Power system dynamic economic dispatching distributed method based on Cauchy convergence sequence

Abstract

A power system dynamic economic dispatching distributed method based on a Cauchy convergence sequence comprises the following steps: aiming at a centralized dynamic economic dispatching problem, starting from a spatial coupling relation, establishing a space-based distributed dispatching mathematical model; based on the distributed scheduling mathematical model, constructing a Cauchy convergence sequence, and proposing a distributed solving strategy of dynamic economic scheduling of the power system based on the Cauchy convergence sequence; and according to a distributed solving strategy, a convergence condition is constructed, and a current iteration point meeting the convergence condition is a close-enough adjacent point of a global optimal point of the centralized dynamic economic dispatching problem. According to the method disclosed by the invention, the convergence of the distributed strategy of the multi-divisible convex optimization problem is ensured, the characteristics of simple structure and convenience in implementation of the traditional ADMM method are perfectly inherited, and compared with the ADMM method, additional calculation is not needed.

Description

Technical Field

[0001] The invention relates to the technical field of power system economic dispatch, and in particular to a distributed method for dynamic economic dispatch of a power system based on a Cauchy convergence sequence. Background Art

[0002] The problem of economic dispatch of power systems has always been a core issue in the field of power system research. The classic static economic dispatch problem takes into account the constraints of power balance and active output of units, but does not consider the limit of ramp rate. Dynamic economic dispatch (DED) seeks to minimize the total cost within a given period of time to determine the best dispatch strategy while ensuring that multiple operating conditions, including power balance, generator output, and ramp rate, are met. This strategy makes power grid energy management more in line with actual operational needs.

[0003] A variety of mathematical optimization methods have been used to solve the DED problem. Some heuristic optimization algorithms that require global information to obtain the optimal solution, such as particle swarm optimization, whale optimization algorithm, and crow search algorithm, usually obtain local optimal solutions and cannot guarantee the global optimal solution. There are also some centralized methods, such as gradient projection algorithm and Maclaurin series Lagrangian method, which can find the global optimal solution. However, similar to heuristic algorithms, these methods require longer computing cycles due to centralized solutions. For small systems, these methods can be implemented, but as the size of the system increases, they will face the "curse of dimensionality". Therefore, traditional technologies face great difficulties in dealing with such problems.

[0004] In order to solve the above problems, distributed algorithms are a good choice for solving DED problems. Compared with centralized algorithms, they solve smaller problems, so distributed algorithms do not need to face the "curse of dimensionality". For the traditional centralized DED model, it can be decomposed into a series of sub-optimization problems with coupling relationships in time and space. From a temporal perspective, the centralized DED model is regarded as a coupling of sub-optimization problems (ED problems of a single time section) with inequality relationships (climbing / landslide constraints). From a spatial perspective, the centralized DED model is regarded as a coupling of sub-optimizations (all cycle outputs of a single unit including climbing / landslide constraints) with linearly separable equality relationships (power balance). It is obvious that the difficulty of decoupling equality coupling relationships is lower than that of inequality coupling relationships.

[0005] The alternating direction method (ADMM) is an important method for solving convex optimization problems with linearly separable equation relationships. The article entitled "Optimal Dispatching of Electric-Gas Integrated Energy System Based on Improved Alternating Direction Multiplier Method" is published in Transactions of China Electrotechnical Society, Vol. 39, No. 09, 2024, pp. 2797-2809. This article transforms the optimal dispatching model of the electric-gas integrated energy system into distributed solutions for the power system subproblem and the natural gas system subproblem. The article entitled "Distributed Optimization Model of Shared Energy Storage Participating in Grid Ancillary Services Based on ADMM" is published in Electric Power Automation Equipment, Vol. 44, No. 02, 2024, pp. 1-8. This article establishes an optimization model for the combination of active distribution network and shared energy storage; problem decomposition and distributed optimization solution are realized based on ADMM. However, the above two papers deal with convex optimization problems with linearly separable equation relationships. The article entitled “The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent”, CHEN C, HE B, YEY, et al., Mathematical Programming, 2016, 155(1): 57-79. (“The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent”, Mathematical Programming, 2016, Vol. 155, No. 01, pp. 57-79) clearly points out that a condition that can fully ensure the convergence of the direct extension of ADMM requires that the coefficient matrix given in the model is orthogonal; this sufficient condition is only of theoretical significance. When it comes to DED problems, it cannot be met, so ADMM cannot be directly used to achieve distributed solution. Summary of the invention

[0006] In order to overcome the limitations of various technical solutions given in the background technology, the present invention proposes a distributed method for dynamic economic dispatch of power systems based on the Cauchy convergence sequence, which ensures the convergence of distributed strategies for multi-separable convex optimization problems and perfectly inherits the characteristics of the traditional ADMM method, which is simple in structure and easy to implement, and does not require additional calculations compared with ADMM.

