Voltage/Frequency Regulation With Optimal Load Dispatch in Microgrids using SMC based Distributed Cooperative Control

Majority of the contemporary hierarchical control strategies for microgrids are either centralized or distributed, relying on leader-follower (or master-slave) consensus for secondary frequency and/or voltage regulation. Thus, in either case, these strategies are susceptible to single-point-failure (SPF). This potential research gap motivated the authors to propose a distributed three-layered hierarchical control strategy applicable to droop-controlled islanded AC microgrids. The proposed strategy can simultaneously ensure (i) frequency and voltage regulation of DGs within the prescribed frequency and voltage deviation limits as per IEC 60034-1 standard (i.e., ±2% and ±5%, respectively) without relying on leader-follower consensus at the secondary level, (ii) distributed economic dispatch of active power with minimum error, and (iii) distributed reactive power dispatch with plausible error. The proposed technique is fully distributed and shares the computation and communication burden among the neighboring nodes using a sparse communication network, thus, it is insusceptible to SPF. The feasibility of the proposed technique is guaranteed through various time-domain based numerical simulations executed in Matlab/Simulink under different loading conditions and microgrid expansion.

2018 ✓ ✓ ✓ ☞ Leader-follower consensus is required for the distributed secondary frequency and voltage control, which is prone to SPF. [24] 2018 ✓ ✓ ✗ ☞ Leader-follower consensus is required for the distributed secondary frequency and voltage control, which is prone to SPF. Also, optimal dispatch is neglected. [25] 2019 ✓ ✗ ✓ ☞ It does not consider secondary voltage regulation for AC microgrid subsystem. Moreover, only active loads and, hence, active power flow have been considered. Hence, there is no reactive power flow in the AC microgrid subsystem. The tertiary controller in the upper control layer is centralized and, hence, prone to SPF. [26] 2019 ✓ ✗ ✓ ☞ Only active power is dispatched. Voltage regulation and reactive power dispatch are neglected. Generation cost parameters have not been considered. Moreover, the power lines are assumed to be loss-less (purely inductive). [27] 2020 ✓ ✓ ✓ ☞ MGCC is required for optimal dispatch, voltage and synchronization control, which is prone to SPF. [28] 2020 ✗ ✓ ✓ ☞ Only reactive power is dispatched. Frequency regulation and active power dispatch are neglected. [29] 2020 ✓ ✗ ✓ ☞ Only active power is dispatched. Voltage regulation and reactive power dispatch are neglected. [30] 2020 ✓ ✗ ✓ ☞ Only active power is dispatched. Voltage regulation and reactive power dispatch are neglected. Generation cost parameters have not been considered. Moreover, the power lines are assumed to be loss-less (purely inductive). [31] 2021 ✓ ✗ ✓ ☞ Only active power is dispatched. Voltage regulation and reactive power dispatch are neglected. Generation cost parameters have not been considered. Moreover, the power lines are assumed to be loss-less (purely inductive). [32] 2021 ✓ ✗ ✓ ☞ Only active power is dispatched. Voltage regulation and reactive power dispatch are neglected. Generation cost parameters have not been considered. Moreover, the power lines are assumed to be loss-less (purely inductive). [33] 2021 ✓ ✓ ✓ ☞ Leader-follower consensus is required for the distributed secondary frequency and voltage control, which is prone to SPF. Moreover, the optimal dispatch is focused on the minimization of overall network power losses only, and generation cost minimization has been neglected. [34] 2021 The primary control is generally enforced as a decen- , [17], [18]. 9 The secondary layer can be implemented as a (i) traditional 14 Thus, it bears higher cost and heavy computation burden. It 15 is highly sensitive to failures; thus, it is unreliable and may 16 ultimately lead to to the single-point-failure (SPF) [11], [19]. 17 The decentralized secondary controller also bears higher cost 18 as well as higher reliability [19]. The distributed secondary 19 controller requires a sparse communication network for com-20 munication among (local controllers of) neighboring DGs. 21 Thus, it is more reliable, bears lower cost, less sensitive to 22 failures and obviates the requirement of a central computa-23 tion and communication unit [11], [19]. For a more detailed 24 comparison of various secondary control strategies, the read-25 ers are referred to [12], [13]. Traditionally, the tertiary layer 26 can also be implemented as a centralized controller [15] and 27 is susceptible to the same issues stated with reference to the 28 centralized secondary controller.

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The main obligation of the hierarchical control of a micro- This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. to demonstrate that the proposed strategy is also applicable 45 to a large AC microgrid system. 46 The rest of the paper is structured as follows: the communication between energy nodes of the microgrid is ex-48 plained in Section II using graph theory. The configuration 49 of the microgrid testbench is described in Section III. The Matlab/Simulink. Finally, Section VI presents concluding 55 remarks to this paper.

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In distributed control of microgrids, an energy node (or DG) 58 is considered as an agent. Hence, their mutual interaction can 59 be mathematically and graphically described using the multi- cation lines (one-way or two-way), which implies that the 67 graph can be directed (one-way) or undirected (two-way).

