Voltage and frequency control of islanded microgrids: a plug-and-play ...
Voltage and frequency control of islanded microgrids: a plug-and-play approach Stefano Riverso†∗ , Fabio Sarzo† and Giancarlo Ferrari-Trecate† † Dipartimento
di Ingegneria Industriale e dell’Informazione, Universit`a degli Studi di Pavia ∗ [email protected], Corresponding author
Abstract—In this paper we propose a new decentralized control scheme for Islanded microGrids (ImGs) composed by the interconnection of Distributed Generation Units (DGUs). Local controllers regulate voltage and frequency at the Point of Common Coupling (PCC) of each DGU and they are able to guarantee stability of the overall ImG. The control design procedure is decentralized, since, besides two global scalar quantities, the synthesis of a local controller uses only information on the corresponding DGU and lines connected to it. Most important, our design procedure enables Plug-and-Play (PnP) operations: when a DGU is plugged in or out, only DGUs physically connected to it have to retune their local controllers. We study the performance of the proposed controllers simulating different scenarios in MatLab/Simulink and using indexes proposed in IEEE standards.
I. I NTRODUCTION In recent years, research on Islanded microGrids (ImG) has received major attention. ImGs are self-sufficient microgrids composed by several Distributed Generation Units (DGUs) and designed to operate safely and reliably in absence of a connection with the main grid. Besides fostering the use of renewable generation, ImGs bring distributed generation sources close to loads and allow power to be delivered to rural areas, remote lands, islands or harsh environments [1], [2]. The interest in ImGs is also motivated by microgrids that normally operate in grid-connected mode and that can be switched offgrid for guaranteeing users remain powered in presence of grid faults. For grid-connected microgrids, voltage and frequency are set by the main grid. However, in islanded mode, voltage and frequency control must be performed by DGUs. This is a challenging task, especially if one allows for (a) meshed topology with the goal increasing redundancy and robustness to line faults; (b) decentralized regulation of voltage and frequency, meaning that each DGU is equipped with a local controller and controllers do not communicate in real-time. As reviewed in [2], many available decentralized regulators are based on droop control [3], [4], [5], [6]. The main drawback of applying the droop method to ImGs is that frequency and amplitude deviations can be heavily affected by loads. Stability is another critical issue in ImGs controlled in a decentralized way [2]. The key challenge is to guarantee stability is not spoiled by the interaction among DGUs and, in the The research leading to these results has received funding from the European Union Seventh Framework Programme [FP7/2007-2013] under grant agreement n◦ 257462 HYCON2 Network of excellence.
context of droop control, this problem has been investigated only recently [7]. For regulators not based on droop control, almost all studies focused on radial microgrids (i.e. a DGU is connected to at most two other DGUs) while control of ImGs with meshed topology is still largely unexplored [2]. In this paper we consider the design of decentralized regulators for meshed ImGs with a view on decentralization of the synthesis procedure. More specifically, we develop a Plug-and-Play (PnP) design algorithm where the synthesis of a local controller for a DGU requires parameters of transmission lines connected to it, the knowledge of two global scalar parameters, but not specific information about any other DGU. This implies that when a DGU is plugged in or out, only DGUs physically connected to it have to retune their local controllers. PnP control design for general linear constrained systems has been proposed in [8], [9]. PnP design for ImGs is however different since it is based on the concept of neutral interactions [10] rather than on robustness against subsystem coupling. Furthermore, for achieving neutral interactions among DGUs, we exploit Quasi-Stationary Line (QSL) approximations of line dynamics [11]. Our theoretical results are backed up by numerical simulations using realistic models of Voltage Source Converters (VSCs), associated filters and transformers. As a first testbed, we consider two radially connected DGUs [12] and show that, in spite of QSL approximations, PnP controllers exhibit very good performances in terms of voltage tracking and robustness to nonlinear loads (the latter is measured as in the IEEE standards [13]). Additional experiments with unbalanced loads are given in [14] . Then, we consider a meshed ImG comprising 10 DGUs and simulate the plugging-in of a new DGUs. In spite of the presence of loops in the ImG topology, numerical results show that the addition of the new DGU is promptly compensated by local controllers. The paper is organized as follows. In Section II we present dynamical models of ImGs and introduce the adopted line approximation. In Section III we exploit the notion of neutral interactions for designing decentralized controllers and discuss how to perform PnP operations. In Section IV we discuss simulation results and Section V is devoted to conclusions. II. M ICROGRID MODEL In this section, we present dynamical models of ImGs used in this paper. For the sake of clearness, we first introduce an ImG consisting of two parallel DGUs and then generalize
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Fig. 1: Electrical scheme of an ImG composed of two radially connected DGUs with unmodeled loads.
the model to ImGs composed of N DGUs. As in [15], [12], [16], [17], [18], we consider the microgrid in Figure 1 where two DGUs, generally denoted with i and j, are connected through a three-phase line with non-zero impedance (Rij , Lij ). Each DGU provides real and reactive power for its local loads connected to the PCC. We assume loads are unknown and, similarly to [18], we treat load currents IL as disturbances for the DGUs. As shown in Figure 1, at the PCC of each area we use a shunt capacitance Ct for attenuating the impact of high-frequency harmonics of the load voltage. For ∗ ∈ {i, j}, the model of DGU ∗ in a dq-frame rotating with speed ω0 is given by the following state equations where ◦ ∈ {i, j} and ◦= 6 ∗ dV∗,dq + jω0 V∗,dq = dt k∗ 1 1 It∗,dq + I∗◦,dq − IL∗,dq Ct∗ Ct∗ Ct∗ dIt∗,dq + jω0 It∗,dq = dt Rt∗ k∗ 1 − It∗,dq − V∗,dq + Vt∗,dq Lt∗ Lt∗ Lt∗
dI
As in equation (T1.10) in [11], we set ij,dq = 0 and dt = 0 (see also [19] and references therein). Then, (2) gives the QSL model dIji,dq dt
I¯∗◦,dq =
V∗,dq V◦,dq − (R∗◦ + jω0 L∗◦ ) (R∗◦ + jω0 L∗◦ )
(1b)
(2)
Each state in (1) and (2) can be split in two parts (the real component d- and the imaginary component q- of dq reference frame, respectively). Note that by setting ∗ = i and ∗ = j in (2) one obtains two opposite line currents Iij and Iji so as to have a reference current entering in each DGU. In order to guarantee that Iij (t) = −Iji (t), ∀t ≥ 0, we introduce the following modeling assumption. Assumption 1. Initial states verify I∗◦,dq (0) = −I◦∗,dq (0). Moreover it holds L∗◦ = L◦∗ and R∗◦ = R◦∗ . In the next section, we propose an approximate model that allows one to describe each DGU as a dynamical system affected directly by state of the other DGU, hence avoiding the need of using the line current in the DGU state equations.