[0007] The technical solution of the present invention is implemented as follows: a distributed method for dynamic economic dispatch of power system based on Cauchy convergence sequence,

[0008] Step A: Aiming at the centralized dynamic economic dispatch problem, starting from the spatial coupling relationship, a space-based distributed dispatch mathematical model is established;

[0009] Step B: Based on the distributed dispatching mathematical model of step A, construct a Cauchy convergence sequence, and propose a distributed solution strategy for dynamic economic dispatch of power system based on the Cauchy convergence sequence;

[0010] Step C: According to the distributed solution strategy of step B, a convergence condition is constructed, and the current iteration point that meets the convergence condition is a sufficiently close neighboring point of the global optimal point of the centralized dynamic economic scheduling problem.

[0011] Furthermore, in step A, the space-based distributed scheduling mathematical model is as follows:

[0012]

[0013] P i ∈Ω i ,i=1,2,…,N

[0014] in:

[0015]

[0016] P i =(P i,1 ,P i,2 ,…,P i,S ) T

[0017]

[0018] b=(P D,1 ,P D,2 ,…,P D,S ) T

[0019]

[0020] A i =I S×S

[0021] Among them, f i,s (P i,s ) is expressed as the cost of the i-th generator set in time period s, P i,s is the generator output power of the i-th generator set in time period s, i is an integer, and 1≤i≤N, N represents the total number of generator sets, S represents the total number of time periods, and parameter a i ,b i ,c i is the coefficient of the given quadratic function; P D,s is the power demand in time period s, Indicates that the total output power of all generator sets in time period s must be equal to the power demand P in that time period D,s , I S×Srepresents the identity matrix with S rows and S columns;

[0022] P i,min ≤P i,s ≤P i,max is the output limit of generator set i, and the feasible domain is [P i,min ,P i,max ], P i,min represents the minimum output of generator set i, P i,max represents the maximum output of generator set i; represents the climbing characteristic constraint of the generator set, It represents the landslide characteristic constraint of the generating unit; is the ramp rate limit of the ith generator set (the superscript i represents the ith generator set, and the subscript u represents the ramp rate), is the landslide rate limit of the i-th generator set (the superscript i represents the i-th generator set, and the subscript d represents landslide).

[0023] Furthermore, in step B, the distributed solution strategy for dynamic economic dispatch of power system based on Cauchy convergence sequence is:

[0024]

[0025] Among them, the superscript k represents the kth iteration; L is a positive integer less than the maximum number of iterations, c0>0 is the initial value of parameter c; λ represents the Lagrange multiplier, the superscript T is the matrix transpose symbol, β i >(Ni)c k .

[0026] Furthermore, in step C, the convergence condition is constructed as:

[0027] The convergence index ε>0.

[0028] Furthermore, step A: for the centralized dynamic economic dispatch model, starting from the spatial coupling relationship, a space-based distributed dispatch mathematical model is established.

[0029] Furthermore, for the centralized dynamic economic dispatch problem, the corresponding dispatch model is as follows:

[0030]

[0031] P i,min ≤P i,s ≤P i,max ,s=1,2,…,S

[0032]

[0033] Among them, F is the minimum total power generation cost, F is a double summation: the inner summation is the summation of all generator sets i from 1 to N, and the outer summation is the summation of all time periods s from 1 to S; considering that each generator set i is located in a different geographical space, the equality constraint is used As the coupling condition of the distributed scheduling mathematical model, it is only necessary to decouple the equality constraint condition to establish the space-based distributed scheduling mathematical model.

[0034] Furthermore, β i =(N-i+1)c k .

