CN103500997B

CN103500997B - Electric power system dispatching method based on hybrid multi-objective lambda iteration method and Newton method

Info

Publication number: CN103500997B
Application number: CN201310422944.3A
Authority: CN
Inventors: 吴青华; 詹俊鹏; 周孝信
Original assignee: South China University of Technology SCUT
Current assignee: South China University of Technology SCUT
Priority date: 2013-09-16
Filing date: 2013-09-16
Publication date: 2015-07-01
Anticipated expiration: 2033-09-16
Also published as: CN103500997A

Abstract

The invention discloses a power system scheduling method based on hybrid multi-objective lambda iteration method and Newton's method, comprising the following steps: 1) obtaining relevant data of each generating set in the power system; Mathematical optimization model; 3) Based on the multi-objective Kuhn-Tucker optimal condition, the multi-objective λ iterative method is used to solve the EED problem without considering the transmission line loss, and the Pareto optimal solution is obtained; 4) each Pareto optimal The optimal solution is used as the initial solution, and Newton’s method is used to solve the EED problem considering the loss of the transmission line to obtain the optimal solution set; 5) The multi-objective decision-making method is used to determine the final solution in the optimal solution set; 6) The final solution is used as an instruction through automatic power generation The control device is sent to the relevant power plant or unit, and the control of the generating power of the unit is realized through the automatic control and adjustment device. The method of the invention has small calculation amount, short calculation time and high convergence precision, and greatly improves the economy and efficiency of power system power generation.

Description

Power System Scheduling Method Based on Hybrid Multi-objective Lambda Iteration Method and Newton Method

技术领域technical field

本发明涉及一种电力系统调度方法，尤其是一种基于混合多目标λ迭代法和牛顿法的电力系统调度方法，属于电力系统的运行、分析和调度领域。The invention relates to a power system dispatching method, in particular to a power system dispatching method based on a hybrid multi-objective lambda iteration method and a Newton method, and belongs to the field of power system operation, analysis and dispatch.

背景技术Background technique

经济调度是电力系统的一个重要的基本问题^[1][2][3]，随着近年来环境污染成为一个全球性问题，作为污染物排放主要来源的电力企业被要求降低污染物排放。以中国为例，全国SO2和NOx的总排放中，火电机组分别占了42.5％和38.0％。为降低排放，一种方式是安装脱硫和脱氮装置，另一种方式则是在发电优化调度中选择排放较小的方案。为此传统的经济调度变成了一个多目标优化问题，即本发明中的环境经济调度问题[4]，该问题要求同时降低发电成本和污染物排放值。Economic dispatch is an important basic problem in power system ^[1][2][3] , as environmental pollution has become a global problem in recent years, power companies, which are the main source of pollutant emissions, are required to reduce pollutant emissions. Taking China as an example, thermal power units accounted for 42.5% and 38.0% of the total national SO2 and NOx emissions, respectively. In order to reduce emissions, one way is to install desulfurization and denitrification devices, and the other way is to choose a solution with less emissions in the optimal dispatch of power generation. For this reason, the traditional economic dispatch becomes a multi-objective optimization problem, that is, the environmental economic dispatch problem in the present invention [4], which requires simultaneously reducing power generation cost and pollutant discharge value.

为了求解该问题，学者们采用了加权法^[5][6][7][8]，加权法的优势在于其方法简单，但是其权值的设置却并不容易，需要根据经验进行设置，且对于不同问题需设置不同的权值，即使同一个问题但参数(如系统负荷)不同也需设置不同的权值。此项不足降低了其在解决实际问题中的实用性。另一种解决该问题的方法是多目标进化算法^{[4][9][10][11][12][13][14][15][16][17]}。多目标进化算法的优势在于其可用于求解各种非凸的不连续的不可导的复杂优化问题，但是这些算法在缺乏问题具体信息的情况下对解空间进行随机搜索其效率低，计算量大。随着所述问题机组数量增加，其解空间愈加复杂，这些算法耗费的计算量急剧增加且求解精度不能得到保证。多目标进化算法对于约束条件的处理大多采用罚函数法，罚函数值对于不同系统不尽相同，需要根据经验进行调整设置，降低了其在解决实际问题中的实用性。In order to solve this problem, scholars have adopted the weighting method ^[5][6][7][8] . The advantage of the weighting method is that its method is simple, but the setting of its weight value is not easy, and it needs to be set according to experience. And different weights need to be set for different problems, even if the same problem has different parameters (such as system load), different weights need to be set. This shortcoming reduces its practicality in solving practical problems. Another approach to solve this problem is the multi-objective ^{evolutionary algorithm [4][9][10][11][12][13][14][15][16][17].} The advantage of multi-objective evolutionary algorithm is that it can be used to solve various non-convex, discontinuous, non-differentiable complex optimization problems, but these algorithms perform random searches on the solution space in the absence of specific information about the problem, which is inefficient and requires a large amount of calculation. . As the number of units of the problem increases, its solution space becomes more and more complex, and the amount of computation consumed by these algorithms increases sharply, and the solution accuracy cannot be guaranteed. The multi-objective evolutionary algorithm mostly adopts the penalty function method to deal with the constraint conditions. The value of the penalty function is different for different systems. It needs to be adjusted and set according to experience, which reduces its practicability in solving practical problems.

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发明内容Contents of the invention

本发明的目的是为了解决上述现有技术需要设置不同目标函数之间的加权值，约束条件对应的罚函数值对于不同系统不尽相同，需要根据经验进行调整设置，而且求解精度不能得到保证，所耗费计算时间也较长的缺陷，提供一种适合求解大规模电力系统问题，可以提高经济效率的基于混合多目标λ迭代法和牛顿法的电力系统调度方法。The purpose of the present invention is to solve the above-mentioned prior art that needs to set weighted values between different objective functions. The penalty function values corresponding to the constraint conditions are different for different systems, and need to be adjusted and set according to experience, and the solution accuracy cannot be guaranteed. It takes a long time to calculate, and it provides a power system scheduling method based on hybrid multi-objective lambda iteration method and Newton method, which is suitable for solving large-scale power system problems and can improve economic efficiency.

本发明的目的可以通过采取如下技术方案达到：The purpose of the present invention can be achieved by taking the following technical solutions:

基于混合多目标λ迭代法和牛顿法的电力系统调度方法，其特征在于包括以下步骤：The power system scheduling method based on hybrid multi-objective λ iterative method and Newton method is characterized in that it comprises the following steps:

1)获得具有多台发电机组的电力系统中每台机组的出力上限与下限数据、出力-燃料费用函数的系数数据、出力-排放函数的系数数据、输电线路损耗的B系数数据和系统总负荷数据；1) Obtain the output upper limit and lower limit data of each unit in the power system with multiple generator sets, the coefficient data of the output-fuel cost function, the coefficient data of the output-emission function, the B coefficient data of the transmission line loss and the total system load data;

2)根据步骤1)所获得的数据，建立电力系统环境经济调度问题的数学优化模型；2) According to the data obtained in step 1), establish a mathematical optimization model of the power system environmental economic dispatch problem;

3)根据步骤2)所建立的模型，基于多目标库恩塔克最优条件，采用多目标λ迭代法(MλI)求解不考虑输电线路损耗的电力系统环境经济调度问题，得到一组帕雷托最优解；3) According to the model established in step 2), based on the multi-objective Kuhn-Tucker optimal condition, the multi-objective λ iterative method (MλI) is used to solve the environmental economic dispatching problem of the power system without considering the loss of transmission lines, and a set of Paley support the optimal solution;

4)将步骤3)得到的每一个帕雷托最优解作为初始解，采用牛顿法求解考虑输电线路损耗的电力系统环境经济调度问题，得到最优解集；4) Use each Pareto optimal solution obtained in step 3) as an initial solution, and use Newton's method to solve the power system environmental economic dispatch problem considering the transmission line loss, and obtain the optimal solution set;

5)采用多目标决策方法在步骤4)得到的最优解集中确定最终解；5) adopt the multi-objective decision-making method to determine the final solution in the optimal solution set that step 4) obtains;

6)将步骤5)确定的最终解作为指令通过自动发电控制装置发送给相关发电厂或机组，通过发电厂或机组的自动控制调节装置，实现对机组发电功率的控制。6) The final solution determined in step 5) is sent as an instruction to the relevant power plant or unit through the automatic power generation control device, and the control of the generating power of the unit is realized through the automatic control and adjustment device of the power plant or unit.

