0_damp2 )
, fdamp2 = 1 − e μdamp2 ≥ μ0_damp2 , 1 (μQS −μ0_QS ) , μQS ≤ μ0_QS . fQS = 1 − e τQS
(42-e) (42-f)
From (42) one observes that the definitions of assessment indices are consistent with the constraints of the feasible range of the virtual impedance. This ensures that the comprehensive assessment is defined inside the feasible range. B. Virtual Impedance Optimization To optimize virtual resistances and virtual reactances of all droop-controlled DGs of an islanded microgrid, index J must be maximized. Since J represents the overall system performance, the maximum J indicates that the overall system
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TABLE I E LECTRICAL PARAMETERS OF THE M ICROGRID
TABLE II C ONTROL PARAMETERS OF DG U NITS
Fig. 6.
Schematic diagram of the studied microgrid.
performance is optimal within the feasible range. Therefore, the optimization problem is formulated as Max J = c1 fvol + c2 fdec + c3 fdamp + c4 fQS , ⎧ g(x, Z0 ) = 0, ⎪ ⎪ ⎪ ⎪ μLV ≥ μ0_LV , ⎪ ⎪ ⎪ ⎪ ⎨ μHV ≤ μ0_HV , μdec ≥ μ0_dec , s.t. ⎪ ⎪ μ ⎪ damp1 ≥ μ0_damp1 , ⎪ ⎪ ⎪ μ ⎪ damp2 ≥ μ0_damp2 , ⎪ ⎩ μQS ≤ μ0_QS ,
(43)
where g(x, Z0 ) = 0 is the power flow constraint. The virtual impedance optimization problem is a non-linear and non-convex optimization problem which also includes eigenvalues constraints. In contrast to the classical gradient-based optimization methods, the heuristic optimization technique provides an effective solution method for this class of problems. In this study, a particle swarm optimization (PSO) [21] is adopted to solve the virtual impedance optimization problem. PSO has been widely used as one of the promising optimization technique due to its implementation simplicity and computational efficiency. Compared with other heuristic optimization techniques, such as genetic algorithm and ant colony optimization, PSO provides a flexible and wellbalanced mechanism to enhance the global and local exploration abilities [21]. VII. S TUDY R ESULTS AND D ISCUSSION A. Study System A single-line diagram of a 9-bus studied microgrid is presented in Fig. 6 [22]. It includes four DG units and five loads which are connected to a 0.38kV and 50Hz distribution system. Each DG unit is connected to its load bus through the coupling inductance Lc . Load1∼Load4 are the local loads of DG units. Line1∼Line4 are the feeder lines between each DG unit and the PCC bus. Load5 is located at the PCC bus. The microgrid is connected to the utility grid through a circuit breaker (CB) and a 10kV/0.38kV /Yg transformer. The
microgrid is operated in the islanded mode, i.e., the CB is open. The structure of each DG unit is as that of Fig. 1. Each three-phase load is represented by a series RL branch at each phase. Each feeder line is modelled by a lumped, series RL branch at each phase. It should be noted that when there are other types of DG units in the microgrid, e.g., solar_PV, wind power units or diesels, their impacts on the system performances can be considered by incorporating their steady-state and dynamic models into the power flow analysis (Section III) and the small-signal dynamic stability analysis (Section IV). However, the basic concept of the feasible range and optimal value of the virtual impedance remains the same. Electrical and control parameters of the system are shown in Table I and Table II, respectively. The modulation strategy of the inverter is a SPWM with the modulation frequency of 3500Hz. Table I indicates that X/R ratios of the lines are small [23] and thus the output active and reactive power components of the DG units are tightly coupled in the absence of the virtual impedance. From Table II one observes that the ratios of the four DG capacities are 4:6:5:3. Thus the ratios of droop coefficients of the DGs are also selected as 4:6:5:3, for both the active and reactive power components. Kpv and Kiv are the proportional and integral parameters of the voltage controller presented in Fig. 4, and Kpc and Kic are the proportional and integral parameters of the current controller in Fig. 1 [18]. The difference of voltage and current control parameters of the DG units reflect their different response times. Parameters of normalization functions in (42) for the study system of Fig. 6 are shown in Table III. In Table III, μ0 and μs in fLV and fHV are per unit values and μ0 is also the constraint value of the feasible range of the virtual impedance. B. Feasible Range of Virtual Impedance The microgrid operating point, for the study results of this subsection and the following Sections VII-C and VII-D, corresponds to the maximum active power load scenario, Table I, since it constitutes the most demanding operational condition
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TABLE III PARAMETERS OF N ORMALIZATION F UNCTIONS IN (36)
