Distributed accelerating optimization method for smart grid economic dispatch based on directed unbalanced topology

Abstract

The present application belongs to the field of smart grid economic dispatch, and particularly relates to a kind of smart grid economic dispatch distributed acceleration optimization method based on directed unbalanced topology, including considering a group of n generators connected through smart grid, constructing economic dispatch problem model, in the model, optimal distribution meeting demand is obtained under the premise of meeting system constraints;The present application combines momentum term with extended distributed gradient descent method, updates Lagrange multiplier in a distributed manner, and ensures that the agent can obtain more information from its in-degree neighbor to speed up the convergence speed of the algorithm, according to the small gain theorem, as long as the step length alpha and momentum coefficient beta do not exceed a certain boundary, the algorithm can quickly converge to the global optimal solution of economic dispatch problem under the condition that the generator cost function is smooth and strongly convex.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of smart grid economic dispatch, and particularly relates to a smart grid economic dispatch distributed acceleration optimization method based on a directed unbalanced topology. BACKGROUND

[0002] With the continuous development of society, the development of various industries almost cannot do without the support of electricity, and the development of national economy has been closely related to the stable and safe operation of the power system. The economic dispatch problem of the power system is one of the important problems in the operation of the power system, which is essentially an optimization problem, and its goal is to minimize the total generation cost while meeting the demand for supply and demand balance and the output limit of the generator. In view of the irreplaceable advantages of multi-agent system in distributed computing, many researchers take it as a carrier for distributed optimization to carry out theoretical research and application promotion. The multi-agent system distributed optimization realizes the global optimization goal of the system through the interaction and cooperation of multiple agents, wherein the global objective function is the sum of the local objective functions of all agents, and the local objective function of each agent is only known by itself. The economic dispatch problem can be regarded as a problem of processing the sum of the optimization local objective functions, and each agent only has its local objective function, and the constraints of the agent variable are composed of global coupling equality constraints and local linear constraints.

[0003] There are many centralized methods for solving the economic dispatch problem, such as iterative method, linear programming method and Newton method, and some complex techniques such as genetic algorithm and particle swarm optimization method are also included in the traditional centralized method, which can solve non-convex cost functions and more stringent constraints. Generally speaking, most of the traditional methods need a control center that can access the entire system state. Although this centralized method is simple to implement, it may be limited in performance, such as high communication requirements and cost, heavy computing burden, limited flexibility and scalability, etc. With the rapid development of science and technology and the continuous expansion of network scale, the traditional power grid gradually develops into the future smart grid, and the traditional centralized control and optimization technology is difficult to solve the problem of large-scale complex network, therefore, developing a more suitable distributed optimization algorithm for smart grid to solve general convex optimization problems is the current research focus.

[0004] Some known distributed optimization methods for different networks usually rely on distributed consensus protocols (local computation and local information exchange) to take the generator incremental cost as the consensus variable, each agent maintains some variables and updates them through information exchange with neighboring agents. In the methods currently proposed, such as the leader-follower consensus algorithm, the leader collects the mismatch between system demand and power generation, and then updates the system marginal cost, but the algorithm cannot run in a fully distributed manner. In order to avoid the dependence on the leader, a two-level incremental cost consensus algorithm and a distributed algorithm based on consensus and innovation terms are proposed, but all these works only involve undirected networks. Undirected communication networks require that information must be exchanged bidirectionally. Since directed communication networks are lower in cost than undirected communication networks, it is advantageous to utilize the ability of low-cost communication networks in future smart grids. For directed graphs, a (sub) gradient push method (SP) is proposed, which eliminates the imbalance of directed networks by using column stochastic matrices. Since SP is based on a step-decreasing (sub) gradient descent, it also exhibits a slow sublinear convergence rate. In order to speed up the convergence rate, the push-sum strategy is introduced into the distributed inexact gradient tracking method, which uses a fixed step size to solve the economic dispatch problem. There are also research works that extend the fixed step size to a non-uniform step size, but most of the methods use row and column stochastic matrices simultaneously to obtain linear convergence on directed networks. It is worth noting that although these methods are suitable for directed networks, they all require each agent to have exact out-degree information of itself. However, this requirement is likely to be unrealistic in the actual operation of smart grids. In the distributed optimization methods using row stochastic matrices and non-uniform step sizes, such as the distributed primal-dual augmented (sub) gradient algorithm, but without using momentum terms, the convergence rate is slow, while using momentum terms can enable agents to obtain more information from network neighbors in order to converge faster. In the proposed distributed accelerated optimization algorithm with Nesterov momentum terms, a faster convergence rate than the centralized gradient descent method (CGD) is presented, but it is only applicable to undirected networks.

