Opinions on Fast Distributed Optimization for Large-scale Scheduling of Heterogeneous Flexibility Resources Provisionally Accepted

Man Tan^1* Xiang Gao² Yutong Liu³

• ¹Hunan University, China
• ²Shenzhen Polytechnic, China
• ³Shandong University, China

With the high proportion integration of distributed energy sources such as renewable energy and energy storage systems, the traditional distribution network has evolved from a passive power supply network to an active network with bidirectional power flow [1]. The operation and scheduling of active distribution networks (ADNs) have undergone great challenges due to intrinsic intermittence and volatility from renewable energy resources [2,3]. This has led to the necessity to fully utilize support and adjust capability of flexibility resources such as distributed energy storage and electric vehicles for reliable power supply [4,5]. Considering the properties of large quantities, decentralized location and diverse stakeholders for heterogeneous flexibility resources, the traditional centralized control strategy faces various challenges in the form of system reliability, mass communication and information privacy [6]. Hence, distributed optimization is proposed to purge the globally unified control of distribution networks that would enable efficient management of flexibility resources through distributed clustering [7][8][9]. However, conventional distributed algorithms have slow convergence properties owing to the gradient-based update process and communication delays [10], which cannot satisfy the fast real-time scheduling of ADNs. Therefore, this paper focus on providing insightful perspectives and discussions on the fast distributed optimization for large-scale scheduling of heterogeneous flexibility resources.The main contributions of this paper are twofold as listed: (1) A bi-level distributed scheduling model of large-scale heterogeneous flexibility resources is proposed to minimize the overall operational cost of ADNs and promote accommodation of renewable energy resources. (2) A fast distributed asynchronous optimization method is presented to accelerate convergence speed for real-time scheduling of ADNs, and the correctness and superiority of the proposed method is demonstrated by case studies.
The optimized scheduling of ADNs needs to take into account potential benefits of different dispatching entities such as distribution networks and diversified flexibility resources for minimizing the overall operational costs, while stimulating the incorporation of renewable energy. The objective function aims to minimize the costs associated with purchasing electricity, renewable energy curtailment and dispatching flexibility resources for the purpose of economy enhancement as follows, where T denotes the total number of scheduling periods; t  denotes the duration of each scheduling period; buy t  , RES.curt  , DR  , ESS  represents the purchase price of electricity in time period t , the cost coefficient for penalty of renewable energy curtailment, the unit dispatch cost of controllable loads and the cost coefficient for charging and discharging of energy storage system, respectively; RES  , DR  , ESS  respectively represents the set of renewable energy, demand response and energy storage system in ADNs; buy t P denotes the purchased active power from the main grid in time period A bi-level distributed scheduling strategy is proposed to minimize overall operational cost of ADNs, which is poised to meet the needs of individual economies and privacy preservation for agents with diverse flexible resources [11]. At the upper level, the scheduling and control center of ADNs serves as a decision-maker to achieve synergies among multiple flexibility resources via information and energy exchange, thereby maximizing the overall economics of scheduling for distribution networks with high renewables. At the lower level, nodes integrated with controllable flexibility resources achieves autonomous operation through the fully utilization of inherent adjustment capacity. The proposed hierarchical optimization scheduling model can be solved by Alternating Direction Method of Multipliers (ADMM) algorithm, which is a popular and efficient method to deal with distributed optimization problems with stable robustness and convergence [12,13]. The distributed scheduling model of large-scale heterogeneous flexibility resources can be decomposed into the master problem of distribution networks at the upper level and subproblems of controllable flexibility resources at the lower level based on ADMM, as shown in Figure 1(a).
The objective function can be separated according to the bi-level distributed optimization scheduling strategy as follows,where N denotes the set of nodes integrated with flexibility resources, ADN F is the objective function of the master problem for ADN; N,i F is the objective function of subproblem for node i . The optimization variables for the master problem are mainly comprised of the power purchased from or sold to the main grid. The optimization variables for the subproblems include multiple heterogeneous flexibility resources dispatching power consist of energy storage system, photovoltaic generation, wind turbine generation, micro gas turbine and demand response resources. Furthermore, there are coupling relationships between the nodes integrated with controllable flexibility resources and ADNs due to their energy interaction. Hence, the power injected to nodes integrated with controllable flexibility resources are proposed as virtual decoupling variables to establish the consistency coupling constraints as follows [14],,,ˆî t i t -= 0 XZ (5)The variables , ˆit X and , ˆit Z are solved separately at the upper and lower levels, and the optimization results are delivered iteratively between the two levels to solve the model. A Lagrange penalty function is added to the objective functions of the master problem and subproblems, as follows: (7) where k denotes iteration number;  is the penalty coefficient; , j it u is Lagrange multiplier. The proposed distributed scheduling model is solved by optimizing the coupling variables through continuous iterations between the master problem and subproblems. Relevant information of expected interaction energy for ADNs and nodes integrated with controllable flexibility resources is delivered mutually at the two levels. The Lagrange multipliers will be updated after each iteration step as follows,2 1 N, , ,, 1 2 ˆmin() 2 it The primal residual k r and dual residual k s are introduced as convergence criteria [15], which are calculated after each iteration step as follows, N ,, 1 2