(3)
We then replace variables I∗◦,dq in (1a) with the right-hand side of (3). Splitting complex dq quantities in their d and q components one obtains the following model of DGU ∗ (namely ΣDGU ) [∗] x˙ [∗] (t) = A∗∗ x[∗] (t) + B∗ u[∗] (t) + M∗ d[∗] (t) + ξ[∗] (t) y[∗] (t) = C∗ x[∗] (t)
(1a)
For the line ∗◦ we obtain dI∗◦,dq + jω0 I∗◦,dq = dt 1 R∗◦ 1 V◦,dq − I∗◦,dq − V∗,dq L∗◦ L∗◦ L∗◦
A. QSL model
(4)
z[∗] (t) = H∗ y[∗] (t) where x[∗] = [V∗,d , V∗,q , It∗,d , It∗,q ]T , u[∗] = [Vt∗,d , Vt∗,q ]T , d[∗] = [IL∗,d , IL∗,q ]T , z[∗] = [V∗,d , V∗,q ]T are the state, the control input, the exogenous input and the controlled variables. The measurable output is y[∗] (t) and we assume y[∗] = x[∗] . Furthermore ξ[∗] (t) = A∗◦ x[◦] is the coupling with DGU ◦. The matrices of ΣDGU are obtained from (1) and (3) and they [∗] are collected in Appendix A.1 of [14]. Since the model of DGU ∗ does not depend on the dynamics of the line and the line dynamics are asymptotically stable, we have that stability of the ImG depends on the stability of ΣDGU and ΣDGU [i] [j] interconnected through the QSL model (3). We refer to the resulting system as QSL-ImG model. B. QSL model of a microgrid composed by N DGUs Next, we generalize model (4) to ImGs composed of N DGUs. Let D = {1, . . . , N }. Two DGUs i and j are neighbors if there is a transmission line connecting them and we denote with Ni ⊂ D the subset of neighbors of DGU i. Note that, if j ∈ Ni , then i ∈ Nj since the neighboring relation is symmetric. Then, the dynamics of DGU i, can be described by P model (4) setting ξ[i] = j∈Ni Aij x[j] (t). The new matrices of ΣDGU are given in Appendix A.2 of [14]. The overall QSL[i] ImG model is given by x(t) ˙ = Ax(t) + Bu(t) + Md(t)
(5a)
y(t) = Cx(t)
(5b)
z(t) = Hy(t)
(5c)
where x = (x[1] , . . . , x[N ] ) ∈ R4N , u = (u[1] , . . . , u[N ] ) ∈ R2N , d = (d[1] , . . . , d[N ] ) ∈ R2N , y = (y[1] , . . . , y[N ] ) ∈ R4N , z = (z[1] , . . . , z[N ] ) ∈ R2N and matrices A, B, M, C and H are reported in Appendix A.3 of [14].
such that the nominal closed-loop subsystem is asymptotically stable. From Lyapunov theory, we can achieve this aim if there exists a symmetric matrix Pi ∈ R6×6 , Pi ≥ 0 such that ˆi Ki )T Pi + Pi (Aˆii + B ˆi Ki ) < 0. (Aˆii + B
(9)
III. P LUG - AND -P LAY DECENTRALIZED VOLTAGE CONTROL
A. Decentralized control scheme with integrators In order to track a constant set-point zref (t), when d(t) is constant, we augment the ImG model with integrators as in Figure 2. The dynamics of the integrators is y[1] ...
y[N ] zref [1]
− +
.. . zref [N ] + −
R
dt
R
dt
v[1]
u[1]
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z[1] .. .
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.. .
u[N ]
... d[1]
= zref [i] (t) − Hi Ci x[i] (t),
Pi =
(6)
and hence, the DGU model augmented with integrators ˆ DGU ) is (namely Σ [i] ˆi u[i] (t)+ x ˆ˙ [i] (t) = Aˆii x ˆ[i] (t) + B X Aˆij x ˆ[j] (t)
ˆ i dˆ[i] (t)+ M (7)
j∈Ni
yˆ[i] (t) = Cˆi x ˆ[i] (t) ˆ i yˆ[i] (t) z[i] (t) = H where x ˆ[i] = [xT [i] , vi,d , vi,q ]T ∈ R6 is the state, yˆ[i] = x ˆ[i] ∈ 6 R is the measurable output and dˆ[i] = [d[i] , zref [i] ]T ∈ R4 collects the exogenous signals (the current of the load and the reference signals, in dq coordinates). Matrices in (7) can be obtained from the DGU model ΣDGU and (6). [i] B. Decentralized PnP control based on neutral interactions In this section, we present a decentralized control approach that will ensure asymptotic stability for the network ˆ DGU . Furthermore, local controllers of augmented DGUs Σ [i] will be synthesized in a decentralized fashion allowing PnP ˆ DGU with the following operations. We equip each DGU Σ [i] state-feedback controller C[i] :
u[i] (t) = Ki yˆ[i] (t) = Ki x ˆ[i] (t)
(10)
ˆi and Ki , for all ˆ B ˆ and K collect matrices Aˆij , B where A, i, j ∈ D. Note that (9) does not imply (10), i.e. coupling terms might spoil stability of the closed loop QSL-ImG model (see [14] for an example). In order to derive conditions such that (9) guarantees (10), we will exploit the following assumptions.
Fig. 2: Control scheme with integrators for ImG. v˙ [i] (t) = e[i] (t) = zref [i] (t) − z[i] (t)
ˆ + BK) ˆ T P + P(A ˆ + BK) ˆ (A 0 is a parameter common to all matrices Pi , i ∈ D. ηR (iii) It holds Cs Zij2 ≈ 0, ∀i ∈ D, ∀j ∈ Ni , where Zij = ij |Rij + jω0 Lij |. Within electrical networks, usually there are blocks of capacitors, positioned at various PCCs that can be switched in steps so as to modify their total capacitance. Therefore, Assumption 2-(i) is not restrictive. As for Assumption 2-(ii) we show In Section 3.2 of [14] that checking the existence of Pi as in (11) and Ki fulfilling (9) amounts to solve a convex optimization problem. Here, we just highlight that η > 0 and Cs > 0 are the only global parameters that must be known for designing local controllers. Remark 1. Assumption 2-(iii) can be fulfilled in different ηR ways. When an upper bound to all ratios Z 2ij (which depend ij upon line parameters only) is known, it is enough to set the control design parameter η sufficiently small. If, however, lines ηR are mainly inductive, one has Z 2ij ≈ 0 and bigger values of ij η can be used for synthesizing local controllers. Proposition 1. Let Assumption 2 holds. Then, the overall closed-loop QSL-ImG model is asymptotically stable. The proof of Proposition 1 is given in [14]. Performance of controllers C[i] can be enhanced by designing pre-filters of reference signals (C˜[i] ) and local compensator of measurable disturbances (N[i] ). These steps, that are customary in control design, are detailed in in Section 3.3. of [14]. The overall design procedure is summarized in Algorithm 1.