under different active power loads. To facilitate visualization of the feasible range and the optimal virtual impedance, the results of Fig. 7 are deduced based on the assumption that the virtual impedances of all DG units are identical and all feeder lines have the same length. It should be noted that the developments of Section III to Section VI are not subject to these assumptions. Fig. 7 (a) shows the feasible range of the virtual impedance when all lines are 300m long. The real (imaginary) part of the virtual impedance are specified on the R0 -axis (X0 -axis) of Fig. 7 (a) and the boundaries corresponding to voltage limits (20), (21), power decoupling (25), system damping (28), and reactive power sharing (34) are also plotted on the R0 X0 plane. The virtual impedance inside the feasible range can also satisfy the system performance criteria with respect to the node voltage profile, system damping, power decoupling, and reactive power sharing. Fig. 7 (a) concludes: • Further away from one boundary line indicates a “better” performance with respect to the corresponding criterion. • The negative virtual resistance facilitates power decoupling, reactive power sharing, and compliance with the lower voltage limit. However, it increases the DG voltage based on (4). The upper voltage boundary may influence the feasible range, specially subject to a large negative virtual resistance. Fig. 7 (b) and (c) show the feasible ranges of the virtual impedance when the lengths of feeder lines all are either 150m or 50m, respectively. Comparison of Fig. 7 (a), (b) and (c) indicates that when all the line lengths become longer, the feasible range of the virtual reactance does not significantly vary, while the feasible value of the virtual resistance becomes smaller. This is because a longer feeder line length needs a smaller and even negative virtual resistance to mitigate the effect of the line resistance. Fig. 8 corresponds to the scenario when the lengths of line1 to line4 are selected at 150m, 300m, 225m and 50m, and the virtual impedances of all DG units remain identical. Fig. 8 indicates that there is no feasible region for the virtual impedance. C. Optimal Virtual Impedance 1) Equal Feeder Line Lengths and Equal Virtual Impedances: Corresponding to equal feeder line lengths at either 50m, 150m or 300m, i.e., cases of Fig. 7 (c), (b) and (a), Table IV shows the real- and imaginary- part of the optimal virtual impedance and index J (43), and Table V shows the associated weighting factors (43). The weighting factors
Fig. 7.
Feasible range of virtual impedance with equal feeder line length.
TABLE IV O PTIMAL V IRTUAL I MPEDANCE VALUE IN E QUAL F EEDER L INE L ENGTH
are determined based on the subjective preference of the microgrid operator. The locations of the optimal virtual impedances are identified by “O” Fig. 7. 2) Unequal Virtual Impedances: This section in contrast to the previous section determines the optimal virtual impedances of the DG units assuming that they are not necessarily equal. Table VI shows four different feeder line length scenarios
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TABLE VIII O PTIMIZATION R ESULTS IN D IFFERENT L INE L ENGTH S CENARIOS
Fig. 8. Feasible range of virtual impedance with unequal feeder line length.
TABLE IX O PTIMIZATION R ESULTS OF S CENARIO 2 W ITH O PTIMAL V IRTUAL I MPEDANCE
TABLE V W EIGHTING FACTORS OF O BJECTIVE F UNCTION
TABLE VI F OUR T YPICAL L INE L ENGTH S CENARIOS TABLE X R ESULTS OF S CENARIO 2 W ITHOUT V IRTUAL I MPEDANCE
TABLE VII W EIGHTING FACTORS OF O BJECTIVE F UNCTION TABLE XI P OWER F LOW R ESULTS FOR S CENARIO 2 OF TABLE VIII
that are considered and Table VII shows the weighting factors of the optimization cost function (43). The optimal virtual impedances, corresponding to the four scenarios of Table VI are given in Table VIII. The values of index J, Table VIII, indicate that in all cases J is larger than 0.91, which reveals the overall system performance is highly desirable. Table IX (Table X) provide values of μ and fx from (42-a) to (42-f) for scenario 2 of Table VIII, with (without) considering the optimal virtual impedance. Table IX indicates that (i) the system performance meets all the required indices, (ii) the error of reactive power sharing is 0.999kVar and corresponds to a high degree of reactive power sharing, and (iii) the power decoupling degree, μdec , is 1.976 and indicates a high degree of power decoupling. Table X indicates that without the use of the virtual impedance, the system does not meet the performance requirements in term of power decoupling, system damping and reactive power sharing. D. Simulation Results Time-domain simulation studies in the PSCAD/EMTDC platform are carried out to verify the results of the proposed virtual impedance design approach. Table XI compares the power flow results obtained from PSCAD and the proposed power flow analysis for the case
of Scenario 2 of Table VIII. Table XI shows that the maximum deviation of the two methods, associated with the real and imaginary components of all the nodes voltages, are 0.14% and 1.28%, respectively. The good agreement between the results demonstrates the accuracy and the validity of the proposed approach. Fig. 9 compares the output real power components of the DG units obtained from PSCAD (solid line) and the proposed small-signal dynamic model (dashed line, index “lin”) for the case of Scenario 2 of Table VIII when the load power of load5 at time 0.5s is changed from 145kW+10kVar to 165kW+10kVar. The good agreement between the results also demonstrates the accuracy and the validity of the proposed small-signal dynamic model.
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Fig. 9. Output active power of the four DG units for Scenario 2 of Table VIII.
Fig. 12. Simulation results when a 2.5kW+20kVar shunt reactor is connected to the PCC bus at time 1s.
Fig. 10.
Output active power of the DG units.
Fig. 11.
Lowest node voltage of the system.