T k k k i t i t r it r   = = -   XZ (9) N 1 , , s 1 2 ( ) T k k k i t i t it s  -  = = -  XX (10) where r  , s  refers to the convergence threshold for primal residual and dual residual, respectively. If the convergence criterion is not satisfied, the next iteration will continue with updated Lagrange multipliers along with latest data on coupling and decoupling variables. Otherwise, the iteration process will be terminated to obtain the optimal scheduling determination of heterogeneous flexibility resources with minimal operational costs for ADNs.
Considering the time-varying nature of communication network and the varied responsiveness of heterogeneous flexibility resources [11], traditional synchronized computation is incapable to satisfy the fast real-time scheduling of ADNs owing to the increased communication overhead and limited convergence speed. Specifically, under the synchronous protocol, the optimization model for the master problem is triggered at each iteration only if the scheduling center of ADNs receives the information from all nodes [16]. The master problem and computationally fast subproblems will remain idle most of the time, thereby impeding the full utilization of parallel computing resources. Hence, the distributed asynchronous optimization is adopted to improve the convergence efficiency, which allows the master problem to execute next iterative updates without the reception of complete information from all nodes [17], as shown in Figure 1(b). Initially, k D is proposed to denote the index set of nodes from which the scheduling center receives coupling information during iteration k. The variables information of node i is uploaded to the scheduling center if k iD  . If a node fails to deliver information promptly due to communication delays or slow response speed, the data of the last iteration will be used instead to execute the next optimization updates for the master problem as follows, , ,,kk it k it kk it iD iD -    =     X X X(11)Two asynchronous constraints are set in the computation process to guarantee the convergence of asynchronous optimization algorithm [17]. On the one hand, to ensure the efficacy of each iteration, the master problem proceeds to next iteration only if the number of nodes in k D is larger than the set threshold 1  . On the other hand, taking into account the hazard of unbounded delays on algorithm convergence [17,18], the inactive iteration of every node as well as k iD  must be less than the set maximum tolerable delay  . This means that the coupling variable information of per node used by the center must be at most τ iterations old [19]. k k i kk i iD d d i D -    =  +  (12)When the both two conditions cannot be satisfied simultaneously, the scheduling center must wait until the updated information from the unusual nodes is received. The master problem and subproblems, with smaller idle time, are frequently updated in comparison with the synchronous optimization. However, the benefit of improved update frequency can outweigh the cost of increased number of iterations, enabling the asynchronous algorithm to converge in the shortest possible time [18].In order to further accelerate convergence speed, this paper also proposes a method to curtail the feasibility domains of the master problem [20,21], as shown in Figure 1(c). The feasibility of interactive power information delivered by the master problem is examined during the subproblem of nodes at the lower level. The optimization model of subproblem for node i can be expressed as follows, DS,( ˆˆ: )i t i t i t ii ii it F s t G H  = 0 X M M Z , (13)where i M denotes the optimization variables for subproblem of node i ; () ii G M denotes inequality constraints set for subproblem of node i ;,, , () ˆît tt i iH X , Z denotes the coupling equational constraints of the two levels; , it  denotes the dual multipliers of coupling equational constraints. If the expected interaction power delivered by scheduling center of the master problem is not feasible for subproblem of node i , the relaxation factor i S is introduced to transform the subproblem as follows, D ,( ˆ) i t i t i t i i i i ii it S s t F S S G H -   = 0 Z , X M  (14), , S, , ˆ min . . 0, 0 ( ) 0 :With the optimization solution of relaxed subproblem, the node i can provide feedback on feasible cutset constraints to the master problem as follows [22], , ,, 1 , ˆ+ ( ) 0 ˆTT i i t i i t t t i t SH =   XZ μ , (15)where ˆi S denotes the optimal value of the objective function of relaxed subproblem. Otherwise, no constraints are returned to the master problem if the subproblem is found to be feasible. Therefore, the objective function and constraint conditions are both restricted through the feedback of feasible cutset constraints after feasibility examination. Consequently, an improvement in the convergence speed was observed owing to a reduction in the feasibility domains of the master problem.
To validate the effectiveness of the proposed fast distributed optimization method for large-scale scheduling of heterogeneous flexibility resources in this paper, the IEEE33 bus distribution system is used as a specimen for case studies. The quantity of energy storage systems, photovoltaic generation, wind turbine generation, micro gas turbine and demand response resources in the distribution system is defined to be 2, 2, 1, 1 and 2, respectively. The proposed model is solved by centralized algorithm, general synchronous distributed algorithm and fast distributed asynchronous algorithm, respectively, to verify the preeminence of the presented method through comparative analysis. The comparison between results of the operational costs for ADNs and convergence properties under different algorithms is shown in Table 1.