Algorithm 1 Design of controller C[i] and compensators C˜[i] ˆ DGU and N[i] for subsystem Σ [i] ˆ DGU as in (7) Input: DGU Σ [i] Output: Controller C[i] and, optionally, pre-filter C˜[i] and compensator N[i]
(Scenario 2). Parameters values for all DGUs are given in Appendix C of [14]. We highlight that they are comparable to those used in [15], [12] and [18]. Simulations have been conducted in MatLab/Simulink using the SimPowerSystem Toolbox and the PnPMPC-toolbox [20]. A. Scenario 1
IV. S IMULATION RESULTS In this section, we study performance brought about by PnP controllers described in Section III by using the ImG in Figure 1 with two DGUs (Scenario 1) and an ImG with 10 DGUs
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In this section, we discuss the operations for updating the controllers when DGUs are added to or removed from an ImG. The goal is to preserve stability of the new closed-loop system. As a starting point, we consider a microgrid composed by ˆ DGU , i ∈ D equipped with local controllers C[i] subsystems Σ [i] and compensators C˜[i] and N[i] , i ∈ D produced by Algorithm 1. Plugging-in operation Consider the plug-in of a new DGU ˆ DGU described by matrices, AˆN +1 N +1 , B ˆN +1 , CˆN +1 , Σ [N +1] ˆ N +1 , H ˆ N +1 and {AˆN +1 j }j∈N M . In particular, NN +1 N +1 ˆ DGU identifies the DGUs that are directly coupled to Σ [N +1] through transmission lines and {AˆN +1 j }j∈NN +1 are the corresponding coupling terms. For designing controller C[N +1] and compensators C˜[N +1] and N[N +i] , we execute Algorithm ˆ DGU , j ∈ NN +1 , have the new 1. We note that DGUs Σ [j] DGU ˆ neighbor Σ . Therefore, the redesign of controllers C[j] [N +1] ˜ and compensators C[j] and N[j] , ∀j ∈ NN +1 is needed because matrices Aˆjj , j ∈ NN +1 change. In conclusion, ˆ DGU is allowed only if Algorithm 1 does the plug-in of Σ [N +1] not stop in Step A when computing controllers C[k] for all k ∈ NN +1 ∪{N +1}. Note that, the redesign is not propagated further in the network, i.e. even without changing controllers C[i] , C˜[i] and N[i] , i 6∈ {N + 1} ∪ NN +1 asymptotic stability of the new overall closed-loop QSL-ImG model is ensured. Unplugging operation We consider the unplugging of DGU ˆ DGU , k ∈ D. Matrix Aˆjj of each Σ ˆ DGU , j ∈ Nk changes Σ [k] [j] ˆ DGU from the network. For this due to the disconnection of Σ [k] reason, for each j ∈ Nk , the redesign through Algorithm 1 of controllers C[j] and compensators C˜[j] and N[j] , j ∈ NN +1 , is ˆ DGU is allowed only if all these needed and unplugging of Σ [k] operations can be successfully terminated. As for the pluggingin operation, the re-design of local controllers C[j] , j ∈ / Nk is not required.
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C. PnP operations
For the sake of simplicity, we set i = 1 and j = 2 for the ImG in Figure 1 . Controllers for DGUs 1 and 2 have been designed running Algorithm 1 with i = 1 and i = 2. 1) Voltage tracking for DGU 1: In the first test, we assess the performance in tracking step changes in the voltage reference at P CC1 . For each DGU, we use an RL parallel load with constant parameters R = 76 Ω and L = 111.9 mH. The d and q components of the voltage at P CC1 are initially set at 0.2 per-unit (pu) and 0.6 pu and those of P CC2 are set at 0.5 pu and 0.7 pu, respectively. The reference signals of DGU 1, i.e. V1,d ref and V1,q ref , are affected by two step changes: the d component of the load voltage steps up to 0.3 pu at t = 0.5 s and the q component steps down to 0.5 pu at t = 1.5 s. Figure 3 shows the dynamic responses of the two DGUs to these changes. In particular, Figures 3a and 3b show good tracking performances with small interactions between the two DGUs. Figures 3c and 3d show the instantaneous voltage at P CC1 in the abc frame, during the two step changes of the reference signals. Note that the proposed decentralized control strategy ensures an excellent tracking of the references in about two cycles.
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(A) Find Ki such that (9) holds for a matrix Pi structured as in (11) by solving the optimization problem O in Section 3.2 of [14]. If it is not feasible stop C[i] cannot be designed). Optional steps (B) Design the asymptotically stable local pre-filter C˜[i] and compensator N[i] as in Section 3.3. of [14].
