Multi-Agent Based Load Balancing Dispatch for Power Systems with Renewable Energy

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(1)Proceedings of the 51st ISCIE International Symposium on Stochastic Systems Theory and Its Applications Fukushima, Nov. 1-2, 2019. Multi-Agent Based Load Balancing Dispatch for Power Systems with Renewable Energy Kenta Hanada1 , Takayuki Wada1 , Izumi Masubuchi2 , Toru Asai3 , and Yasumasa Fujisaki1 1. Graduate School of Information Science and Technology, Osaka University 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan 2 Graduate School of System Informatics, Kobe University 1-1 Rokkodai, Nada, Kobe, Hyogo 657-8501, Japan 3 Graduate School of Engineering, Nagoya University Furo, Chikusa, Nagoya, Aichi 464-8603, Japan. E-mail: {k-hanada, t-wada, fujisaki}@ist.osaka-u.ac.jp, [email protected], [email protected]. Abstract. 2. Multi-agent based load balancing dispatch is considered for power systems with renewable energy. It is assumed that each agent generates uncertain amount of power due to renewable energy resources. A suitable communication gain of the distributed algorithm is derived for the desired consensus in mean square or with bounded error variance.. 1. Introduction. Load Balancing Dispatch with Renewable Energy. We consider a power dispatch with load balancing which is a problem to find a power generation plan such that every load ratio is equal to the same value for all agents and the sum of the power output to be generated is equal to a given demand [2, 3]. Let V = {1, 2, . . . , N } be the set of agents and N be the number of agents. We formulate the power dispatch as a decision problem: Find x1 , x2 , . . . , xN. Since it is crucial for our society to maintain stable power supply, we need appropriate operation plans in order to avoid malfunctions or black out of the power systems. A power dispatch with load balancing provide such a plan in terms of the quality of service (QoS) [1]. It is formulated as a problem to find a power generation plan such that every load ratio is equal to the same value for all agents and the sum of the power output to be generated is equal to a given demand [2, 3, 4, 5, 6]. The existing studies [2, 3] deal with the multi-agent based distributed load balancing dispatch under noisy environment. That is, the noise is added when an agent communicates with its neighbors, while the agents are assumed that they have no disturbance. In this paper, we consider a multi-agent based load balancing dispatch for power systems with renewable energy. We model the vibrations or fluctuations of power output caused by solar or wind turbines as probabilistic disturbances. That is, the noise is added to each agent. We propose a stochastic distributed algorithm and analyze the convergence properties. This paper organized as follows. In Section 2, we formulate the load balancing dispatch with renewable energy. In Section 3, we consider the multi-agent based distributed algorithm for the problem. We analyze the convergence of the algorithm with diminishing communication gain in the first part of Section 4. We also analyze the one with static communication gain in the second part of Section 4. In Section 5, we show a numerical example. Finally, we have concluding remarks in Section 6.. – 13 –. s.t.. x2 xN x1 = = ··· = , ψ1 ψ2 ψN. N ∑. xi = l,. (1). i=1. where xi ∈ R is a power output plan to be generated at agent i ∈ V, ψi ∈ R is a positive capacity factor of agent i, and l ∈ R is a given positive constant demand. Note that the desired load ratio β = xi /ψi (i ∈ V) satisfies β = l/(ψ1 + ψ2 + · · · + ψN ). In this paper, we suppose that agents generate power output x î ∈ R according to the power generation plan xi via renewable energy resources such as solar, wind, or geothermy. Hence, the actual power output x î generated by agent i can be modeled as uncertain amount x î = xi +wi , where E[wi ] = 0, and Var[wi ] ≤ vi2 < ∞ for all i ∈ V. Here E[a] and Var[a] denote the expectation and the variance of a random variable a. This paper presents the multi-agent based load balancing algorithm for power systems with renewable energy. The aim of this distributed algorithm is to achieve the power generation sequence such that the load ratio asymptotically converges to the same value, while the sum of the power output to be generated is equal to a given demand at each time step.. 3. Multi-agent System. Since we deal with a multi-agent based algorithm, we introduce a communication graph. Let G = (V, E) be a digraph which represents a network, where E ⊆ V ×V is the set of directed edges. We assume that the digraph.