[0035] Beneficial effects of the present invention:

[0036] The present invention provides a distributed method for dynamic economic dispatch of power systems based on convergence sequences. Compared with the traditional ADMM method, it ensures the convergence of the distributed strategy for multi-separable convex optimization problems, and perfectly inherits the characteristics of the traditional ADMM method of simple structure and easy implementation. Compared with ADMM, it does not require additional calculations.

[0037] The distributed solution strategy of the present invention adds a strictly convex quadratic function to the objective function of the i-th sub-optimization problem of the classic ADMM If the original function is a convex function, then the convex function plus a strictly convex function, the new function is still a convex function. At the same time, the strictly convex quadratic function is very convenient to solve and will not increase the difficulty of solving the problem. It also perfectly inherits the characteristics of the traditional ADMM method, which is simple in structure and easy to implement. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0039] Figure 1 It is a trend diagram of the convergence condition of the distributed method of the present invention under different initial penalty parameters;

[0040] Figure 2 This is a diagram showing the output stacking of the generator sets in the distributed method of the present invention. DETAILED DESCRIPTION

[0041] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0042] A distributed method for dynamic economic dispatch of power system based on Cauchy convergence sequence, comprising the following steps:

[0043] Step A: For the centralized dynamic economic dispatch problem, the corresponding dispatch model is as follows:

[0044]

[0045] P i,min ≤P i,s ≤P i,max ,s=1,2,…,S

[0046]

[0047] Where F is the minimum total power generation cost, F is a double summation: the inner summation is the summation of all generator sets i from 1 to N, and the outer summation is the summation of all time periods s from 1 to S; f i,s (P i,s ) is expressed as the cost of the i-th generator set in time period s (the cost is expressed in the form of a quadratic function, where the parameter a i ,b i ,c i is the coefficient of the given quadratic function). i,s is the generator output power of the i-th generator set at time s, i is an integer, and 1≤i≤N, where N represents the total number of generator sets. Indicates that the total output power of all generator sets in time period s must be equal to the power demand P in that time period D,s .P i,min ≤P i,s ≤P i,max is the output limit of generator set i, and the feasible domain is [P i,min ,P i,max ], P i,min represents the minimum output of generator set i, P i,max represents the maximum output of generator set i; represents the climbing characteristic constraint of the generator set, Represents the landslide characteristic constraint of the generator set. is the ramp rate limit of the ith generator set (the superscript i represents the ith generator set, and the subscript u represents the ramp rate), is the landslide rate limit of the ith generator set (the superscript i represents the ith generator set, and the subscript d represents landslide). S represents the total number of time cycles.

[0048] Based on the spatial coupling relationship of the traditional centralized dynamic economic dispatch mathematical model, a space-based distributed dispatch mathematical model can be established; its spatial coupling relationship is: Considering that there are a total of S time periods, there are a total of S equality constraints based on the spatial coupling relationship. Based on the time coupling relationship, a time-based distributed scheduling mathematical model can be established for the traditional centralized dynamic economic dispatch mathematical model; its time coupling relationship is the unit climbing / slip characteristic constraint. Considering that there are a total of N generators and S time periods, there are a total of 2×N×(S-1) inequality constraints based on the time coupling relationship. Considering that 2×N×(S-1) is much larger than S, and considering that the difficulty of handling inequality constraints is much greater than that of equality constraints, it is natural to choose a space-based distributed dispatch mathematical model.

[0049] For the centralized dynamic economic dispatch model, from the spatial coupling relationship Starting from this, the space-based distributed scheduling mathematical model is established as follows:

[0050]

[0051] P i ∈Ω i ,i=1,2,…,N

[0052] in:

[0053]

[0054] P i =(P i,1 ,P i,2 ,…,P i,S ) T

[0055]

[0056]

[0057] A i =I S×S

[0058] Step B: Directly use ADMM to solve the space-based distributed scheduling model in step A:

[0059]

[0060] Where c is a given positive number, and the superscript T is the matrix transpose symbol.

[0061] The above ADMM-based solution to the space-based distributed scheduling model strategy is equivalent to solving the variational inequality:

[0062]

[0063] in

[0064]

[0065] The superscript k indicates the kth iteration, the superscript * indicates the optimal solution of the EDE scheduling model, λ indicates the Lagrange multiplier, and I S×S represents the identity matrix with S rows and S columns.