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Fig. 3: Scenario 1 - Performance of PnP decentralized voltage control in terms of set-point tracking for DGU 1. 2) Impact of a nonlinear load: In this test, we study the performance of our controllers in presence of a highly nonlinear load. Voltage references are initially set to Vd,ref = 0.8 pu and Vq,ref = 0.6 pu for both DGUs. At the beginning of the simulation, we connect at P CC1 and P CC2 the RL load
described in Section IV-A1. At t = 0.5 s, the load connected at P CC2 is suddenly replaced by a three-phase six-pulse diode rectifier. The rectifier produces a DC output voltage that feeds a purely resistive load with R = 120 Ω. We highlight that this is a standard test for assessing robustness of microgrid operations to nonlinearities (see, for example, Section VI.C in [18] and [13]). The results of the simulation are shown in Figure 4. In particular, Figure 4a shows the dq components of the load voltage at P CC2 which confirm the good tracking performance of the controller in spite of the inclusion of the rectifier. From Figure 4b one can notice that, except for short transients, local controllers successfully regulate the output sinusoidal waveforms at the desired levels. Figure 4c provides a plot of the Total Harmonic Distortion (THD) (expressed in %) of load voltage at P CC2 . We note that, after the connection of the rectifier, the THD value grows. However, the average value of THD after t = 0.5 is equal to 4.5% which is below the maximum limit (5%) recommended by IEEE standards in [13]. Considering that the rectifier input currents are highly distorted, as shown in Figure 4d, the control architecture ensures that the load is fed with high-quality voltages. V2d
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Fig. 4: Scenario 1 - Performance of PnP decentralized voltage control in presence of a highly nonlinear load. B. Scenario 2 In this second scenario, we consider the ImG depicted in black in Figure 5. Differently from Scenario 1, some DGUs have more than one neighbor and it is also present a loop that further complicates voltage regulation. ˆ DGU , i ∈ D = {1, . . . , 10}, we For each subsystem Σ [i] execute Algorithm 1 in order to design controllers C[i] and compensators C˜[i] and N[i] (see [14] for details). For evaluating the PnP capabilities of our control approach, ˆ DGU with Σ ˆ DGU and we simulate the connection of DGU Σ [11] [1] DGU ˆ Σ , as shown in Figure 5. Therefore, we have N11 = [6]
ˆ DGU , {1, 6}. As described in Section III-C, only subsystems Σ [j] j ∈ N11 must update their controllers C[j] and compensators C˜[j] and N[j] . This is done by re-executing Algorithm 1 for ˆ DGU , j ∈ N11 . Then, we execute Algorithm 1 each DGU Σ [j] for synthesizing C[11] , C˜[11] and N[11] for the new DGU. Since ˆ DGU is Algorithm 1 never stops in Step A, the addition of Σ [11] allowed and local controllers can be replaced by the new ones.
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Fig. 5: Scenario 2 - Scheme of the microgrid composed by 10 ˆ DGU (in gray). DGUs (in black) and plugging-in of Σ [11] ˆ DGU is executed at time The real-time plugging-in of Σ [11] t = 1.6 s. Before this event, references for DGUs 1-10 ˆ DGU is are those described in Section 4.2.1 of [14] and Σ [11] assumed to work isolated, tracking a reference voltage with dq components Vd,ref = 1 pu and Vq,ref = 0.5 pu, respectively. In order to test tracking performances after the addition of ˆ DGU , at time t = 1.8 s we change the d component of Σ [11] ˆ DGU to 0.6 pu. Figure 6 shows the voltage reference for Σ [11] ˆ DGU and its the dq component of the load voltages for Σ [11] DGU DGU ˆ ˆ neighbors Σ[1] and Σ[6] . In particular, from Figures 6a and 6b, we note that right after the plugging-in time (t = 1.6 ˆ DGU and Σ ˆ DGU deviate from s), the load voltages of Σ [1] [6] the respective reference signals. However, this deviation is immediately compensated and, after a short transient, the load voltages at P CC1 and P CC6 converge to their reference ˆ DGU : values. Similar remarks can be done for the new DGU Σ [11] as shown in Figure 6c, there is a short transient at the time of the plugging-in, that is effectively compensated by the control action. Moreover, the controller C[11] and compensators C˜[11] and N[11] ensure desired tracking when the reference signal Vd,ref steps down at t = 1.8 s. ˆ DGU Additional simulations illustrating the unplugging of Σ [2] are provided in [14]. V. C ONCLUSIONS In this paper, we presented a decentralized control scheme for guaranteeing voltage and frequency stability in ImGs. A key feature of our approach is that plugging-in and -out of DGUs requires to update only a limited number of local controllers. Furthermore, a global model of the ImG is not required in any design step. Numerical results in Section IV and in [14] confirm the feasibility of PnP control even for
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Fig. 6: Scenario 2 - Performance of decentralized voltage control during the plug-in operation (t = 1.6 s) and a change in the reference for DGU 11 (t = 1.8 s).
ImGs with meshed topology. Further studies will focus on coupling local voltage controllers with a higher control layer devoted to power flow regulation. R EFERENCES [1] R. H. Lasseter, “MicroGrids,” in Proceedings of IEEE Power Engineering Society Winter Meeting, vol. 1, New York, NY, USA, January 27-31, 2002, pp. 305–308.
[2] J. M. Guerrero, M. Chandorkar, T. Lee, and P. C. Loh, “Advanced Control Architectures for Intelligent Microgrids Part I: Decentralized and Hierarchical Control,” IEEE Transactions on Industrial Electronics, vol. 60, no. 4, pp. 1254–1262, 2013. [3] P. Piagi and R. H. Lasseter, “Autonomous control of microgrids,” in Proceedings of IEEE Power & Energy Society General Meeting, Montreal, QC, Canada, June 18-22, 2006. [4] J. M. Guerrero, J. Matas, L. G. De Vicu˜na, M. Castilla, and J. Miret, “Decentralized Control for Parallel Operation of Distributed Generation Inverters Using Resistive Output Impedance,” IEEE Transactions on Industrial Electronics, vol. 54, no. 2, pp. 994–1004, 2007. [5] A. Mehrizi-Sani and R. Iravani, “Potential-Function Based Control of a Microgrid in Islanded and Grid-Connected Modes,” IEEE Transactions on Power Systems, vol. 25, no. 4, pp. 1883–1891, 2010. [6] J. M. Guerrero, J. C. Vasquez, J. Matas, L. G. De Vicu˜na, and M. Castilla, “Hierarchical Control of Droop-Controlled AC and DC Microgrids A General Approach Toward Standardization,” IEEE Transactions on Industrial Electronics, vol. 58, no. 1, pp. 158–172, 2011. [7] J. W. Simpson-Porco, F. D¨orfler, and F. Bullo, “Synchronization and power sharing for droop-controlled inverters in islanded microgrids,” Automatica, vol. 49, no. 9, pp. 2603–2611, 2013. [8] S. Riverso, M. Farina, and G. Ferrari-Trecate, “Plug-and-Play Decentralized Model Predictive Control for Linear Systems,” IEEE Transactions on Automatic Control, vol. 58, no. 10, pp. 2608–2614, 2013. [9] ——, “Plug-and-Play Model Predictive Control based on robust control invariant sets,” Automatica, p. Accepted, 2014. [10] J. Lunze, Feedback control of large scale systems. Upper Saddle River, NJ, USA: Prentice Hall, Systems and Control Engineering, 1992. [11] V. Venkatasubramanian, H. Schattler, and J. Zaborszky, “Fast TimeVarying Phasor Analysis in the Balanced Three-Phase Large Electric Power System,” IEEE