[0005] Therefore, it is necessary to develop a distributed accelerated optimization algorithm with low communication requirements and strong robustness to deal with the economic dispatch problem of smart grids closer to real life. SUMMARY

[0006] In view of the above problems, the present application provides a distributed accelerated optimization method for economic dispatch of smart grids based on directed imbalanced topology, considering a group of n generators connected through a smart grid, constructing an economic dispatch problem model, in which the optimal allocation meeting the demand is obtained under the premise of meeting the system constraints. The economic dispatch problem model includes:

[0007]

[0008] Constraints:

[0009]

[0010] where C(x) represents the total cost of generators, C i (x i ) represents the cost function of generator i, d i represents the load demand for generators; represents the minimum generation of generator i, x i represents the current generation of generator i, represents the maximum generation of generator i.

[0011] Further, in the process of solving the economic dispatch problem model, the Lagrange function is introduced to model the economic dispatch problem with constraints into an unconstrained problem, and then this unconstrained problem is converted into its dual form, and the Lagrange dual problem is solved by applying the gradient ascent method.

[0012] Further, the process of introducing the Lagrange function to model the economic dispatch problem with constraints into an unconstrained problem includes:

[0013]

[0014] where, represents the economic dispatch problem model of the unconstrained problem, and λ is the Lagrange multiplier.

[0015] Further, the process of converting this unconstrained problem into its dual form includes:

[0016]

[0017] where Φ i (λ) is the Lagrange dual function, and represents x i represents the current generation of generator i, X i represents the set of generations of all generators; represents the set of real numbers.

[0018] Further, the standard gradient ascent method for processing the dual problem includes:

[0019]

[0020]

[0021] where λ t+1 represents the Lagrange multiplier at the t+1 iteration, γ tdenotes the step size of the tth iteration of the gradient ascent algorithm; denotes the power generation of generator i obtained by solving the correspondence between the solution of the dual problem and the solution of the economic dispatch problem model; i (λ) = -Φ i (λ).

[0022] Further, a consensus protocol is constructed to solve the optimal solution of the dual problem, and the process includes:

[0023]

[0024]

[0025]

[0026]

[0027]

[0028] wherein, denotes the power generation of generator i in the t+1th iteration; denotes the inverse function of the gradient of the generator cost function; denotes the consensus value of generator i in the t+1th iteration; denotes the momentum term coefficient; h denotes an adjustable parameter; r ij denotes the connection weight between generator i and generator j; denotes the uncoordinated step size; denotes the gradient of Φ i (λ) at time t; denotes the i-th term of Φ denotes the consensus value at time ; and denotes the inner neighbor set of generator i.

[0029] Further, the connection weight r ij between generator i and generator j denotes:

[0030]

[0031]

[0032]

[0033] wherein, 0 < η < 1 is a constant, denotes the inner neighbor set of generator i, denotes the generator set.

[0034] Furthermore, the uncoordinated step size of generator i at time t is expressed as: And satisfy Where ρ and ω are adjustable parameters.

[0035] The beneficial effects of this invention include:

[0036] 1. Compared with other methods for solving economic scheduling problems in directed networks, this invention only uses row random matrices. Since constructing row random weight matrices does not require each generator to know the out-degree information of all its internal neighbors, it is easier to implement in a distributed manner compared to column random matrices or double random matrices.

[0037] 2. This invention obtains the solution to the original problem by solving the dual problem of the original problem, extending the centralized gradient descent method to a distributed one, that is, the update of the Lagrange multiplier λ in the Lagrange function is distributed, which is more suitable for distributed constraint optimization problems on directed networks.

[0038] 3. This invention incorporates the momentum term into the distributed gradient descent method, ensuring that the generator can obtain more information from its internal neighbors, thereby accelerating the convergence speed.

[0039] 4. The present invention is more flexible in step size selection because it allows for inconsistent step sizes, which means that the range of step size selection is wider than most existing methods. Attached Figure Description

[0040] Figure 1 This is a flowchart of an embodiment of the present invention;

[0041] Figure 2 These are the cost parameters and power generation capacity of each generator in the embodiments of the present invention;

[0042] Figure 3 This is a communication topology diagram according to an embodiment of the present invention;

[0043] Figure 4 This is a graph showing the output power variation of each generator in an embodiment of the present invention;

[0044] Figure 5 This is a graph showing the output power variation of each generator in a prior art embodiment;

[0045] Figure 6 This is a graph showing the changes in the consistency variable (incremental cost) according to an embodiment of the present invention;

[0046] Figure 7 This is a graph showing the changes in consistency variables (incremental cost) in existing technology embodiments;

[0047] Figure 8 This is a generator supply and demand balance diagram according to an embodiment of the present invention;

[0048] Figure 9 Generator supply-demand balance chart of prior art embodiments. DETAILED DESCRIPTION

[0049] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by a person of ordinary skill in the art without creative work fall within the protection scope of the present application.