[0066] where ||·|| G represents the distance metric under the G matrix, (x is a given column vector), Represents the measure of the distance between the k-th iteration result and the optimal solution under the matrix G. Similarly, It represents the measure of the distance between the k+1 iteration result and the optimal solution under the matrix G.

[0067] Considering the specific representation of the matrix G, it is impossible to determine whether ||·|| G Whether it meets the definition of norm, we use a positive definite matrix M to replace the matrix G, ||·|| M is called the M-norm, (x is a given column vector), ||·|| M This conforms to the definition of norm. Now construct a sequence {A k},in At this time, it can be guaranteed holds, and the equality sign is only in P k+1 =P k When it holds, then the sequence {A k} is a monotonically decreasing sequence with a lower bound, then the sequence {A k}converges and must have an infimum.

[0068] Because the sequence {A k} converges and must have an infimum, so we know Therefore, we can see that when k→∞ According to the squeeze principle, According to the basic properties of the norm, we know that the M norm must be greater than or equal to zero. Therefore, we know that for There exists a positive integer Q such that when k>Q, ||P k -P k+1 || M <ε. According to the Cauchy convergence criterion, we know that the sequence {Pk} is convergent. This means that if there is a positive definite matrix M replacing the matrix G to achieve the distributed computing of the strategy, then the sequence {P k} is convergent, which means that the strategy is convergent.

[0069] The purpose of constructing the positive definite matrix M is to ensure that the algorithm converges and is serially distributed. Therefore, the constructed positive definite matrix M retains the non-diagonal elements on the basis of the matrix G, and only the diagonal elements can be changed. A sufficient condition is that the non-diagonal elements of the M matrix are equal to the non-diagonal elements of the G matrix, the diagonal elements of the M matrix are taken so that the M matrix is ​​a diagonally dominant matrix, and the parameter β i >(Ni)c. The constructed matrix M is a positive definite matrix, which retains the serial distribution characteristics of the ADMM method and can theoretically guarantee the convergence of the strategy. The positive definite matrix M can be written as:

[0070]

[0071] Therefore, the distributed solution strategy for dynamic economic dispatch of power system based on Cauchy convergence sequence can be expressed as:

[0072]

[0073] Observe any sub-optimization problem, take the first sub-optimization problem as an example, f1(P1) is the original optimization target, -λ T,k A1P1 is the Lagrangian function part, To augment the Lagrangian part, we can also consider the external penalty function, In order to ensure the convergence of the strategy, it is recommended to take β1 = Nc k In the entire strategy iteration process, the only parameter that needs to be designed is c k .

[0074] Considering External penalty function, so it is natural to design c k is a strictly monotonically increasing sequence, which speeds up convergence. But at the same time, The matrix M is a fixed positive definite matrix. If c k If it is a strictly monotonically increasing sequence, the matrix M is a variable matrix, and the convergence of the algorithm cannot be guaranteed theoretically. Therefore, we design the parameter c k It is a strictly monotonically increasing sequence in the first finite number of iterations and remains unchanged afterwards. Then the sequence is greater than zero and has upper and lower bounds. In this way, after a finite number of iterations, the matrix M is the first fixed positive definite matrix, which can ensure the convergence of the strategy and effectively improve the convergence characteristics of the strategy. A reasonable design scheme is:

[0075]

[0076] Where L is a positive integer less than the maximum number of iterations, c0>0 is c k The initial value of .

[0077] Step C: According to the distributed solution strategy, the original gap is expressed as: The dual gap of the i-th sub-optimization problem is expressed as: For the dual gap of the i-th sub-optimization problem obtained at the k-th iteration, the parameter c k and β i are all positive numbers with upper and lower bounds, so the convergence condition can be simplified to:

[0078]

[0079] A sufficiently small ε will inevitably ensure that the primal-dual gap approaches 0, that is, the current solution obtained in step B is close to the optimal solution of the original problem, and the simplified convergence condition must satisfy:

[0080]

[0081] This index is exactly the convergence judgment index of the external penalty method using a variable penalty parameter. The convergence index of the distributed method for dynamic economic dispatch of power system based on convergence sequence proposed by the present invention meets the convergence judgment index of the external penalty method, realizing the integration of the two.