具体的，步骤1)所述电力系统中机组数量为n_g台，则步骤2)所述电力系统环境经济调度问题的数学优化模型的建立过程，具体如下：Concretely, the number of units in the power system described in step 1) is n _g units, then the establishment process of the mathematical optimization model of the power system environmental economic dispatching problem described in step 2) is as follows:

2.1)机组i的出力-燃料费用函数如下式所示：2.1) The output-fuel cost function of unit i is shown in the following formula:

${f f}_{fuel fuel} (({x x}_{i i})) = = {a a}_{i i} + + {b b}_{i i} {x x}_{i i} + + {c c}_{i i} {x x}_{i i}^{22} - - - - - - ((11))$

其中，1≤i≤n_g，f_fuel(x_i)为机组i的出力-燃料费用函数，x_i为机组的有功出力，a_i、b_i和c_i为机组i的出力-燃料费用系数；Among them, _1≤i≤ng , f _fuel (xi ₎ is the output-fuel cost function of unit i, x _i is the active output of the unit, a _i , b _i and ci are the output-fuel cost coefficients of unit _i ;

2.2)机组i的出力-排放函数如下式所示：2.2) The output-discharge function of unit i is shown in the following formula:

${f f}_{emi emi} (({x x}_{i i})) = = {α α}_{i i} + + {β β}_{i i} {x x}_{i i} + + {γ γ}_{i i} {x x}_{i i}^{22} + + {ϵ ϵ}_{i i} {e e}^{{ξ ξ}_{i i} {x x}_{i i}} - - - - - - ((22))$

其中，f_emi(x_i)为机组i的出力-排放函数，α_i、β_i、γ_i、ε_i和ξ_i为机组i的出力-排放函数系数；Among them, f _emi (xi ₎ is the output-emission function of unit i, and α _i , β _i , γ _i , ε _i and ξ _i are the output-emission function coefficients of unit i;

2.3)机组i的出力上限与下限约束如下式所示：2.3) The output upper limit and lower limit constraints of unit i are shown in the following formula:

${x x}_{i i}^{min min} \leq \leq {x x}_{i i} \leq \leq {x x}_{i i}^{max max} - - - - - - ((33))$

其中，和分别为机组i的出力上限与下限；in, and are the output upper limit and lower limit of unit i respectively;

2.4)考虑输电线路损耗的负荷平衡约束如下式所示：2.4) Considering the load balance constraint of transmission line loss is shown in the following formula:

${Σ Σ}_{i i = = 11}^{{n no}_{g g}} {x x}_{i i} - - {x x}_{D D.} - - {x x}_{L L} = = 00 - - - - - - ((44))$

其中，x_D为系统总负荷，x_L为输电线路损耗，x_L采用如下式所示的B系数法计算：Among them, x _D is the total load of the system, x _L is the transmission line loss, and x _L is calculated by the B coefficient method shown in the following formula:

${x x}_{L L} = = {Σ Σ}_{i i = = 11}^{{n no}_{g g}} {Σ Σ}_{j j = = 11}^{{n no}_{g g}} {x x}_{i i} {B B}_{ij ij} {x x}_{j j} + + {Σ Σ}_{i i = = 11}^{{n no}_{g g}} {B B}_{i i 00} {x x}_{i i} + + {B B}_{0000} - - - - - - ((55))$

其中，B_ij、B_i0和B₀₀为输电线路损耗的B系数；Among them, B _ij , B _i0 and B ₀₀ are B coefficients of transmission line losses;

2.5)考虑输电线路损耗的电力系统环境经济调度问题的数学优化模型如下：2.5) The mathematical optimization model of the power system environmental economic dispatch problem considering transmission line loss is as follows:

$Minimize Minimize : : f f ((x x)) = = [\begin{matrix} {f f}_{11} ((x x)) \\ {f f}_{22} ((x x)) \end{matrix}] = = [\begin{matrix} {Σ Σ}_{i i = = 11}^{{n no}_{g g}} {f f}_{fuel fuel} (({x x}_{i i})) \\ {Σ Σ}_{i i = = 11}^{{n no}_{g g}} {f f}_{emi emi} (({x x}_{i i})) \end{matrix}] - - - - - - ((66))$

s.t. (3)(4)(5)s.t. (3)(4)(5)

其中，f_fuel(x_i)和f_emi(x_i)分别由式(1)和式(2)表示。Wherein, f _fuel ( _xi ) and f _emi ( _xi ) are represented by formula (1) and formula (2), respectively.

具体的，步骤3)所述采用多目标λ迭代法求解不考虑输电线路损耗的电力系统环境经济调度问题，得到帕雷托最优解，具体如下：Specifically, in step 3), the multi-objective λ iterative method is used to solve the environmental economic dispatching problem of the power system without considering the transmission line loss, and the Pareto optimal solution is obtained, as follows:

3.1)给定固定出力机组的一个出力值；3.1) An output value of a fixed output unit is given;

3.2)对于一个很大的λ值，求得每台机组对应的有功出力，计算发电负荷不平衡量；3.2) For a large λ value, obtain the corresponding active output of each unit, and calculate the unbalanced amount of power generation load;

3.3)对于一个很小的λ值，求得每台机组对应的有功出力，计算发电负荷不平衡量；3.3) For a very small λ value, obtain the corresponding active output of each unit, and calculate the unbalanced amount of power generation load;

3.4)若步骤3.2)和步骤3.3)计算的发电负荷不平衡量同号，则不存在潜在帕雷托最优解，返回步骤3.1)；若步骤3.2)和步骤3.3)计算的发电负荷不平衡量不同号，则采用二分法修改λ值，直至找到一个λ值，其对应的每台机组有功出力满足负荷平衡，保存该λ值和每台机组的有功出力，各机组的有功出力即为帕雷托最优解。3.4) If the power generation load unbalance calculated in step 3.2) and step 3.3) have the same sign, then there is no potential Pareto optimal solution, return to step 3.1); if the power generation load unbalance calculated in step 3.2) and step 3.3) is different number, then use the dichotomy method to modify the λ value until a λ value is found, and the corresponding active output of each unit satisfies the load balance, save the λ value and the active output of each unit, and the active output of each unit is Pareto Optimal solution.

具体的，所述发电负荷不平衡量为各机组总发电功率与系统总负荷之间的差值。Specifically, the unbalanced power generation load is the difference between the total power generated by each unit and the total load of the system.

具体的，所述每台机组对应的有功出力使每两台机组之间满足等差比微增率法则。Specifically, the active power output corresponding to each unit satisfies the law of incremental rate of equal difference ratio between every two units.