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The elements of set E(G) are indicated as (V i (G), V j (G)), 69 implying an edge (or allowed flow of information) from ith 70 to jth agent. Associated with each edge,

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The correctness and soundness of the proposed idea is val-90 idated in Matlab/Simulink using a three-phase microgrid 91 simulation testbench, which is shown in Fig. 1   The primary control is provided at each DG using the con-12 ventional droop technique. It consists of P ωand QV -droop 13 controls that are, respectively, expressed in (1) below [40],

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[41]: 15 where the instantaneous voltages for each phase (i.e., v a , v b 40 and v c ), as depicted in Fig. 3, are given in (2) below [37]: where V i,pk equals the product of the modulation index, where (K tf , η th , K f v , K ev and K m ) > 0 are constants, 53 specified in Table 5, in Appendix A.

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The secondary control is provided at each DG using the 56 distributed technique that concurrently ensures: where σ i is the active power error between the neighboring 1 agents, a ij ≥ 0 is the ijth elements of the adjacency matrix, 2 A, z pi ∈ R N is an auxiliary (or intermediate) state variable 3 for active power control, u P i is the twisting-based sliding 4 mode control law for active power, κ 1 , κ 2 > 0 are the ad- varying signal indicating the optimal active power dispatch 8 reference (with a bounded derivative), which is generated 9 and provided by the distributed tertiary controller, expressed 10 in (17), to the distributed secondary active power controller, 11 described in (4), while P D represents the combined active 12 power demand.

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The auxiliary state variable, z pi , in (4), is initialized such 14 VOLUME 4, 2016 5 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.  that: Taking time-derivative of P i,set in (4), the following 2 closed-loop system is obtained: In (6), P i,set indicates the active power reference signal 5 generated by the distributed secondary active power con-  The principal roles of the distributed secondary active 10 power tracking control algorithm, given in (4), are as follows: where ψ i is the voltage error between the neighboring 27 agents, z vi ∈ R N is an auxiliary state variable for sec- This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.
the local input reference voltage signal (with a bounded 2 derivative), and Q D is the total reactive power demand.

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The auxiliary state variable, z vi , in (7), is initialized such 4 that: Taking time-derivative of V i,set in (7), the following 6 closed-loop system is obtained: In (9), V i,set indicates the reference voltage signal com- The principal roles of the distributed secondary voltage 13 tracking control algorithm, given in (7), are as follows:  This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. tracking control algorithm, expressed in (7), because it also 1 has the same form.

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The discontinuous signum function, appearing in the con-3 trol algorithm (4), introduces discontinuity in it. Therefore, 4 the solution of the stated control algorithm can be interpreted 5 in the Filippov sense [45]. 6 Lemma 1. Suppose the communication graph, G, is con- Proof. It follows from (4) that: Because, the graph, G, shown in Fig. 1, is undirected, it 12 follows that:  Objective function: imated by a quadratic type function, generation cost, α i , β i , and γ i > 0 are the generation cost 42 coefficients, P i is the active power output of ith DG with its 43 lower and upper limits, respectively, indicated by P i,min and 44 P i,max , and P D represents the total active power demand.

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Among several typical approaches for centralized solution 46 of ELD problem, one is the Lagrange multiplier technique. 47 In this stated method, the system constraints are integrated 48 into the objective function. The Lagrangian function, L, for 49 working out the ELD problem can be written as under: where λ indicates the Lagrange multiplier associated with the 51 power balance constraint, stated in (13).

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Then, the first-order Karush-Kuhn-Tucker optimization 53 conditions are applied as under: 54 Condition 1 yields: Condition 2 yields: This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. only, and neglects the inequality constraints (i.e., P i,min = 0 6 and P i,max = ∞).

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So, considering the inequality constraint too (i.e., P i,min ≤ 8 P i ≤ P i,max ), the optimal power dispatch conditions, given 9 in (15), can be revised as follows: Note that λ opt (i.e., the optimal incremental cost) is calcu- where K > 0 and (0 < m < 1). Then V (t) = 0 for all t ≥ t s , and the settling time, t s , can be estimated by [51]: Proof. Having Lemmas 2 through 4, it is now easy to under- 40 stand the convergence of the distributed tertiary control law.   Now, using sgn(·) c = sgn(·)|·| c , the differential error for 48 (18),δ(t), can be expressed as follows: Let, V λ be the Lyapunov function candidate, such that Let an undirected graph, G λ , has an associated adjacency This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. of V λ can be described as: Applying Lemma 2, for 0 < c < 1, it follows: Then, applying Lemma 3, it follows: Finally, applying Lemma 4, it follows that the incremental 5 cost consensus is established in a finite-time, T λ , using the 6 control protocol, given in (17), with (0 < c < 1). Further-7 more, in terms of the initial errors, the convergence time, T λ , 8 can be upper bounded as follows: Moreover, or λ i (t) = λ j (t), for all t ≥ T λ This completes the proof. respectively, ± 2 % and ± 5 %, as per IEC 60034-1.

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For performance assessment, four different tests are con- This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. A, indicates various generation cost parameters for each DG. 5 For the distributed secondary controller, the adjustable design 6 parameters, expressed in (4) and (7)     shown in Figs. 10 and 11, respectively, it indicates that the 1 distributed secondary controller is also precisely fulfilling its 2 frequency and voltage regulation tasks, despite load varia-3 tion. Figures 12 and 13, respectively, indicate the incremental 4 costs, λ opt , and the corresponding total generation cost,  This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.    This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.     This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.       His research interests include the analysis and optimal design of next-98 generation electrical machines using smart materials, such as electromagnet, 99 piezoelectric, magnetic shape memory alloy, and so on. This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3183635