Transactions on Automatic Control, vol. 40, no. 11, pp. 1975–1982, 1995. [12] R. Moradi, H. Karimi, and M. Karimi-Ghartemani, “Robust decentralized control for islanded operation of two radially connected DG systems,” in Proceedings of IEEE International Symposium on Industrial Electronics (ISIE 2010), Bari, Italy, 4-7 July, 2010, pp. 2272–2277. [13] IEEE, “IEEE Recommended Practice for Monitoring Electric Power Quality,” IEEE Std 1159-2009, 3 Park Avenue, New York, NY 100165997, USA, 29 June, Tech. Rep. June, 2009. [14] S. Riverso, F. Sarzo, and G. Ferrari-Trecate, “Plug-and-play voltage and frequency control of islanded microgrids with meshed topology,” Dipartimento di Informatica e Sistemistica, Universit`a degli Studi di Pavia, Pavia, Italy, Tech. Rep., 2014. [Online]. Available: http://arxiv.org/ [15] H. Karimi, H. Nikkhajoei, and R. Iravani, “Control of an ElectronicallyCoupled Distributed Resource Unit Subsequent to an Islanding Event,” IEEE Transactions on Power Delivery, vol. 23, no. 1, pp. 493–501, 2008. [16] A. H. Etemadi, E. J. Davison, and R. Iravani, “A Decentralized Robust Control Strategy for Multi-DER Microgrids - Part I: Fundamental Concepts,” IEEE Transactions on Power Delivery, vol. 27, no. 4, pp. 1843–1853, 2012. [17] ——, “A Decentralized Robust Control Strategy for Multi-DER Microgrids - Part II: Performance Evaluation,” IEEE Transactions on Power Delivery, vol. 27, no. 4, pp. 1854–1861, 2012. [18] M. Babazadeh and H. Karimi, “A Robust Two-Degree-of-Freedom Control Strategy for an Islanded Microgrid,” IEEE Transactions on Power Delivery, vol. 28, no. 3, pp. 1339–1347, 2013. [19] H. Akagi, E. H. Watanabe, and M. Aredes, Instantaneous power theory and applications to power conditioning. Hoboken, New Jersey, USA: John Wiley & Sons, IEEE Press series on Power Engineering, 2007. [20] S. Riverso, A. Battocchio, and G. Ferrari-Trecate, “PnPMPC: a toolbox for MatLab,” 2012. [Online]. Available: http://sisdin.unipv.it/pnpmpc/pnpmpc.php
di Ingegneria Industriale e dell’Informazione, Universit`a degli Studi di Pavia ∗ [email protected], Corresponding author
Abstract—In this paper we propose a new decentralized control scheme for Islanded microGrids (ImGs) composed by the interconnection of Distributed Generation Units (DGUs). Local controllers regulate voltage and frequency at the Point of Common Coupling (PCC) of each DGU and they are able to guarantee stability of the overall ImG. The control design procedure is decentralized, since, besides two global scalar quantities, the synthesis of a local controller uses only information on the corresponding DGU and lines connected to it. Most important, our design procedure enables Plug-and-Play (PnP) operations: when a DGU is plugged in or out, only DGUs physically connected to it have to retune their local controllers. We study the performance of the proposed controllers simulating different scenarios in MatLab/Simulink and using indexes proposed in IEEE standards.
I. I NTRODUCTION In recent years, research on Islanded microGrids (ImG) has received major attention. ImGs are self-sufficient microgrids composed by several Distributed Generation Units (DGUs) and designed to operate safely and reliably in absence of a connection with the main grid. Besides fostering the use of renewable generation, ImGs bring distributed generation sources close to loads and allow power to be delivered to rural areas, remote lands, islands or harsh environments [1], [2]. The interest in ImGs is also motivated by microgrids that normally operate in grid-connected mode and that can be switched offgrid for guaranteeing users remain powered in presence of grid faults. For grid-connected microgrids, voltage and frequency are set by the main grid. However, in islanded mode, voltage and frequency control must be performed by DGUs. This is a challenging task, especially if one allows for (a) meshed topology with the goal increasing redundancy and robustness to line faults; (b) decentralized regulation of voltage and frequency, meaning that each DGU is equipped with a local controller and controllers do not communicate in real-time. As reviewed in [2], many available decentralized regulators are based on droop control [3], [4], [5], [6]. The main drawback of applying the droop method to ImGs is that frequency and amplitude deviations can be heavily affected by loads. Stability is another critical issue in ImGs controlled in a decentralized way [2]. The key challenge is to guarantee stability is not spoiled by the interaction among DGUs and, in the The research leading to these results has received funding from the European Union Seventh Framework Programme [FP7/2007-2013] under grant agreement n◦ 257462 HYCON2 Network of excellence.
context of droop control, this problem has been investigated only recently [7]. For regulators not based on droop control, almost all studies focused on radial microgrids (i.e. a DGU is connected to at most two other DGUs) while control of ImGs with meshed topology is still largely unexplored [2]. In this paper we consider the design of decentralized regulators for meshed ImGs with a view on decentralization of the synthesis procedure. More specifically, we develop a Plug-and-Play (PnP) design algorithm where the synthesis of a local controller for a DGU requires parameters of transmission lines connected to it, the knowledge of two global scalar parameters, but not specific information about any other DGU. This implies that when a DGU is plugged in or out, only DGUs physically connected to it have to retune their local controllers. PnP control design for general linear constrained systems has been proposed in [8], [9]. PnP design for ImGs is however different since it is based on the concept of neutral interactions [10] rather than on robustness against subsystem coupling. Furthermore, for achieving neutral interactions among DGUs, we exploit Quasi-Stationary Line (QSL) approximations of line dynamics [11]. Our theoretical results are backed up by numerical simulations using realistic models of Voltage Source Converters (VSCs), associated filters and transformers. As a first testbed, we consider two radially connected DGUs [12] and show that, in spite of QSL approximations, PnP controllers exhibit very good performances in terms of voltage tracking and robustness to nonlinear loads (the latter is measured as in the IEEE standards [13]). Additional experiments with unbalanced loads are given in [14] . Then, we consider a meshed ImG comprising 10 DGUs and simulate the plugging-in of a new DGUs. In spite of the presence of loops in the ImG topology, numerical results show that the addition of the new DGU is promptly compensated by local controllers. The paper is organized as follows. In Section II we present dynamical models of ImGs and introduce the adopted line approximation. In Section III we exploit the notion of neutral interactions for designing decentralized controllers and discuss how to perform PnP operations. In Section IV we discuss simulation results and Section V is devoted to conclusions. II. M ICROGRID MODEL In this section, we present dynamical models of ImGs used in this paper. For the sake of clearness, we first introduce an ImG consisting of two parallel DGUs and then generalize
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Fig. 1: Electrical scheme of an ImG composed of two radially connected DGUs with unmodeled loads.