具体的，步骤5)所述多目标决策方法采用逼近于理想值的排序方法，具体如下：Concrete, step 5) described multi-objective decision-making method adopts the sorting method approaching ideal value, specifically as follows:

5.1)首先，计算标幺化的加权决策矩阵v_ij：5.1) First, calculate the per-unit weighted decision matrix v _ij :

${v v}_{ij ij} = = {ω ω}_{i i} (({f f}_{i i}^{+ +} - - {f f}_{ij ij})) / / (({f f}_{i i}^{+ +} - - {f f}_{i i}^{- -})),, i i = = 1,2,3 1,2,3,, \cdot \cdot \cdot \cdot \cdot &Center Dot;,, {N N}_{obj obj},, j j = = 1,2,3 1,2,3,, \cdot \cdot \cdot &Center Dot; \cdot &Center Dot;,, J J$

其中， $f_{i}^{-} = \min_{j} f_{ij}, f_{i}^{+} = \max_{j} f_{ij}, ω_{i} = \frac{1}{N_{obj}} Σ_{j = i}^{N_{obj}} \frac{1}{j}, i = 1,2,3, \cdot \cdot \cdot, N_{obj},$ f_ij为最优解集中的第j个解的第i个目标函数值，N_obj为多目标优化中目标的个数，J为最优解集中解的个数；in, $f_{i}^{-} = \min_{j} f_{ij}, f_{i}^{+} = \max_{j} f_{ij}, ω_{i} = \frac{1}{N_{obj}} Σ_{j = i}^{N_{obj}} \frac{1}{j}, i = 1,2,3, &Center Dot; &Center Dot; &Center Dot;, N_{obj},$ f _ij is the i-th objective function value of the j-th solution in the optimal solution set, N _obj is the number of objectives in multi-objective optimization, and J is the number of solutions in the optimal solution set;

5.2)分别计算最理想点A⁺和最不理想点A^-：5.2) Calculate the most ideal point A ⁺ and the least ideal point A ^- respectively:

${A A}^{+ +} = = {{{v v}_{11}^{+ +},, {v v}_{22}^{+ +},, \cdot &Center Dot; \cdot &Center Dot; \cdot \cdot,, {v v}_{N N}^{+ +}}}$

${A A}^{- -} = = {{{v v}_{11}^{- -},, {v v}_{22}^{- -},, \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot;,, {v v}_{N N}^{- -}}}$

其中， $v_{i}^{+} = \max_{j} v_{ij}, v_{i}^{-} = \min_{j} v_{ij};$ in, $v_{i}^{+} = \max_{j} v_{ij}, v_{i}^{-} = \min_{j} v_{ij};$

5.3)分别计算每一个最优解到最理想点的距离D⁺和到最不理想点的距离D^-：5.3) Calculate the distance D ⁺ from each optimal solution to the most ideal point and the distance D ^- to the least ideal point respectively:

${D D.}_{j j}^{+ +} = = \sqrt{{Σ Σ}_{i i = = 11}^{N N} {(({v v}_{ij ij} - - {v v}_{i i}^{+ +}))}^{22}}$

${D D.}_{j j}^{- -} = = \sqrt{{Σ Σ}_{i i = = 11}^{N N} {(({v v}_{ij ij} - - {v v}_{i i}^{- -}))}^{22}}$

其中，j＝1，2，3，…J；Among them, j=1, 2, 3, ... J;

5.4)计算每个最优解的距离比R_j：5.4) Calculate the distance ratio R _j of each optimal solution:

${R R}_{j j} = = {D D.}_{j j}^{- -} / / (({D D.}_{j j}^{- -} + + {D D.}_{j j}^{+ +}))$

其中，j＝1，2，3，…J；Among them, j=1, 2, 3, ... J;

5.5)将R_j最大的最优解选择为最终的机组检修及出力方案。5.5) Select the optimal solution with the largest R _j as the final unit maintenance and output plan.

本发明相对于现有技术具有如下的有益效果：Compared with the prior art, the present invention has the following beneficial effects:

1、本发明方法根据多目标库恩塔克条件，求得的解是全局最优解而非近似最优解；其计算量与机组数量成线性关系，适合求解大规模电力系统问题。1. According to the multi-objective Kuhn-Tucker condition, the solution obtained by the method of the present invention is a global optimal solution rather than an approximate optimal solution; its calculation amount is linearly related to the number of units, and is suitable for solving large-scale power system problems.

2、本发明方法在适用于大量机组的大规模电力系统问题中，计算量小，计算时间短，收敛精度高，在含有大量机组的大规模电力系统中其在精度和计算量方面的优势明显，能极大地提高电力系统发电的经济性和效率。2. The method of the present invention is suitable for large-scale power system problems with a large number of units, with small calculation amount, short calculation time and high convergence accuracy. It has obvious advantages in accuracy and calculation amount in large-scale power systems containing a large number of units , can greatly improve the economy and efficiency of power system power generation.

附图说明Description of drawings

图1为本发明方法的电力系统调度方法流程示意图。Fig. 1 is a schematic flow chart of a power system dispatching method according to the method of the present invention.

图2为本发明方法与MOPSO算法在6机组EED问题上得到的帕雷托前沿曲线图。Fig. 2 is the Pareto front curve obtained by the method of the present invention and the MOPSO algorithm on the EED problem of 6 units.

图3为本发明方法与MOPSO算法在14机组EED问题上得到的帕雷托前沿曲线图。Fig. 3 is the Pareto front curve obtained by the method of the present invention and the MOPSO algorithm on the EED problem of 14 units.

图4为本发明方法与MOPSO算法在140机组EED问题上得到的帕雷托前沿曲线图。Fig. 4 is the Pareto front curve obtained by the method of the present invention and the MOPSO algorithm on the EED problem of 140 units.

具体实施方式Detailed ways

实施例1：Example 1:

如图1所示，本实施例的基于混合多目标λ迭代法和牛顿法的电力系统调度方法包括以下步骤：As shown in Figure 1, the power system scheduling method based on the hybrid multi-objective lambda iteration method and Newton method of the present embodiment includes the following steps:

1)获得具有n_g台发电机组的电力系统中每台机组的出力上限与下限数据和出力-燃料费用函数的系数数据a_i，b_i，c_i、出力-排放函数的系数数据α_i，β_i，γ_i，ε_i，ξ_i、输电线路损耗的B系数数据B_ij，B_i0，B₀₀和系统总负荷数据x_D；1) Obtain the output upper limit and lower limit data of each unit in the power system with n _g generator sets and Coefficient data a _i , bi _, _ci of output-fuel cost function, coefficient data α _i , β _i , γ _i , ε _i , ξ _i of output-emission function, B coefficient data B _ij of transmission line loss, B _i0 , B ₀₀ and total system load data x _D ;

2)根据步骤1)所获得的数据，建立电力系统环境经济调度问题的数学优化模型即EED模型(以下表述将环境经济调度问题和环境经济调度模型分别写作EED问题和EED模型)，具体如下：2) According to the data obtained in step 1), establish the mathematical optimization model of the power system environmental economic dispatching problem, that is, the EED model (the following expressions will write the environmental economic dispatching problem and the environmental economical dispatching model as EED problem and EED model respectively), as follows:

2.1)机组i的出力-燃料费用函数如下式的二次多项式所示：2.1) The output-fuel cost function of unit i is shown in the quadratic polynomial of the following formula:

2.2)机组i的出力-排放函数如下式的二次多项式与指数幂之和所示：2.2) The output-discharge function of unit i is shown as the sum of the quadratic polynomial and the exponent power of the following formula:

${x x}_{i i}^{min min} {\leq \leq x x}_{i i} \leq \leq {x x}_{i i}^{max max} - - - - - - ((33))$