[0082] The specific embodiments are as follows:

[0083] In one embodiment, a 10-machine 8-cycle power system economic dispatch problem is given, and the generator set data is shown in Table 1. Table 1 contains 8 columns of information, the first column is the unit number; the second and third columns are the maximum and minimum outputs of the generator set; the generator set cost function is a quadratic function, and the 4th to 6th columns are the corresponding generator set quadratic function coefficients; the 7th and 8th columns are the generator set climbing and landslide characteristic constraints. The total load of the 8-cycle system is 1036MW, 1110MW, 1258MW, 1406MW, 148MW 0, 1628MW, 1702MW, 1776MW. Assume that the initial value of the generator set output is 0.5×(maximum value + minimum value). Convergence index ε=10 -4 The maximum number of iterations is set to 50. k , and its monotonically increasing sequence parameter L=30.

[0084] Figure 1From the changing trend of the convergence conditions under different initial penalty parameters, it can be seen that the distributed solution strategy for dynamic economic dispatch of power system based on Cauchy convergence sequence proposed in the present invention can effectively realize the distributed solution of dynamic economic dispatch problems and is robust to parameters. Figure 2 The horizontal axis is time, representing 8 time periods, and the vertical axis represents the unit output. Each bar represents the total unit output in a time period. The bar is divided into 10 parts, and each part from bottom to top represents the output of units 1-10 in this time period. The black solid line is the actual load of the system in 8 time periods. The figure shows that the dispatch result obtained by using the distributed solution strategy for dynamic economic dispatch of power system based on Cauchy convergence sequence proposed by the present invention fully meets the system load demand.

[0085] Table 1.10 Unit data

[0086]

[0087]

[0088] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A distributed method for dynamic economic dispatch of power system based on Cauchy convergence sequence, characterized by: Step A: Aiming at the centralized dynamic economic dispatch problem, starting from the spatial coupling relationship, a space-based distributed dispatch mathematical model is established; Step B: Based on the distributed dispatching mathematical model of step A, construct a Cauchy convergence sequence, and propose a distributed solution strategy for dynamic economic dispatch of power system based on the Cauchy convergence sequence; Step C: According to the distributed solution strategy of step B, a convergence condition is constructed, and the current iteration point that meets the convergence condition is a sufficiently close neighboring point of the global optimal point of the centralized dynamic economic scheduling problem.

2. The distributed method for dynamic economic dispatch of power system based on Cauchy convergence sequence according to claim 1, characterized in that: In step A, the space-based distributed scheduling mathematical model is as follows: in: Pi=(Pi,1,Pi,2,…,Pi, S ) T b=(P D,1 ,P D,2 ,…,P D,S ) T A i =I S×S Among them, f i,s (P i,s ) is expressed as the cost of the i-th generator set in time period s, P i,s is the generator output power of the i-th generator set in time period s, N represents the total number of generator sets, S represents the total number of time periods, and parameter a i ,b i ,c i is the coefficient of the given quadratic function; P i,min represents the minimum output of generator set i, P i,max represents the maximum output of generator set i; is the ramp rate limit of the i-th generator set, is the landslide rate limit of the i-th generator set, P D,s is the power demand in time period s, I S×S represents the identity matrix with S rows and S columns.

3. The distributed method for dynamic economic dispatch of power system based on Cauchy convergence sequence according to claim 2 is characterized by: In step B, the distributed solution strategy for dynamic economic dispatch of power system based on Cauchy convergence sequence is: Among them, the superscript k represents the kth iteration; L is a positive integer less than the maximum number of iterations, c0>0 is the parameter c k The initial value of ;λ represents the Lagrange multiplier, the superscript T is the matrix transpose symbol, β i >(Ni)c k .

4. The distributed method for dynamic economic dispatch of power system based on Cauchy convergence sequence according to claim 3 is characterized by: In step C, the convergence condition is constructed as: The convergence index ε>

0.

5. The distributed method for dynamic economic dispatch of power system based on Cauchy convergence sequence according to claim 2 is characterized by: Step A: Centralized dynamic economic dispatch problem, the corresponding dispatch model is as follows: Among them, F is the minimum total power generation cost; considering that each generator set i is located in a different geographical space, the equation is constrained As the coupling condition of the distributed scheduling mathematical model, it is only necessary to decouple the equality constraint condition to establish the space-based distributed scheduling mathematical model.

6. The distributed method for dynamic economic dispatch of power system based on Cauchy convergence sequence according to claim 3 is characterized by: β i =(N-i+1)c k 。

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