the model to ImGs composed of N DGUs. As in [15], [12], [16], [17], [18], we consider the microgrid in Figure 1 where two DGUs, generally denoted with i and j, are connected through a three-phase line with non-zero impedance (Rij , Lij ). Each DGU provides real and reactive power for its local loads connected to the PCC. We assume loads are unknown and, similarly to [18], we treat load currents IL as disturbances for the DGUs. As shown in Figure 1, at the PCC of each area we use a shunt capacitance Ct for attenuating the impact of high-frequency harmonics of the load voltage. For ∗ ∈ {i, j}, the model of DGU ∗ in a dq-frame rotating with speed ω0 is given by the following state equations where ◦ ∈ {i, j} and ◦= 6 ∗ dV∗,dq + jω0 V∗,dq = dt k∗ 1 1 It∗,dq + I∗◦,dq − IL∗,dq Ct∗ Ct∗ Ct∗ dIt∗,dq + jω0 It∗,dq = dt Rt∗ k∗ 1 − It∗,dq − V∗,dq + Vt∗,dq Lt∗ Lt∗ Lt∗
dI
As in equation (T1.10) in [11], we set ij,dq = 0 and dt = 0 (see also [19] and references therein). Then, (2) gives the QSL model dIji,dq dt
I¯∗◦,dq =
V∗,dq V◦,dq − (R∗◦ + jω0 L∗◦ ) (R∗◦ + jω0 L∗◦ )
(1b)
(2)
Each state in (1) and (2) can be split in two parts (the real component d- and the imaginary component q- of dq reference frame, respectively). Note that by setting ∗ = i and ∗ = j in (2) one obtains two opposite line currents Iij and Iji so as to have a reference current entering in each DGU. In order to guarantee that Iij (t) = −Iji (t), ∀t ≥ 0, we introduce the following modeling assumption. Assumption 1. Initial states verify I∗◦,dq (0) = −I◦∗,dq (0). Moreover it holds L∗◦ = L◦∗ and R∗◦ = R◦∗ . In the next section, we propose an approximate model that allows one to describe each DGU as a dynamical system affected directly by state of the other DGU, hence avoiding the need of using the line current in the DGU state equations.
(3)
We then replace variables I∗◦,dq in (1a) with the right-hand side of (3). Splitting complex dq quantities in their d and q components one obtains the following model of DGU ∗ (namely ΣDGU ) [∗] x˙ [∗] (t) = A∗∗ x[∗] (t) + B∗ u[∗] (t) + M∗ d[∗] (t) + ξ[∗] (t) y[∗] (t) = C∗ x[∗] (t)
(1a)
For the line ∗◦ we obtain dI∗◦,dq + jω0 I∗◦,dq = dt 1 R∗◦ 1 V◦,dq − I∗◦,dq − V∗,dq L∗◦ L∗◦ L∗◦
A. QSL model
(4)
z[∗] (t) = H∗ y[∗] (t) where x[∗] = [V∗,d , V∗,q , It∗,d , It∗,q ]T , u[∗] = [Vt∗,d , Vt∗,q ]T , d[∗] = [IL∗,d , IL∗,q ]T , z[∗] = [V∗,d , V∗,q ]T are the state, the control input, the exogenous input and the controlled variables. The measurable output is y[∗] (t) and we assume y[∗] = x[∗] . Furthermore ξ[∗] (t) = A∗◦ x[◦] is the coupling with DGU ◦. The matrices of ΣDGU are obtained from (1) and (3) and they [∗] are collected in Appendix A.1 of [14]. Since the model of DGU ∗ does not depend on the dynamics of the line and the line dynamics are asymptotically stable, we have that stability of the ImG depends on the stability of ΣDGU and ΣDGU [i] [j] interconnected through the QSL model (3). We refer to the resulting system as QSL-ImG model. B. QSL model of a microgrid composed by N DGUs Next, we generalize model (4) to ImGs composed of N DGUs. Let D = {1, . . . , N }. Two DGUs i and j are neighbors if there is a transmission line connecting them and we denote with Ni ⊂ D the subset of neighbors of DGU i. Note that, if j ∈ Ni , then i ∈ Nj since the neighboring relation is symmetric. Then, the dynamics of DGU i, can be described by P model (4) setting ξ[i] = j∈Ni Aij x[j] (t). The new matrices of ΣDGU are given in Appendix A.2 of [14]. The overall QSL[i] ImG model is given by x(t) ˙ = Ax(t) + Bu(t) + Md(t)
(5a)
y(t) = Cx(t)
(5b)
z(t) = Hy(t)
(5c)
where x = (x[1] , . . . , x[N ] ) ∈ R4N , u = (u[1] , . . . , u[N ] ) ∈ R2N , d = (d[1] , . . . , d[N ] ) ∈ R2N , y = (y[1] , . . . , y[N ] ) ∈ R4N , z = (z[1] , . . . , z[N ] ) ∈ R2N and matrices A, B, M, C and H are reported in Appendix A.3 of [14].
such that the nominal closed-loop subsystem is asymptotically stable. From Lyapunov theory, we can achieve this aim if there exists a symmetric matrix Pi ∈ R6×6 , Pi ≥ 0 such that ˆi Ki )T Pi + Pi (Aˆii + B ˆi Ki ) < 0. (Aˆii + B
(9)
III. P LUG - AND -P LAY DECENTRALIZED VOLTAGE CONTROL
A. Decentralized control scheme with integrators In order to track a constant set-point zref (t), when d(t) is constant, we augment the ImG model with integrators as in Figure 2. The dynamics of the integrators is y[1] ...
y[N ] zref [1]
− +
.. . zref [N ] + −
R
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dt
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.. .