[0050] The present application provides a directed unbalanced topology-based distributed acceleration optimization method for economic dispatch of smart grid, considering a group of n generators connected through smart grid, constructing an economic dispatch problem model, in which the optimal distribution meeting the demand is obtained under the premise of meeting system constraints, the economic dispatch problem model comprising:

[0051]

[0052] Constraint conditions:

[0053]

[0054] Wherein, C(x) represents the total cost of generators, C i (x i ) represents the cost function of generator i, d i represents the load demand of the generator; represents the minimum power generation of generator i, x i represents the current power generation of generator i, represents the maximum power generation of generator i.

[0055] In the present embodiment, considering a group of n generators connected through smart grid, the network corresponding to the generator group is directed and strongly connected, wherein V={1,2,...,n} represents the vertex set (each vertex is a generator), represents the directed edge set, a directed edge is represented as (i,j)∈ε, which means that node i can transmit information to node j, i is called the inner neighbor of j, similarly, if (j,i)∈ε, j is called the outer neighbor of i, the inner neighbor set of node i is defined as the outer neighbor set is defined as If the communication network is The economic dispatch problem model constructed in this embodiment is unbalanced; a graph is unbalanced if there is a path between any two nodes is strongly connected, and the weighted adjacency matrix of the graph is defined as r ij represents the connection weight between the ith node and the jth node.

[0056] The mathematical model of the economic dispatch problem in the power system is as follows:

[0057]

[0058]

[0059]

[0060] where x i is the power generation of the ith generator, d i is the load demand of the generator, C i (x i ) is the cost function of the generator i, the generator cost function is set to be a smooth general quadratic convex function, which is expressed as: where a i , b i , and c i are cost coefficients, and are the minimum and maximum power generation of the generator, respectively, is the total demand constraint.

[0061] The Lagrange function is introduced to convert the original problem with constraints into an unconstrained problem, and then the unconstrained problem is converted into its dual form, and the Lagrange dual problem is solved by applying the gradient ascent method. The Lagrange function is:

[0062]

[0063] where λ is the Lagrange multiplier, and the dual problem of the original problem is constructed as:

[0064]

[0065] where Φ i (λ) is the Lagrange dual function, which is expressed as: For any given where is the set of real numbers, the dual function Φ i (λ) is differentiable at λ, and its gradient is is the solution of the original problem.

[0066] The standard gradient ascent method to solve the dual problem can be described as:

[0067]

[0068] where t is the iteration index. Finally, the dual problem is converted into the following minimization form:

[0069]

[0070] where q i (λ)=-Φ i (λ), which has the same dual optimal solution set as the dual problem.

[0071] Since each cost function is convex and continuously differentiable, it is shown that the Slater condition holds, i.e., there is strong duality between the primal problem and its dual problem, and the set of dual optimal solutions is nonempty, then there is at least one dual problem optimal solution λ * such that C(x * )=Φ(λ * ), where x * is the optimal solution of the primal problem.

[0072] The directed unbalanced topology adopts a row stochastic weight matrix, and the weight satisfies the following conditions:

[0073]

[0074]

[0075]

[0076] where 0 < η < 1 is a constant, denotes the inner neighbor set of generator i, and the weight r ij > η indicates that the interaction strength between generators will not be too weak.

[0077] A suitable form of the decay step is selected: the uncoordinated step in the consistency algorithm satisfies: where ρ, ω are adjustable parameters, denotes the uncoordinated step at time t; the uncoordinated step gives each generator flexibility in choosing the step size, and the uncoordinated step also satisfies

[0078] This embodiment proposes a distributed accelerated optimization algorithm. This algorithm iteratively updates the Lagrange multipliers in a distributed manner to obtain the optimal solution to the Lagrange dual problem. Under the Slater condition, the optimal solution to the primal problem and its linear convergence rate are derived from the optimal solution to the dual problem, thereby achieving optimal scheduling and allocation. The following consensus protocol is designed to update each variable iteratively to obtain the optimal solution to the dual problem, including:

[0079]

[0080]

[0081]

[0082]

[0083]

[0084] in, Indicates inconsistent step size. The momentum term (Nesterov momentum) coefficient. express The i-th term, Dual function Φ i (λ) is the gradient at time t, and the weight matrix R = {r ij} is a row random matrix, Indicates at time The algorithm utilizes a consensus iteration step to determine the consistent value at that time. It is used for asymptotic estimation of the left eigenvector of a row random matrix, overcoming the imbalance in directed networks.