2.4)考虑输电线路损耗的负荷平衡约束如下式所示：2.4) The load balance constraint considering transmission line loss is shown in the following formula:

2.5)考虑输电线路损耗的EED模型如下：2.5) The EED model considering transmission line loss is as follows:

$Minimize Minimize : : f f ((x x)) = = [\begin{matrix} {f f}_{11} ((x x)) \\ {f f}_{22} ((x x)) \end{matrix}] = = [\begin{matrix} {Σ Σ}_{i i = = 11}^{{n no}_{g g}} {f f}_{fue fue 11} (({x x}_{i i})) \\ {Σ Σ}_{i i = = 11}^{{n no}_{g g}} {f f}_{emi emi} (({x x}_{i i})) \end{matrix}] - - - - - - ((66))$

s.t. (3)(4)(5)s.t. (3)(4)(5)

其中，f_fue1(x_i)和f_emi(x_i)分别由式(1)和式(2)表示；Among them, f _fue1 ( _xi ) and f _emi ( _xi ) are expressed by formula (1) and formula (2) respectively;

3)根据步骤2)所建立的EED模型，基于多目标库恩塔克最优条件，采用多目标λ迭代法(MλI)求解不考虑输电线路损耗的EED问题，得到帕雷托最优解，具体如下：3) According to the EED model established in step 2), based on the multi-objective Kuhn-Tucker optimal condition, the multi-objective λ iterative method (MλI) is used to solve the EED problem without considering the transmission line loss, and the Pareto optimal solution is obtained. details as follows:

3.2)对于一个很大的λ值(设该值为-0.00001)，求得每台机组对应的有功出力，计算发电负荷不平衡量；3.2) For a large λ value (set the value to -0.00001), obtain the corresponding active output of each unit, and calculate the unbalanced amount of power generation load;

3.3)对于一个很小的λ值(设该值为-6000)，求得每台机组对应的有功出力，计算发电负荷不平衡量；3.3) For a very small λ value (set the value to -6000), obtain the corresponding active output of each unit, and calculate the unbalanced amount of power generation load;

对于一个含有等式和不等式约束的多目标优化模型可表示为式(7)：For a multi-objective optimization model with equality and inequality constraints, it can be expressed as formula (7):

Minimize：f(x)Minimize: f(x)

s.t. g(x)≤0 (7)s.t. g(x)≤0 (7)

h(x)＝0h(x)=0

若为式(7)的一个帕雷托最优解，则存在一组(λ，u，v)满足式(8)：like is a Pareto optimal solution of formula (7), then there exists a set of (λ, u, v) satisfying formula (8):

$\begin{matrix} &dtri; &dtri; f f ((\overset{&OverBar; &OverBar;}{x x})) λ λ + + &dtri; &dtri; g g ((\overset{&OverBar; &OverBar;}{x x})) u u + + &dtri; &dtri; h h ((\overset{&OverBar; &OverBar;}{x x})) v v = = 00 \\ [[u u]] g g ((\overset{&OverBar; &OverBar;}{x x})) = = 00 \\ λ λ &GreaterEqual; &Greater Equal; 00,, λ λ &NotEqual; &NotEqual; 00,, u u &GreaterEqual; &Greater Equal; 00 \end{matrix} - - - - - - ((88))$

其中若则u_i＝0，其中集合而λ≥0，λ≠0表示λ里面每个元素大于等于0，但不能每个元素同时等于0；Among them, if Then u _i =0, where the set And λ≥0, λ≠0 means that each element in λ is greater than or equal to 0, but each element cannot be equal to 0 at the same time;

EED模型对应的f，g和h可以表示成式(9)：The f, g and h corresponding to the EED model can be expressed as formula (9):

$\begin{matrix} {f f}_{11} ((x x)) = = {Σ Σ}_{i i = = 11}^{{n no}_{g g}} (({a a}_{i i} + + {b b}_{i i} {x x}_{i i} + + {c c}_{i i} {x x}_{i i}^{22})) \\ {f f}_{22} ((x x)) = = {Σ Σ}_{i i = = 11}^{{n no}_{g g}} (({α α}_{i i} + + {β β}_{i i} {x x}_{i i} + + {γ γ}_{i i} {x x}_{i i}^{22} + + {ϵ ϵ}_{i i} {e e}^{{ξ ξ}_{i i} {x x}_{i i}})) \\ {g g}_{i i} ((x x)) = = - - {x x}_{i i} + + {x x}_{i i}^{min min} \leq \leq 00,, i i &Element; &Element; {I I}_{00} \\ {g g}_{(({i i + + n no}_{g g}))} ((x x)) = = {x x}_{i i} - - {x x}_{i i}^{max max} \leq \leq 00,, i i &Element; &Element; {I I}_{00} \\ h h ((x x)) = = {Σ Σ}_{i i = = 11}^{{n no}_{g g}} {x x}_{i i} - - {x x}_{D D.} - - {x x}_{L L} = = 00 \end{matrix} - - - - - - ((99))$

其中，I₀＝{1，2，…，n_g}；Wherein, I ₀ ={1, 2, . . . , n _g };

根据多目标库恩塔克条件，若是一个帕雷托最优解，则：According to the multi-objective Kuhn-Tucker condition, if is a Pareto optimal solution, then:

$\begin{matrix} &dtri; &dtri; f f ((\overset{- -}{x x})) λ λ + + &dtri; &dtri; g g ((\overset{- -}{x x})) u u + + &dtri; &dtri; h h ((\overset{- -}{x x})) v v = = [\begin{matrix} {λ λ}_{11} \frac{{&PartialD; &PartialD; f f}_{11}}{{&PartialD; &PartialD; x x}_{11}} + + {λ λ}_{22} \frac{{&PartialD; &PartialD; f f}_{22}}{{&PartialD; &PartialD; x x}_{11}} + + {Σ Σ}_{i i = = 11}^{{22 n no}_{g g}} {u u}_{i i} \frac{{&PartialD; &PartialD; g g}_{i i}}{{&PartialD; &PartialD; x x}_{11}} + + v v \frac{&PartialD; &PartialD; h h}{{&PartialD; &PartialD; x x}_{11}} \\ {λ λ}_{11} \frac{{&PartialD; &PartialD; f f}_{11}}{{&PartialD; &PartialD; x x}_{22}} + + {λ λ}_{22} \frac{{&PartialD; &PartialD; f f}_{22}}{{&PartialD; &PartialD; x x}_{22}} + + {Σ Σ}_{i i = = 11}^{{22 n no}_{g g}} {u u}_{i i} \frac{{&PartialD; &PartialD; g g}_{i i}}{{&PartialD; &PartialD; x x}_{22}} + + v v \frac{&PartialD; &PartialD; h h}{{&PartialD; &PartialD; x x}_{22}} \\ \cdot &Center Dot; \\ \cdot \cdot \\ \cdot &Center Dot; \\ {λ λ}_{11} \frac{{&PartialD; &PartialD; f f}_{11}}{{&PartialD; &PartialD; x x}_{{n no}_{g g}}} + + {λ λ}_{22} \frac{{&PartialD; &PartialD; f f}_{22}}{{&PartialD; &PartialD; x x}_{{n no}_{g g}}} + + {Σ Σ}_{i i = = 11}^{{22 n no}_{g g}} {u u}_{i i} \frac{{&PartialD; &PartialD; g g}_{i i}}{{&PartialD; &PartialD; x x}_{{n no}_{g g}}} + + v v \frac{&PartialD; &PartialD; h h}{{&PartialD; &PartialD; x x}_{{n no}_{g g}}} \end{matrix}] \\ = = [\begin{matrix} {λ λ}_{11} \frac{{&PartialD; &PartialD; f f}_{11}}{{&PartialD; &PartialD; x x}_{11}} + + {λ λ}_{22} \frac{{&PartialD; &PartialD; f f}_{22}}{{&PartialD; &PartialD; x x}_{11}} + + {u u}_{11} \frac{{&PartialD; &PartialD; g g}_{11}}{{&PartialD; &PartialD; x x}_{11}} + + {u u}_{((11 + + {n no}_{g g}))} \frac{{&PartialD; &PartialD; g g}_{((11 + + {n no}_{g g}))}}{{&PartialD; &PartialD; x x}_{11}} + + v v \frac{&PartialD; &PartialD; h h}{{&PartialD; &PartialD; x x}_{11}} \\ {λ λ}_{11} \frac{{&PartialD; &PartialD; f f}_{11}}{{&PartialD; &PartialD; x x}_{11}} + + {λ λ}_{22} \frac{{&PartialD; &PartialD; f f}_{22}}{{&PartialD; &PartialD; x x}_{22}} + + {u u}_{22} \frac{{&PartialD; &PartialD; g g}_{22}}{{&PartialD; &PartialD; x x}_{22}} + + {u u}_{((22 + + {n no}_{g g}))} \frac{{&PartialD; &PartialD; g g}_{((22 + + {n no}_{g g}))}}{{&PartialD; &PartialD; x x}_{22}} + + v v \frac{&PartialD; &PartialD; h h}{{&PartialD; &PartialD; x x}_{22}} \\ \cdot &Center Dot; \\ \cdot &Center Dot; \\ \cdot &Center Dot; \\ {λ λ}_{11} \frac{{&PartialD; &PartialD; f f}_{11}}{{&PartialD; &PartialD; x x}_{{n no}_{g g}}} + + {λ λ}_{22} \frac{{&PartialD; &PartialD; f f}_{22}}{{&PartialD; &PartialD; x x}_{{n no}_{g g}}} + + {u u}_{{n no}_{g g}} \frac{{&PartialD; &PartialD; g g}_{{n no}_{g g}}}{{&PartialD; &PartialD; x x}_{{n no}_{g g}}} + + {u u}_{(({22 n no}_{g g}))} \frac{{&PartialD; &PartialD; g g}_{(({22 n no}_{g g}))}}{{&PartialD; &PartialD; x x}_{{n no}_{g g}}} + + v v \frac{&PartialD; &PartialD; h h}{{&PartialD; &PartialD; x x}_{{n no}_{g g}}} \end{matrix}] = = 00 \end{matrix} - - - - - - ((1010))$

其中，λ≥0，λ≠0，u≥0；Among them, λ≥0, λ≠0, u≥0;

式(10)可分成式(11)～(13)三种情况进行讨论：Equation (10) can be divided into three situations of Equation (11) ~ (13) for discussion:

若λ₁＝0且λ₂＞0，则：If λ ₁ =0 and λ ₂ >0, then:

$\frac{&PartialD; &PartialD; {f f}_{22}}{&PartialD; &PartialD; {x x}_{11}} = = \frac{&PartialD; &PartialD; {f f}_{22}}{&PartialD; &PartialD; {x x}_{i i}} + + \frac{{B B}_{11} {u u}_{11} - - {B B}_{22} {u u}_{i i}}{{λ λ}_{22}} = = - - \frac{v v}{{λ λ}_{22}} - - - - - - ((1111))$

若λ₂＝0且λ₁＞0，则：If λ ₂ =0 and λ ₁ >0, then:

$\frac{&PartialD; &PartialD; {f f}_{11}}{&PartialD; &PartialD; {x x}_{11}} = = \frac{&PartialD; &PartialD; {f f}_{11}}{&PartialD; &PartialD; {x x}_{i i}} + + \frac{{B B}_{11} {u u}_{11} - - {B B}_{22} {u u}_{i i}}{{λ λ}_{11}} = = - - \frac{v v}{{λ λ}_{11}} - - - - - - ((1212))$

若λ₂＝0且λ₁＞0，则：If λ ₂ =0 and λ ₁ >0, then:

$\begin{matrix} \frac{\frac{&PartialD; &PartialD; {f f}_{11}}{&PartialD; &PartialD; {x x}_{11}} - - \frac{&PartialD; &PartialD; {f f}_{11}}{&PartialD; &PartialD; {x x}_{i i}}}{\frac{&PartialD; &PartialD; {f f}_{22}}{&PartialD; &PartialD; {x x}_{11}} - - \frac{&PartialD; &PartialD; {f f}_{22}}{&PartialD; &PartialD; {x x}_{i i}}} = = \frac{{B B}_{11} {u u}_{11} - - {B B}_{22} {u u}_{i i}}{{λ λ}_{11} ((\frac{&PartialD; &PartialD; {f f}_{22}}{&PartialD; &PartialD; {x x}_{11}} - - \frac{&PartialD; &PartialD; {f f}_{22}}{&PartialD; &PartialD; {x x}_{i i}}))} - - \frac{{λ λ}_{22}}{{λ λ}_{11}},, &ForAll; &ForAll; i i &NotElement; &NotElement; {I I}_{11} \\ \frac{&PartialD; &PartialD; {f f}_{11}}{&PartialD; &PartialD; {x x}_{11}} = = \frac{&PartialD; &PartialD; {f f}_{11}}{&PartialD; &PartialD; {x x}_{i i}} + + \frac{{B B}_{11} {u u}_{11} - - {B B}_{22} {u u}_{i i}}{{λ λ}_{11}},, &ForAll; &ForAll; i i &Element; &Element; {I I}_{11} \end{matrix} - - - - - - ((1313))$

其中， $I_{1} = {i : \frac{&PartialD; f_{2}}{&PartialD; x_{1}} - \frac{&PartialD; f_{2}}{&PartialD; x_{i}} = 0, i &NotEqual; 1};$ in, $I_{1} = {i : \frac{&PartialD; f_{2}}{&PartialD; x_{1}} - \frac{&PartialD; f_{2}}{&PartialD; x_{i}} = 0, i &NotEqual; 1};$

根据x₁和x_i是否位于其出力上限与下限边界，式(11)～(13)可分为4种情况：According to whether _x1 and _xi are located at the upper and lower boundaries of their output, formulas (11)-(13) can be divided into four situations:

a、若 $x_{i} &NotEqual; x_{i}^{\min}$ 且 $x_{i} &NotEqual; x_{i}^{\max}, &ForAll; i &Element; I_{0},$ 则B₁＝0，B₂＝0；a. If $x_{i} &NotEqual; x_{i}^{\min}$ and $x_{i} &NotEqual; x_{i}^{\max}, &ForAll; i &Element; I_{0},$ Then B ₁ =0, B ₂ =0;

b、若( $x_{1} &NotEqual; x_{1}^{\min}$ 且 $x_{1} &NotEqual; x_{1}^{\max}$ )且( $x_{i} = x_{i}^{\min}$ 或 $x_{i} = x_{i}^{\max}, i &NotEqual; 1$ )，则B₁＝0，B₂＝1；b. If ( $x_{1} &NotEqual; x_{1}^{\min}$ and $x_{1} &NotEqual; x_{1}^{\max}$ )and( $x_{i} = x_{i}^{\min}$ or $x_{i} = x_{i}^{\max}, i &NotEqual; 1$ ), then B ₁ =0, B ₂ =1;

c、若且则B₁＝1，B₂＝1；c. If and Then B ₁ =1, B ₂ =1;

d、若且则B₁＝1，B₂＝0；d. If and Then B ₁ =1, B ₂ =0;