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= zref [i] (t) − Hi Ci x[i] (t),
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(6)
and hence, the DGU model augmented with integrators ˆ DGU ) is (namely Σ [i] ˆi u[i] (t)+ x ˆ˙ [i] (t) = Aˆii x ˆ[i] (t) + B X Aˆij x ˆ[j] (t)
ˆ i dˆ[i] (t)+ M (7)
j∈Ni
yˆ[i] (t) = Cˆi x ˆ[i] (t) ˆ i yˆ[i] (t) z[i] (t) = H where x ˆ[i] = [xT [i] , vi,d , vi,q ]T ∈ R6 is the state, yˆ[i] = x ˆ[i] ∈ 6 R is the measurable output and dˆ[i] = [d[i] , zref [i] ]T ∈ R4 collects the exogenous signals (the current of the load and the reference signals, in dq coordinates). Matrices in (7) can be obtained from the DGU model ΣDGU and (6). [i] B. Decentralized PnP control based on neutral interactions In this section, we present a decentralized control approach that will ensure asymptotic stability for the network ˆ DGU . Furthermore, local controllers of augmented DGUs Σ [i] will be synthesized in a decentralized fashion allowing PnP ˆ DGU with the following operations. We equip each DGU Σ [i] state-feedback controller C[i] :
u[i] (t) = Ki yˆ[i] (t) = Ki x ˆ[i] (t)
(10)
ˆi and Ki , for all ˆ B ˆ and K collect matrices Aˆij , B where A, i, j ∈ D. Note that (9) does not imply (10), i.e. coupling terms might spoil stability of the closed loop QSL-ImG model (see [14] for an example). In order to derive conditions such that (9) guarantees (10), we will exploit the following assumptions.
Fig. 2: Control scheme with integrators for ImG. v˙ [i] (t) = e[i] (t) = zref [i] (t) − z[i] (t)
ˆ + BK) ˆ T P + P(A ˆ + BK) ˆ (A 0 is a parameter common to all matrices Pi , i ∈ D. ηR (iii) It holds Cs Zij2 ≈ 0, ∀i ∈ D, ∀j ∈ Ni , where Zij = ij |Rij + jω0 Lij |. Within electrical networks, usually there are blocks of capacitors, positioned at various PCCs that can be switched in steps so as to modify their total capacitance. Therefore, Assumption 2-(i) is not restrictive. As for Assumption 2-(ii) we show In Section 3.2 of [14] that checking the existence of Pi as in (11) and Ki fulfilling (9) amounts to solve a convex optimization problem. Here, we just highlight that η > 0 and Cs > 0 are the only global parameters that must be known for designing local controllers. Remark 1. Assumption 2-(iii) can be fulfilled in different ηR ways. When an upper bound to all ratios Z 2ij (which depend ij upon line parameters only) is known, it is enough to set the control design parameter η sufficiently small. If, however, lines ηR are mainly inductive, one has Z 2ij ≈ 0 and bigger values of ij η can be used for synthesizing local controllers. Proposition 1. Let Assumption 2 holds. Then, the overall closed-loop QSL-ImG model is asymptotically stable. The proof of Proposition 1 is given in [14]. Performance of controllers C[i] can be enhanced by designing pre-filters of reference signals (C˜[i] ) and local compensator of measurable disturbances (N[i] ). These steps, that are customary in control design, are detailed in in Section 3.3. of [14]. The overall design procedure is summarized in Algorithm 1.
Algorithm 1 Design of controller C[i] and compensators C˜[i] ˆ DGU and N[i] for subsystem Σ [i] ˆ DGU as in (7) Input: DGU Σ [i] Output: Controller C[i] and, optionally, pre-filter C˜[i] and compensator N[i]
(Scenario 2). Parameters values for all DGUs are given in Appendix C of [14]. We highlight that they are comparable to those used in [15], [12] and [18]. Simulations have been conducted in MatLab/Simulink using the SimPowerSystem Toolbox and the PnPMPC-toolbox [20]. A. Scenario 1
IV. S IMULATION RESULTS In this section, we study performance brought about by PnP controllers described in Section III by using the ImG in Figure 1 with two DGUs (Scenario 1) and an ImG with 10 DGUs
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In this section, we discuss the operations for updating the controllers when DGUs are added to or removed from an ImG. The goal is to preserve stability of the new closed-loop system. As a starting point, we consider a microgrid composed by ˆ DGU , i ∈ D equipped with local controllers C[i] subsystems Σ [i] and compensators C˜[i] and N[i] , i ∈ D produced by Algorithm 1. Plugging-in operation Consider the plug-in of a new DGU ˆ DGU described by matrices, AˆN +1 N +1 , B ˆN +1 , CˆN +1 , Σ [N +1] ˆ N +1 , H ˆ N +1 and {AˆN +1 j }j∈N M . In particular, NN +1 N +1 ˆ DGU identifies the DGUs that are directly coupled to Σ [N +1] through transmission lines and {AˆN +1 j }j∈NN +1 are the corresponding coupling terms. For designing controller C[N +1] and compensators C˜[N +1] and N[N +i] , we execute Algorithm ˆ DGU , j ∈ NN +1 , have the new 1. We note that DGUs Σ [j] DGU ˆ neighbor Σ . Therefore, the redesign of controllers C[j] [N +1] ˜ and compensators C[j] and N[j] , ∀j ∈ NN +1 is needed because matrices Aˆjj , j ∈ NN +1 change. In conclusion, ˆ DGU is allowed only if Algorithm 1 does the plug-in of Σ [N +1] not stop in Step A when computing controllers C[k] for all k ∈ NN +1 ∪{N +1}. Note that, the redesign is not propagated further in the network, i.e. even without changing controllers C[i] , C˜[i] and N[i] , i 6∈ {N + 1} ∪ NN +1 asymptotic stability of the new overall closed-loop QSL-ImG model is ensured. Unplugging operation We consider the unplugging of DGU ˆ DGU , k ∈ D. Matrix Aˆjj of each Σ ˆ DGU , j ∈ Nk changes Σ [k] [j] ˆ DGU from the network. For this due to the disconnection of Σ [k] reason, for each j ∈ Nk , the redesign through Algorithm 1 of controllers C[j] and compensators C˜[j] and N[j] , j ∈ NN +1 , is ˆ DGU is allowed only if all these needed and unplugging of Σ [k] operations can be successfully terminated. As for the pluggingin operation, the re-design of local controllers C[j] , j ∈ / Nk is not required.