[0085] After selecting appropriate step size and momentum coefficients, the consensus protocol can eventually iterate to find the optimal solution λ of the dual problem. * Since there is strong duality between the primal problem and the dual problem, the optimal solution x of the primal problem can be found. * Specifically, it is expressed as:

[0086]

[0087] It can be proved using the small gain theorem that the uniformity variable λ t Take O(γ t The rate of convergence to 1 is linear. n λ * Due to the strong convexity of the generator cost function and the Lagrange function, it can be finally derived that... At a linear rate O((γ / 2) t Converging to x * .

[0088] According to the small gain theorem, it can be proved that as long as the step size α and the momentum coefficient β do not exceed certain boundaries, those skilled in the art can derive the specific values ​​of the step size α and the momentum coefficient β based on the small gain theorem. This invention will not elaborate on this.

[0089] Based on the above steps, a comparative experiment was conducted between this invention and its existing (pre-improvement) algorithm that did not incorporate a momentum term. For example... Figure 3 The communication topology shown in this embodiment has N=5 generators, meaning that Bus numbers 1 to 5 are generators. i (x i Let be the cost function of generator i. The generator cost function is set as a smooth, general quadratic convex function, expressed as: Among them, the cost coefficient a of the five generators in this invention i ,b i ,c i like Figure 3 As shown, and These represent the minimum and maximum power outputs of each generator; simulation verification was performed using MATLAB. The flowchart of this invention is as follows. Figure 1 As shown in the experimental results, the optimal output power (x) of each generator is... * ) and optimal incremental cost (λ) * All converge to the same optimal solution as the comparison algorithm, such as Figures 4-9 As shown.

[0090] The top part of the figure shows the simulation results of the algorithm of this invention, and the bottom part shows the simulation results of the algorithm before the improvement.

[0091] like Figures 4-5 The optimal output power of each generator is:

[0092]

[0093] like Figures 6-7 The optimal value for incremental cost is: λ * =7.39kW, Figures 8-9 The results show that the total power generation of the system meets the total demand. It can be seen that the improved optimization algorithm of this invention can achieve the same experimental results as the original comparison algorithm, and the convergence speed of the algorithm is significantly improved.

[0094] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A distributed acceleration optimization method for economic dispatch of smart grid based on directed imbalance topology, characterized in that, Consider a set of n generators connected through a smart grid, construct an economic dispatch problem model, in which the optimal allocation that meets the demand is solved under the premise of meeting system constraints, the economic dispatch problem model includes: wherein, represents the total cost of the generators, represents the cost function of the generators , represents the load demand for the generators; represents the minimum power generation of the generator i, represents the current power generation of the generator i, represents the maximum power generation of the generator i; represents the n-dimensional real number set; In the process of solving the economic dispatch problem model, introduce the Lagrange function to model the economic dispatch problem with constraints into an unconstrained problem, and then convert this unconstrained problem into its dual form, solve the Lagrange dual problem by applying the gradient ascent method, which includes: The process of introducing the Lagrange function to model the economic dispatch problem with constraints into an unconstrained problem includes: ； wherein, represents an economic dispatch problem model of an unconstrained problem, is a Lagrange multiplier; The process of converting this unconstrained problem into its dual form includes: wherein , is the Lagrangian dual function, denoted as , denotes the power generation of the current generator i, denotes the set of power generations of all generators; denotes the set of real numbers.

2. The directed imbalance topology based smart grid economic dispatch distributed accelerated optimization method according to claim 1, characterized in that, The standard gradient ascent method for handling the dual problem includes: ； ； wherein, denotes the Lagrange multiplier at the t+1th iteration, denotes the step size of the gradient ascent algorithm at the tth iteration; denotes the power generation of generator i obtained by solving the correspondence between the solution of the dual problem and the solution of the economic dispatch problem model; .

3. The directed imbalance topology based smart grid economic dispatch distributed accelerated optimization method according to claim 2, characterized in that, Construct a consistency protocol to solve the optimal solution of the dual problem, which includes: wherein, the power generation of generator i at iteration t+1; the inverse function of the gradient of the generator cost function; the consistency value of generator i at iteration t+1; denotes the momentum term coefficient; denotes a tunable parameter; denotes the connection weight between generator i and generator j; denotes the uncoordinated step; denotes the gradient at time t; denotes the i-th term of denotes the consistency value at time t; denotes the inner neighbor set of generator i.

4. The directed imbalance topology based smart grid economic dispatch distributed accelerated optimization method according to claim 3, characterized in that, Connection weight between generator i and generator j Denotes: wherein, is a constant, denotes a set of generators of the inner neighbor set, denotes a set of generators.

5. The directed imbalance topology based smart grid economic dispatch distributed accelerated optimization method according to claim 3, characterized in that, The uncoordinated step of generator i at time t is denoted as and satisfies where is an adjustable parameter.

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