式(10)即式(11)～(13)的求解可分成两步：The solution of formula (10) and formula (11)～(13) can be divided into two steps:

在第一步中求解式(11)和(12)：Solve equations (11) and (12) in the first step:

式(11)的求解：可以简单地采用λ迭代法，给定一个初始λ值并确定满足式(11)的每台机组的出力。若则增加λ₀的值，若则减少λ₀的值，直至式(11)和同时得到满足，此时求得的解为使目标函数f₂(x)最小化的解，根据式(11)和的单调递增特性，若对于一个给定的λ₀，有则令若对于一个给定的λ₀，有则令 The solution of formula (11): simply adopt the λ iterative method, given an initial λ value And determine the output of each unit that satisfies formula (11). like Then increase the value of λ ₀ , if Then reduce the value of λ ₀ until formula (11) and At the same time, the solution obtained at this time is the solution that minimizes the objective function f ₂ (x), according to formula (11) and The monotonically increasing property of , if for a given λ ₀ , there is order If for a given λ ₀ , there is order

式(12)的求解：可以简单地采用λ迭代法，给定一个初始λ值并确定满足式(12)的每台机组的出力。若则增加λ₀的值，若则减少λ₀的值，直至式(12)和同时得到满足，此时求得的解为使目标函数f₁(x)最小化的解，根据式(12)和的单调递增特性，若对于一个给定的λ₀，有则令若对于一个给定的λ₀，有则令 The solution of formula (12): simply adopt the λ iterative method, given an initial λ value And determine the output of each unit that satisfies formula (12). like Then increase the value of λ ₀ , if Then reduce the value of λ ₀ until formula (12) and At the same time, the solution obtained at this time is the solution that minimizes the objective function f ₁ (x), according to formula (12) and The monotonically increasing property of , if for a given λ ₀ , there is order If for a given λ ₀ , there is order

在第二步中求解式(13)：求解不考虑输电线路损耗的EED问题的程序伪码如下表1和表2所示(表1中λ_l设为-0.00001，λ_s设为-6000)，表2为表1中的一个子程序伪码表。In the second step, formula (13) is solved: the program pseudocode for solving the EED problem without considering the transmission line loss is shown in Table 1 and Table 2 below (in Table 1, λ _l is set to -0.00001, and λ _s is set to -6000) , Table 2 is a subroutine pseudo-code table in Table 1.

表1多目标λ迭代法程序伪码表Table 1 Pseudo-code list of multi-objective lambda iteration method program

表2根据λ得到机组出力程序伪码表Table 2 Obtaining the pseudo-code table of unit output program according to λ

表2中有两点需进行说明：一是如何判断和是否满足式(13)，先判断以上述b类为例进行说明，即B₁＝0，B₂＝1，若且则式(13)中第二行满足；若 $(\frac{&PartialD; f_{2}}{&PartialD; x_{1}} - \frac{&PartialD; f_{2}}{{&PartialD; x}_{i}^{\min}} < 0),$ 且 $(\frac{&PartialD; f_{1}}{&PartialD; x_{1}} - \frac{&PartialD; f_{1}}{{&PartialD; x}_{i}}) / (\frac{{&PartialD; f}_{2}}{&PartialD; x_{1}} - \frac{&PartialD; f_{2}}{&PartialD; x_{i}}) < λ,$ 则式(13)中的第一行满足；若 $(\frac{&PartialD; f_{2}}{&PartialD; x_{1}} - \frac{&PartialD; f_{2}}{{&PartialD; x}_{i}^{\min}} < 0),$ 且 $(\frac{&PartialD; f_{1}}{&PartialD; x_{1}} - \frac{&PartialD; f_{1}}{{&PartialD; x}_{i}}) / (\frac{{&PartialD; f}_{2}}{&PartialD; x_{1}} - \frac{&PartialD; f_{2}}{&PartialD; x_{i}}) > λ,$ 则式(13)中的第一行满足；同理，可以类似地判断是否满足式(13)。There are two points in Table 2 that need to be explained: one is how to judge and Whether it satisfies formula (13), first judge Take the above category b as an example for illustration, that is, B ₁ =0, B ₂ =1, if and Then the second row in formula (13) satisfies; if $(\frac{&PartialD; f_{2}}{&PartialD; x_{1}} - \frac{&PartialD; f_{2}}{{&PartialD; x}_{i}^{\min}} < 0),$ and $(\frac{&PartialD; f_{1}}{&PartialD; x_{1}} - \frac{&PartialD; f_{1}}{{&PartialD; x}_{i}}) / (\frac{{&PartialD; f}_{2}}{&PartialD; x_{1}} - \frac{&PartialD; f_{2}}{&PartialD; x_{i}}) < λ,$ Then the first row in formula (13) satisfies; if $(\frac{&PartialD; f_{2}}{&PartialD; x_{1}} - \frac{&PartialD; f_{2}}{{&PartialD; x}_{i}^{\min}} < 0),$ and $(\frac{&PartialD; f_{1}}{&PartialD; x_{1}} - \frac{&PartialD; f_{1}}{{&PartialD; x}_{i}}) / (\frac{{&PartialD; f}_{2}}{&PartialD; x_{1}} - \frac{&PartialD; f_{2}}{&PartialD; x_{i}}) > λ,$ Then the first line in formula (13) satisfies; similarly, it can be judged similarly Whether it satisfies formula (13).

二是如何通过式(13)来确定x_i的值。通常有两种方法，一种是直接求解；另一种是通过查表得到，对于一个固定的x₁值，将每一个x_i对应的的值预先存于一张表中，在MλI的求解过程中，直接查表得到x_i值。由于查表法简单快速，在本实施例的计算过程中采用查表法。The second is how to determine the value of x _i through formula (13). There are usually two methods, one is to solve directly; the other is to obtain by looking up the table, for a fixed x ₁ value, the corresponding value of each x _i The value of is pre-stored in a table, and in the process of solving MλI, the value of _xi can be obtained directly by looking up the table. Because the table look-up method is simple and fast, the table look-up method is used in the calculation process of this embodiment.

4)在步骤(3)中，由于输电线路损耗未被考虑，所以在本步骤中将步骤3)得到的每一个帕雷托最优解作为初始解，采用牛顿法求解考虑输电线路损耗的EED问题，得到最优解集，具体如下：4) In step (3), since the transmission line loss is not considered, in this step, each Pareto optimal solution obtained in step 3) is used as the initial solution, and Newton’s method is used to solve the EED considering the transmission line loss problem, the optimal solution set is obtained, as follows:

采用泰勒展开，可将式(8)表示为式(14)：Using Taylor expansion, formula (8) can be expressed as formula (14):

$\begin{matrix} &dtri; &dtri; f f ((\overset{&OverBar; &OverBar;}{x x})) λ λ + + {&dtri; &dtri;}^{22} f f ((\overset{&OverBar; &OverBar;}{x x})) λΔx λΔx + + &dtri; &dtri; f f ((\overset{&OverBar; &OverBar;}{x x})) Δλ Δλ \\ + + &dtri; &dtri; g g ((\overset{&OverBar; &OverBar;}{x x})) u u + + {&dtri; &dtri;}^{22} g g ((\overset{&OverBar; &OverBar;}{x x})) μΔx μΔx + + &dtri; &dtri; g g ((\overset{&OverBar; &OverBar;}{x x})) Δμ Δμ \\ + + &dtri; &dtri; h h ((\overset{&OverBar; &OverBar;}{x x})) v v + + {&dtri; &dtri;}^{22} h h ((\overset{&OverBar; &OverBar;}{x x})) vΔx vΔx + + &dtri; &dtri; h h ((\overset{&OverBar; &OverBar;}{x x})) Δv Δv = = 00 \\ - - [[μ μ]] g g ((\overset{&OverBar; &OverBar;}{x x})) - - [[μ μ]] &dtri; &dtri; g g {((\overset{&OverBar; &OverBar;}{x x}))}^{T T} Δx Δx - - [[g g ((\overset{&OverBar; &OverBar;}{x x}))]] Δμ Δμ = = 00 \\ h h ((\overset{&OverBar; &OverBar;}{x x})) + + &dtri; &dtri; h h {((\overset{&OverBar; &OverBar;}{x x}))}^{T T} Δx Δx = = 00 \end{matrix} - - - - - - ((1414))$