V1,dq (pu)
C. PnP operations
For the sake of simplicity, we set i = 1 and j = 2 for the ImG in Figure 1 . Controllers for DGUs 1 and 2 have been designed running Algorithm 1 with i = 1 and i = 2. 1) Voltage tracking for DGU 1: In the first test, we assess the performance in tracking step changes in the voltage reference at P CC1 . For each DGU, we use an RL parallel load with constant parameters R = 76 Ω and L = 111.9 mH. The d and q components of the voltage at P CC1 are initially set at 0.2 per-unit (pu) and 0.6 pu and those of P CC2 are set at 0.5 pu and 0.7 pu, respectively. The reference signals of DGU 1, i.e. V1,d ref and V1,q ref , are affected by two step changes: the d component of the load voltage steps up to 0.3 pu at t = 0.5 s and the q component steps down to 0.5 pu at t = 1.5 s. Figure 3 shows the dynamic responses of the two DGUs to these changes. In particular, Figures 3a and 3b show good tracking performances with small interactions between the two DGUs. Figures 3c and 3d show the instantaneous voltage at P CC1 in the abc frame, during the two step changes of the reference signals. Note that the proposed decentralized control strategy ensures an excellent tracking of the references in about two cycles.
V1,abc (pu)
(A) Find Ki such that (9) holds for a matrix Pi structured as in (11) by solving the optimization problem O in Section 3.2 of [14]. If it is not feasible stop C[i] cannot be designed). Optional steps (B) Design the asymptotically stable local pre-filter C˜[i] and compensator N[i] as in Section 3.3. of [14].
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Fig. 3: Scenario 1 - Performance of PnP decentralized voltage control in terms of set-point tracking for DGU 1. 2) Impact of a nonlinear load: In this test, we study the performance of our controllers in presence of a highly nonlinear load. Voltage references are initially set to Vd,ref = 0.8 pu and Vq,ref = 0.6 pu for both DGUs. At the beginning of the simulation, we connect at P CC1 and P CC2 the RL load
described in Section IV-A1. At t = 0.5 s, the load connected at P CC2 is suddenly replaced by a three-phase six-pulse diode rectifier. The rectifier produces a DC output voltage that feeds a purely resistive load with R = 120 Ω. We highlight that this is a standard test for assessing robustness of microgrid operations to nonlinearities (see, for example, Section VI.C in [18] and [13]). The results of the simulation are shown in Figure 4. In particular, Figure 4a shows the dq components of the load voltage at P CC2 which confirm the good tracking performance of the controller in spite of the inclusion of the rectifier. From Figure 4b one can notice that, except for short transients, local controllers successfully regulate the output sinusoidal waveforms at the desired levels. Figure 4c provides a plot of the Total Harmonic Distortion (THD) (expressed in %) of load voltage at P CC2 . We note that, after the connection of the rectifier, the THD value grows. However, the average value of THD after t = 0.5 is equal to 4.5% which is below the maximum limit (5%) recommended by IEEE standards in [13]. Considering that the rectifier input currents are highly distorted, as shown in Figure 4d, the control architecture ensures that the load is fed with high-quality voltages. V2d
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Fig. 4: Scenario 1 - Performance of PnP decentralized voltage control in presence of a highly nonlinear load. B. Scenario 2 In this second scenario, we consider the ImG depicted in black in Figure 5. Differently from Scenario 1, some DGUs have more than one neighbor and it is also present a loop that further complicates voltage regulation. ˆ DGU , i ∈ D = {1, . . . , 10}, we For each subsystem Σ [i] execute Algorithm 1 in order to design controllers C[i] and compensators C˜[i] and N[i] (see [14] for details). For evaluating the PnP capabilities of our control approach, ˆ DGU with Σ ˆ DGU and we simulate the connection of DGU Σ [11] [1] DGU ˆ Σ , as shown in Figure 5. Therefore, we have N11 = [6]
ˆ DGU , {1, 6}. As described in Section III-C, only subsystems Σ [j] j ∈ N11 must update their controllers C[j] and compensators C˜[j] and N[j] . This is done by re-executing Algorithm 1 for ˆ DGU , j ∈ N11 . Then, we execute Algorithm 1 each DGU Σ [j] for synthesizing C[11] , C˜[11] and N[11] for the new DGU. Since ˆ DGU is Algorithm 1 never stops in Step A, the addition of Σ [11] allowed and local controllers can be replaced by the new ones.
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DG3 DG8 DG5 DG1
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Fig. 5: Scenario 2 - Scheme of the microgrid composed by 10 ˆ DGU (in gray). DGUs (in black) and plugging-in of Σ [11] ˆ DGU is executed at time The real-time plugging-in of Σ [11] t = 1.6 s. Before this event, references for DGUs 1-10 ˆ DGU is are those described in Section 4.2.1 of [14] and Σ [11] assumed to work isolated, tracking a reference voltage with dq components Vd,ref = 1 pu and Vq,ref = 0.5 pu, respectively. In order to test tracking performances after the addition of ˆ DGU , at time t = 1.8 s we change the d component of Σ [11] ˆ DGU to 0.6 pu. Figure 6 shows the voltage reference for Σ [11] ˆ DGU and its the dq component of the load voltages for Σ [11] DGU DGU ˆ ˆ neighbors Σ[1] and Σ[6] . In particular, from Figures 6a and 6b, we note that right after the plugging-in time (t = 1.6 ˆ DGU and Σ ˆ DGU deviate from s), the load voltages of Σ [1] [6] the respective reference signals. However, this deviation is immediately compensated and, after a short transient, the load voltages at P CC1 and P CC6 converge to their reference ˆ DGU : values. Similar remarks can be done for the new DGU Σ [11] as shown in Figure 6c, there is a short transient at the time of the plugging-in, that is effectively compensated by the control action. Moreover, the controller C[11] and compensators C˜[11] and N[11] ensure desired tracking when the reference signal Vd,ref steps down at t = 1.8 s. ˆ DGU Additional simulations illustrating the unplugging of Σ [2] are provided in [14]. V. C ONCLUSIONS In this paper, we presented a decentralized control scheme for guaranteeing voltage and frequency stability in ImGs. A key feature of our approach is that plugging-in and -out of DGUs requires to update only a limited number of local controllers. Furthermore, a global model of the ImG is not required in any design step. Numerical results in Section IV and in [14] confirm the feasibility of PnP control even for
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(a) d and q components of the voltage at P CC1 . 1.2 V6,d
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(c) d and q components of the voltage at P CC11 .
Fig. 6: Scenario 2 - Performance of decentralized voltage control during the plug-in operation (t = 1.6 s) and a change in the reference for DGU 11 (t = 1.8 s).
ImGs with meshed topology. Further studies will focus on coupling local voltage controllers with a higher control layer devoted to power flow regulation. R EFERENCES [1] R. H. Lasseter, “MicroGrids,” in Proceedings of IEEE Power Engineering Society Winter Meeting, vol. 1, New York, NY, USA, January 27-31, 2002, pp. 305–308.
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