(2) G is weakly connected with no self loop. Then, we consider the following type of distributed power dispatch algorithm: xi [k + 1] = xi [k] + ui [k], ∑ ∑ ui [k] = − pij [k] + pji [k], (i,j)∈E. w[k] = [ w1 [k] w2 [k] · · · Let a deviation x ˜[k] ∈ R be. x ˜[k] = x[k] − βΨ −1 1N = x[k] − (2). (j,i)∈E. where k ∈ N is discrete time, x[k] ∈ R is a power generation plan for agent i, ui [k] is the amount of increasing/decreasing power output to be generated at agent i, and pij [k] ∈ R is the amount of power output to be adjusted between agents i and j. We suppose that the amount of adjustments pij [k] is a control variable only for agent i, not for agent j in the equation (2). On the other hand, pji [k] in (2) is determined by in-degree neighbors of agent i, that is, agent j decides the amount of adjustments pji [k]. Note that the above update rule preserves sum of xi [k], that is, ∑N ∑N xi [k + 1] = i=1 xi [k] for any k, which follows ∑i=1 ∑N N i=1 ui [k] = 0. Thus, if i=1 xi [1] = l is satisfied, the constraint for demands (1) always holds for any time k. We define the control variable pij [k] as ) ( ˆj [k] x î [k] x − , pij [k] = µ[k]aij ψi ψj. where γ = Ψ −1/2 1N /∥Ψ −1/2 1N ∥ and S ∈ RN ×(N −1) is the orthonormal complement of γ. Applying these transformations to the compact form (3), we have ( ) ξ1 [k + 1] = IN −1 − µ[k]S ⊤ Ψ 1/2 LΨ 1/2 S ξ1 [k] − µ[k]S ⊤ Ψ 1/2 LΨ w[k], l , ξ2 [k + 1] = ξ2 [k] = ξ2 [1] = ∥Ψ −1/2 1N ∥ ˜[k] = ξ1 [k], ξ˜1 [k] = S ⊤ Ψ 1/2 x ξ˜2 [k] = γ ⊤ Ψ 1/2 x ˜[k] = ξ2 [k] −. where µ[k] ∈ R is a communication gain to be determined later, aij ∈ R is a positive static weight corresponding to the edge (i, j) ∈ E, aij = 0 if (i, j) ∈ / E, and x î [k] ∈ R is an actual power output generated by agent i at time k. Note that the actual power output x î [k] would be generated by renewable energy resources according to the power generation plan xi [k], that is, the disturbance wi [k] represents the fluctuations of power output caused by renewable energy resources. We assume that E[wi [k]] = 0, Var[wi [k]] ≤ vi2 < ∞, and the disturbance wi [k] follows an independent and identically distribution with respect to i and k. In order to represent the distributed dispatch algorithm (2) as a compact form, we define L = diag((A + A⊤ )1N ) − (A + A⊤ ), where A is an adjacency matrix whose i, j-th element is aij if (i, j) ∈ E or 0 otherwise, 1N is the N -dimensional vector whose elements are all 1, and diag(a) denotes the diagonal matrix whose diagonal elements correspond to the column vector a. By using the matrix L, the consensus algorithm (2) can be rewritten as (3). lΨ −1 1N , ∥Ψ −1/2 1N ∥2. where ∥a∥ is the Euclidean norm. We then employ a state coordinate transformation [ ] [ ⊤ ] ξ1 [k] S = Ψ 1/2 x[k], ξ2 [k] γ⊤ [ ] [ ] ξ1 [k] x[k] = Ψ −1/2 S γ , ξ2 [k] [ ] [ ⊤ ] ξ˜1 [k] S = Ψ 1/2 x ˜[k], γ⊤ ξ˜2 [k]. x î [k] = xi [k] + wi [k],. x[k + 1] = (IN − µ[k]LΨ ) x[k] − µ[k]LΨ w[k],. wN [k] ]⊤ ∈ RN .. l = 0. ∥Ψ −1/2 1N ∥. Since ( ) ⊤ ˜[k]∥2 = (˜ x[k]) Ψ 1/2 SS ⊤ + γγ ⊤ Ψ 1/2 x ˜[k] ∥Ψ 1/2 x 2 2 ˜ ˜ = ∥ξ1 [k]∥ + ∥ξ2 [k]∥ = ∥ξ˜1 [k]∥2 = ∥ξ1 [k]∥2 ,. (4). we can evaluate the convergence of x ˜[k] as that of ξ1 [k].. 4. Convergence Analysis. Let us define the transition matrix Φ(k, m) as  ( ) ⊤ 1/2 IN LΨ 1/2 S )  −1 − µ[k − 1]S Ψ  (    × IN(−1 − µ[k − 2]S ⊤ Ψ 1/2 LΨ 1/2 S ) Φ(k, m) = · · · × IN −1 − µ[m]S ⊤ Ψ 1/2 LΨ 1/2 S    if k > m   IN −1 otherwise. Then, the solution of ξ1 [k] can be written as ξ1 [k] =Φ(k, 1)ξ1 [1] −. k−1 ∑. µ[m]Φ(k, m + 1)S ⊤ Ψ 1/2 LΨ w[m].. m=1 2. Since E[w[k]] = 0 for any k, the expectation of ∥ξ1 [k]∥ can be evaluated with a deterministic term and a stochastic term, i.e., [ ] 2 E ∥ξ1 [k]∥. where x[k] = [ x1 [k] x2 [k] · · · [ Ψ = diag( 1/ψ1 1/ψ2. xN [k] ]⊤ ∈ RN , ]⊤ · · · 1/ψN ) ∈ RN ×N. – 14 –.