式(14)可写成式(15)的紧凑形式：Equation (14) can be written as a compact form of Equation (15):

A×Δy＝b (15)A×Δy=b (15)

其中in

Δy＝[Δx Δλ Δμ Δv]^T (16)Δy=[Δx Δλ Δμ Δv] ^T (16)

$A A = = [\begin{matrix} M m & &dtri; &dtri; f f ((\overset{&OverBar; &OverBar;}{x x})) & &dtri; &dtri; g g ((\overset{&OverBar; &OverBar;}{x x})) & &dtri; &dtri; h h ((\overset{&OverBar; &OverBar;}{x x})) \\ - - [[μ μ]] &dtri; &dtri; g g {((\overset{&OverBar; &OverBar;}{x x}))}^{T T} & 00 & - - [[g g]] ((\overset{&OverBar; &OverBar;}{x x})) & 00 \\ &dtri; &dtri; h h {((\overset{&OverBar; &OverBar;}{x x}))}^{T T} & 00 & 00 & 00 \end{matrix}] - - - - - - ((1717))$

$b b = = [\begin{matrix} - - {r r}_{dual dual} \\ - - {r r}_{cent cent} \\ - - h h ((\overset{&OverBar; &OverBar;}{x x})) \end{matrix}] = = [\begin{matrix} - - &dtri; &dtri; f f ((\overset{&OverBar; &OverBar;}{x x})) λ λ - - &dtri; &dtri; g g ((\overset{&OverBar; &OverBar;}{x x})) u u - - &dtri; &dtri; h h ((\overset{&OverBar; &OverBar;}{x x})) v v \\ [[μ μ]] g g ((\overset{&OverBar; &OverBar;}{x x})) \\ - - h h ((\overset{&OverBar; &OverBar;}{x x})) \end{matrix}] - - - - - - ((1818))$

在式(17)中：In formula (17):

$M m = = {&dtri; &dtri;}^{22} f f ((\overset{&OverBar; &OverBar;}{x x})) λ λ + + {&dtri; &dtri;}^{22} g g ((\overset{&OverBar; &OverBar;}{x x})) μ μ + + {&dtri; &dtri;}^{22} h h ((\overset{&OverBar; &OverBar;}{x x})) v v - - - - - - ((1919))$

采用牛顿法求解考虑输电线路损耗的EED问题的程序伪码列于下表3中。The program pseudo code for solving the EED problem considering the transmission line loss by Newton's method is listed in Table 3 below.

表3采用牛顿法求解EED问题的程序伪码表Table 3 Pseudo-code list of programs for solving EED problems using Newton's method

5)采用多目标决策方法在步骤4)得到的最优解集中确定最终解；所述多目标决策采用逼近于理想值的排序方法(Technique for Order Preference by Similarity to IdealSolution，TOPSIS)。TOPSIS包括以下步骤：5) Using a multi-objective decision-making method to determine the final solution in the optimal solution set obtained in step 4); the multi-objective decision-making method adopts a sorting method (Technique for Order Preference by Similarity to IdealSolution, TOPSIS) that is close to the ideal value. TOPSIS includes the following steps:

${v v}_{ij ij} = = {ω ω}_{i i} (({f f}_{i i}^{+ +} - - {f f}_{ij ij})) / / (({f f}_{i i}^{+ +} - - {f f}_{i i}^{- -})),, i i = = 11,, 22,, 33,, \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot;,, {N N}_{obj obj},, j j = = 11,, 22,, 33,, \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot;,, J J - - - - - - ((2020))$

其中， $f_{i}^{-} = \min_{j} f_{ij}, f_{i}^{+} = \max_{j} f_{ij},$ $ω_{i} = \frac{1}{N_{obj}} Σ_{j = i}^{N_{obj}} \frac{1}{j}, i = 1, 2, 3, \cdot \cdot \cdot, N_{obj},$ f_ij为最优解集中的第j个解的第i个目标函数值，N_obj为多目标优化中目标的个数，J为最优解集中解的个数；in, $f_{i}^{-} = \min_{j} f_{ij}, f_{i}^{+} = \max_{j} f_{ij},$ $ω_{i} = \frac{1}{N_{obj}} Σ_{j = i}^{N_{obj}} \frac{1}{j}, i = 1, 2, 3, \cdot \cdot \cdot, N_{obj},$ f _ij is the i-th objective function value of the j-th solution in the optimal solution set, N _obj is the number of objectives in multi-objective optimization, and J is the number of solutions in the optimal solution set;

${A A}^{+ +} = = {{{v v}_{11}^{+ +},, {v v}_{22}^{+ +},, \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot;,, {v v}_{N N}^{+ +}}}$

其中，j＝1，2，3，…J；Among them, j=1, 2, 3, ... J;

5.5)将R_j最大的最优解选择为最终的机组检修及出力方案；5.5) Select the optimal solution with the largest R _j as the final unit maintenance and output plan;

实施例2：Example 2:

为验证本发明上述实施例1的基于混合多目标λ迭代法和牛顿法的电力系统调度方法的有效性，本实施例分别以一个6机组，一个14机组和一个140机组的系统为例，所述三个系统的数据可分别从上述背景技术中的文献[15]和[16]中获得，其中6机组系统的总负荷为283.4MW，14机组系统的总负荷为950MW，140机组的总负荷为49342MW。In order to verify the effectiveness of the power system dispatching method based on the mixed multi-objective lambda iteration method and the Newton method in the above-mentioned embodiment 1 of the present invention, this embodiment takes a system of 6 units, a system of 14 units and a system of 140 units as examples, and the The data of the three systems mentioned above can be obtained from the documents [15] and [16] in the above-mentioned background technology, in which the total load of the 6-unit system is 283.4MW, the total load of the 14-unit system is 950MW, and the total load of the 140-unit system It is 49342MW.

为了进行比较，同时采用MOPSO算法求解所述EED问题。MOPSO中种群个数和最大迭代次数设置如表4所示。用MOPSO求解所述EED问题时，采用罚函数法以处理约束条件，即将式(4)所示的等式约束的不平衡量的绝对值乘以罚系数加到对应的目标函数中，所采用的罚系数也列于下表4中。For comparison, the MOPSO algorithm is used to solve the EED problem. The number of populations and the maximum number of iterations in MOPSO are set as shown in Table 4. When solving the EED problem with MOPSO, the penalty function method is adopted to process the constraints, that is, the absolute value of the unbalanced quantity of the equation constraints shown in formula (4) is multiplied by the penalty coefficient and added to the corresponding objective function, and the adopted The penalty factors are also listed in Table 4 below.