Fig. 10 (a) and (b) show the PSCAD-based simulation results when the optimal virtual impedance (point “O” of Fig. 7 (b)) and a non-optimal virtual impedance (point A of Fig. 7 (b)) which does not satisfy the system damping criterion are used in the control of DG units. Comparison of the corresponding results reveals the damping effect of the optimal virtual impedance on an oscillatory mode of the system that results in fluctuation of real-power components. Fig. 11 shows the PCC bus voltage (lowest node voltage) for the study system when the virtual impedance at time t=2.5s is changed from its optimal value (point “O” on Fig. 7 (b)) to a non-optimal value (point B on Fig. 7 (b)) which does not satisfy the lower voltage limit. Fig. 11 shows that the voltage decreases to 0.914, which is lower than the allowable value of 0.93. This is consistent with the position of point B in Fig. 7 (b). Fig. 12 shows the simulation results for the case study of Scenario 2 of Table VIII, when a 2.5kW+20kVar shunt reactor is connected to the PCC bus at time 1s. The total reactive power load of the system increases by 40%, from 50kVar to 70kVar, after the shunt reactor connection.
Fig. 12 (a) shows that the output active power components of the DG units almost remain unchanged. Fig. 12 (b) shows that the output reactive power components of the DG units increase to 15.1kVar, 23.0kVar, 18.8kVar and 10.3kVar respectively due to the extra reactive power consumption by the shunt reactor. The error of reactive power sharing increases from 0.999kVar (Table IX) to 1.193kVar. Fig. 12(c) shows that the PCC bus voltage decreases to 0.96 per unit due to the connection of the shunt reactor. The results of Fig. 12 indicate that the overall system performance is degraded after the shunt reactor connection. However, the system performance satisfies the node voltage limits (20) and the reactive power sharing constraint (34). Since the operating condition is changed, a new optimal system performance in the new operating condition can be obtained by incorporating the shunt reactor model into the system model and then updating the optimal virtual impedance. VIII. C ONCLUSION This paper presents a methodology to (i) specify the feasible range of the virtual impedance and (ii) identify the virtual impedance that provides the overall optimal performance for a microgrid. This paper also provides power flow and linearized dynamics models of the microgrid that incorporates the impacts of the virtual impedance. The analytical and time-domain simulation study results conclude that: • The virtual impedance has an impact on the power flow results. It can increase or decrease the node voltages, depending on the value of the virtual impedance. • The virtual impedance can enhance the damping of oscillatory modes through virtually increasing the electrical distance between one DG unit and the other DG units. • The feasible range of the virtual reactance is not highly sensitive to the variation of the feeder lengths, whereas the feasible value of the virtual resistance becomes smaller as the feeder lengths increase.
WU et al.: FEASIBLE RANGE AND OPTIMAL VALUE OF THE VIRTUAL IMPEDANCE FOR DROOP-BASED CONTROL
• A feasible range for the virtual impedance may not exist if the lengths of feeders within the microgrid are significantly different. A PPENDIX Matrices Ainvi , Binvi and Biwcom of (18) are ⎡ APi 0 ⎢ B C 0 V1i Pvi ⎢ ⎢ D C B CVi B C1i V1i Pvi C1i Ainvi = ⎢ ⎢ BLCL1i DC1i DV1i CPvi ⎢ ⎣ + BLCL2i T −1 0 0 BLCL1i DC1i CVi Vi + BLCL3i CPwi ⎤ 0 BPi ⎥ 0 BV1i EPVi + BV2i ⎥ 0 BC1i DV1i EPVi + BC1i DV2i + BC2i ⎥ ⎥, ALCLi + BLCL1i DC1i DV1i EPvi ⎦ BLCL1i CCi + BLCL1i (DC1i DV2i + DC2i ) (A1) ⎤ ⎡ ⎡ ⎤ 0 BPωcom ⎥ ⎢ ⎢ ⎥ 0 ⎥, Biωcom= ⎢ 0 ⎥. Binvi = ⎢ (A2) ⎦ ⎣ 0 ⎦ ⎣ 0 −1 0 BLCL2i TS Vector x in (13) is x = vlD1 , . . . vlDNPQ , vlQ1 ,
|Eo1 |, . . . EoNDG , δ2 , voD1 ,
...
voDNDG ,
voQ1 ,
...
vlQNPQ ,
...
δNDG ,
...
voQNDG ,
ω . (A3)
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Chen Shen (M’98–SM’07) received the B.E. degree in electrical power engineering and the Ph.D. degree in electrical power engineering from Tsinghua University in 1993 and 1998, respectively. Since 2009, he has been a Professor in the Electrical Engineering Department with Tsinghua University, where he is currently the Director of Power System Research Institute with the Department of Electrical Engineering. His research focuses on power systems analysis and control, including fast modeling and simulation of smart grids, stability analysis of power systems with wind generation, emergency control and risk assessment of power systems, planning, simulation operation, and control for microgrids. Reza Iravani (M’87–SM’00–F’03) received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering. He is currently a Professor with the University of Toronto, Toronto, Canada. His research interests include power electronics, power system dynamics and control.