(3) 2 ( ). · Ψ 1/2 LΨ 1/2 Tr Ψ 1/2 V Ψ 1/2. 2. = ∥Φ(k, 1)ξ1 [1]∥  2  k−1. ∑. + E  µ[m]Φ(k, m + 1)S ⊤ Ψ 1/2 LΨ w[m]  . (5). m=1. Let λ2 ∈ R be the second smallest eigenvalue of the matrix LΨ and λN ∈ R be the largest eigenvalue of the matrix LΨ . This means that ∥Ψ 1/2 LΨ 1/2 ∥ = λN . To prove main results, we prepare the following lemma [3]. Lemma 1. The matrix LΨ satisfies. IN −1 − 1 S ⊤ Ψ 1/2 LΨ 1/2 S ≤ 1 − λ2 .. λN λN. 1 , λ2 (k0 + k). k0 ≥. k−1 ∑ 1 λ2N Tr (V Ψ ) 2 2 λ2 m=1 (k0 + k − 1). ≤. 1 λ2 · N2 Tr (V Ψ ) , k0 + k − 1 λ 2. where V = diag([v1 v2 · · · vN ]⊤ ). Thus we have  2  k−1. ∑. lim E  µd [m]Φ(k, m + 1)S ⊤ Ψ 1/2 LΨ w[m] . k→∞ m=1. 1 λ2 · N2 Tr (V Ψ ) = 0. k→∞ k0 + k − 1 λ2. ≤ lim. 4.1 Diminishing Gain Let us select the diminishing gain µd [k] as µd [k] =. ≤. With (4), (5), (6), and (7), Theorem 1 holds.. λN − 1, λ2. 4.2 Constant Gain Let us choose the constant gain. where k0 ∈ N. Then we have the first main result. Theorem 1. Let the communication gain µd [k] be 1/(λ2 (k + k0 )), where k0 ≥ λN /λ2 − 1. Then, for any initial state x[1], the deviation x ˜[k] satisfies. Proof. From Lemma 1, we obtain. 1 1 ⊤ 1/2 1/2 IN −1 − S Ψ LΨ S ≤ 1 − ,. λ2 (k0 + k) k0 + k. Proof. Since µc [k] = 1/λN , we see ( )k−m 1 ⊤ 1/2 1/2 Φ(k, m) = IN −1 − S Ψ LΨ S . λN. k0 + m − 1 . k0 + k − 1. From Lemma 1, the norm of Φ(k, m) satisfies. k0 ∥Φ(k, 1)ξ1 [1]∥ ≤ ∥ξ1 [1]∥ k0 + k − 1. ∥Φ(k, m)∥ ≤. and thus we can evaluate the deterministic term as k02. 2. lim ∥Φ(k, 1)ξ1 [1]∥ ≤ lim. k→∞. (. 2. 2. (k0 + k − 1). ∥Φ(k, 1)ξ1 [1]∥ ≤. (6). We can also compute the stochastic term as  2 . k−1 ∑. µd [m]Φ(k, m + 1)S ⊤ Ψ 1/2 LΨ w[m]  E . m=1 ( k−1 ∑ = Tr µ2 [m]Φ(k, m + 1)S ⊤ Ψ 1/2 LΨ V m=1. Ψ L⊤ Ψ 1/2 S (Φ(k, m + 1)) µ2 [m] ∥Φ(k, m + 1)∥. ⊤. ( )k−m λ2 1− . λN. Then we have. ∥ξ1 [1]∥. = 0.. ≤. Then we have the second main result.. lim E[∥˜ x[k]∥] < ∞.. Then we have. k−1 ∑. 1 . λN. k→∞. which means that the norm of Φ(k, m) satisfies. k→∞. µc [k] =. Theorem 2. Let the communication gain µc [k] be 1/λN . Then, for any initial state x[1], the deviation x ˜[k] satisfies. lim E [∥˜ x[k]∥] = 0.. k→∞. ∥Φ(k, m)∥ ≤. (7). λ2 1− λN. )k−1 ∥ξ1 [1]∥. and thus we can evaluate the deterministic term as )2(k−1) ( λ2 2 2 lim ∥Φ(k, 1)ξ1 [1]∥ ≤ lim 1 − ∥ξ1 [1]∥ k→∞ k→∞ λN = 0. We can also compute the stochastic term as  2  k−1. ∑. µc [m]Φ(k, m + 1)S ⊤ Ψ 1/2 LΨ w[m]  E . ). m=1. 2. ≤ Tr (V Ψ ). m=1. – 15 –. ( )2(k−m−1) λ2 1− λN m=1 k−1 ∑. (8).

(4) λ2N Tr (V Ψ ) , 2λN λ2 − λ22. 250. The state x[k]. ≤. which is bounded independently of k. Thus we have  2  k−1. ∑. µc [m]Φ(k, m + 1)S ⊤ Ψ 1/2 LΨ w[m]  lim E . k→∞ 2λN λ2 − λ22. Tr (V Ψ ) < ∞.. 100 101. 102. 103. 104. The number of iterations k. (9). Fig. 1: Behavior of each element of the state x[k]. With (4), (5), (8), and (9), Theorem 2 holds. These theorems claim that the proposed distributed algorithm can achieve the desired consensus in mean square or with bounded error variance if we select a suitable communication gain. Notice also that µd [1] = µc [k] when we select k0 = λ2 /λN −1 for µd [k]. That is, the diminishing gain µd [k] is always less than or equal to the constant gain µc [k].. The deviation x ˜[k]. ≤. 150. 50 100. m=1. λ2N. 200. 50 0 -50 -100 100. 101. 102. 103. 104. The number of iterations k. 5. Numerical Examples. In this section, we show a numerical example. We basically followed the settings of [2, 3], where N = 4. The settings are shown as follows:. Fig. 2: Behavior of each element of the deviation x ˜[k]. V ={1, 2, 3, 4}, E ={(1, 2), (2, 1), (2, 3), (3, 2), (3, 4), (4, 3)}, ( ) Ψ =diag [1/120 1/150 1/200 1/280]⊤ , a12 =a21 = 3, a23 = a32 = 7, a34 = a43 = 9, x1 [1] =100, x2 [1] = 120, x3 [1] = 89, x4 [1] = 230.. References [1] A. J. Wood, B. F. Wollenberg, and G. B. Sheblé: Power Generation, Operation, and Control, Wiley, 2014.. Note that a desired value of the weighted consensus was βΨ−1 1 = [ 86.24 107.80. 143.73. 201.23 ]⊤ .. The largest eigenvalue λN of LΨ is 0.2514 and the second smallest eigenvalue λ2 of LΨ is 0.0378. We used the upper bound of the variance for the agents as v12 = 20,. v22 = 40,. v32 = 35,. v42 = 7.. We selected the communication gain µd [k] as 1 . µd [k] = 0.0378(6 + k) ˜[k], Figs 1 and 2 depict the behavior of x[k] and x respectively. These figures indicate that our proposed algorithm works well.. 6. Concluding Remarks. We have considered a multi-agent based load balancing dispatch for power systems with renewable energy. We have derived that a suitable communication gain of the distributed algorithm for the desired consensus in mean square or with bounded error variance. Acknowledgment: This research was supported by JST CREST Grant Number JPMJCR15K2, Japan.. – 16 –. [2] L. Y. Wang, C. Wang, G. Yin, and Y. Wang: Weighted and Constrained Consensus for Distributed Power Dispatch of Scalable Microgrids, Asian Journal of Control, Vol. 17, No. 5, pp. 17251741, 2015. [3] K. Hanada, T. Wada, I. Masubuchi, T. Asai, and Y. Fujisaki: Multi-agent Consensus for Distributed Power Dispatch with Load Balancing, Asian Journal of Control, 10.1002/asjc.2257 (Online), 2019. [4] H. Xin, Z. Qu, J. Seuss, and A. Maknouninejad: A Self-Organizing Strategy for Power Flow Control of Photovoltaic Generators in a Distribution Network, IEEE Transactions on Power Systems, Vol. 26, No. 3, pp. 1462–1473, 2011. [5] H. Yang, L. Yin, Q. Li, W. Chen, and L. Zhou: Multiagent-Based Coordination Consensus Algorithm for State-of-Charge Balance of Energy Storage Unit, Computing in Science & Engineering, Vol. 20, No. 2, pp. 64–77, 2018. [6] S. Baros and M. D. Ilić: Distributed Torque Control of Deloaded Wind DFIGs for Wind Farm Power Output Regulation, IEEE Transactions on Power Systems, Vol. 32, No. 6, pp. 4590–4599, 2017..

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