系统system 种群个数Population number 最大迭代次数The maximum number of iterations f₁对应的罚系数Penalty coefficient corresponding to f ₁ f₂对应的罚系数Penalty coefficient corresponding to f ₂ 6机组6 units 5050 100100 50005000 33 14机组14 units 400400 400400 20002000 44 140机组140 units 800800 600600 2000020000 4040

表4 MOPSO算法在3个系统中的参数和罚系数设置值Table 4 Parameters and penalty coefficient setting values of MOPSO algorithm in three systems

图2表示了本发明实施例1的方法求解所述6机组得到的解，同时也表示了采用MOPSO算法求解所述问题得到的结果，图3表示了本发明实施例1的方法求解所述14机组系统得到的解，同时也表示了采用MOPSO算法求解所述问题得到的结果。Fig. 2 has represented the solution that the method for the embodiment of the present invention 1 solves described 6 units and obtains, also represented the result that adopts MOPSO algorithm to solve described problem to obtain simultaneously, Fig. 3 has represented the method for the embodiment of the present invention 1 to solve described 14 The solution obtained by the unit system also shows the result obtained by using the MOPSO algorithm to solve the problem.

可以看到，与采用MOPSO算法得到的结果，采用扩展λ迭代求解方法能够得到同时具有更小发电费用值和更小输电线路损耗值的解，表明了收敛精度高。It can be seen that, compared with the results obtained by using the MOPSO algorithm, the extended λ iterative solution method can obtain a solution with a smaller power generation cost value and a smaller transmission line loss value at the same time, indicating that the convergence accuracy is high.

为了验证本发明实施例1的方法在求解大规模电力系统中有效性，还用于求解一个含有140机组的电力系统中的EED问题，同时也列出了采用MOPSO算法求解所述问题得到的结果，如图4所示。In order to verify the effectiveness of the method in Embodiment 1 of the present invention in solving large-scale power systems, it is also used to solve the EED problem in a power system containing 140 units, and the results obtained by using the MOPSO algorithm to solve the problem are also listed ,As shown in Figure 4.

本发明实施例1的方法在一台处理器为Core^TM i7-2600 CPU 3.40GHz的个人计算机上实现，其求解6机组系统、14机组系统以及140机组系统的结果，得到的Δ指标和所花费的时间列于下表5中，同时在表5中列出CλIN和MOPSO求解得到的Δ指标和所花费的时间(注：在求解6机组和14机组系统时考虑了输电线路损耗，但在求解140机组系统时由于缺乏数据并未考虑输电线路损耗，表5、图2、图3、图4中MλI和CλIN的区别：MλI表示的是多目标λ迭代法，即仅含步骤3而未含步骤4；CλIN表示的是混合多目标λ迭代法和牛顿法，即同时包含了步骤3和步骤4)。In the method of embodiment 1 of the present invention, a processor is It is implemented on a personal computer with Core ^TM i7-2600 CPU 3.40GHz. The results of solving the 6-unit system, the 14-unit system and the 140-unit system are listed in Table 5 below. List the Δ index and the time spent in solving CλIN and MOPSO (Note: The transmission line loss is considered when solving the 6-unit and 14-unit system, but the transmission line loss is not considered when solving the 140-unit system due to lack of data , the difference between MλI and CλIN in Table 5, Figure 2, Figure 3, and Figure 4: MλI represents the multi-objective λ iterative method, that is, only step 3 but not step 4; CλIN represents the mixed multi-objective λ iterative method and Newton's method, which includes both steps 3 and 4).

综上所述，本发明提供了一种简单高效的电力系统多目标调度方法，其计算量小，计算时间短，收敛精度高，在含有大量机组的大规模电力系统中其在精度和计算量方面的优势明显，极大地提高了电力系统发电的经济性和效率。In summary, the present invention provides a simple and efficient multi-objective scheduling method for power systems, which has a small amount of calculation, short calculation time, and high convergence accuracy. The advantages in this aspect are obvious, which greatly improves the economy and efficiency of power system power generation.

表5三种不同方法在3个系统中求解EED问题得到的结果的Δ指标和所花费的时间Table 5 The Δ index and the time spent on the results obtained by three different methods for solving the EED problem in the three systems

以上所述，仅为本发明专利优选的实施例，但本发明专利的保护范围并不局限于此，任何熟悉本技术领域的技术人员在本发明专利所公开的范围内，根据本发明专利的技术方案及其发明构思加以等同替换或改变，都属于本发明专利的保护范围。The above is only a preferred embodiment of the patent of the present invention, but the scope of protection of the patent of the present invention is not limited thereto. Anyone familiar with the technical field within the scope disclosed by the patent of the present invention, according to the scope of the patent of the present invention Equivalent replacements or changes to the technical solutions and their inventive concepts all fall within the scope of protection of the invention patent.

Claims

1. The power system scheduling method based on the hybrid multi-target lambda iteration method and the Newton method is characterized by comprising the following steps of:

1) acquiring output upper limit and lower limit data, output-fuel cost function coefficient data, output-emission function coefficient data, transmission line loss B coefficient data and system total load data of each unit in an electric power system with a plurality of generator sets;

2) establishing a mathematical optimization model of the environmental economic dispatching problem of the power system according to the data obtained in the step 1);

3) solving the environmental economic dispatching problem of the power system without considering the loss of the power transmission line by adopting a multi-target lambda iteration method based on the multi-target Coueta gram optimal condition according to the model established in the step 2) to obtain a group of pareto optimal solutions;

4) taking each pareto optimal solution obtained in the step 3) as an initial solution, and solving the power system environmental economic scheduling problem considering the loss of the power transmission line by adopting a Newton method to obtain an optimal solution set;

5) determining a final solution in the optimal solution set obtained in the step 4) by adopting a multi-objective decision method;

6) and (3) sending the final solution determined in the step 5) as an instruction to a relevant power plant or unit through an automatic power generation control device, and realizing the control of the power generation power of the unit through an automatic control adjusting device of the power plant or unit.

2. The power system scheduling method based on the hybrid multi-objective lambda iterative method and the Newton method according to claim 1, wherein: step 1) the number of the units in the power system is n_gAnd 2) establishing a mathematical optimization model of the power system environmental economic dispatching problem in the step 2), specifically as follows:

2.1) the output-fuel cost function for unit i is given by:

f_{fuel} (x_{i}) = a_{i} + b_{i} x_{i} + c_{i} x_{i}^{2} - - - (1)

wherein i is more than or equal to 1 and less than or equal to n_g，f_fuel(x_i) As a function of output-fuel cost for unit i, x_iIs the active power output of the unit, a_i、b_iAnd c_iThe output-fuel cost coefficient of the unit i is obtained;

2.2) the output-emission function of unit i is given by:

<math> <mrow> <msub> <mi>f</mi> <mi>emi</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>α</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>β</mi> <mi>i</mi> </msub> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mi>i</mi> </msub> <msubsup> <mi>x</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mo>&Element;</mo> <mi>i</mi> </msub> <msup> <mi>e</mi> <mrow> <msub> <mi>ξ</mi> <mi>i</mi> </msub> <msub> <mi>x</mi> <mi>i</mi> </msub> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </math>

wherein f is_emi(x_i) As a function of the discharge of the unit i, alpha_i、β_i、γ_i、∈_iAnd xi_iThe output-emission function coefficient of the unit i is obtained;

2.3) the upper and lower limits of the output of the unit i are constrained as follows:

wherein,andrespectively representing the upper limit and the lower limit of the output of the unit i;

2.4) the load balancing constraint considering the transmission line loss is shown as follows:

wherein x is_DFor the total load of the system, x_LFor transmission line losses, x_LThe B coefficient is calculated by the following formula:

wherein, B_ij、B_i0And B₀₀The B coefficient is the loss of the transmission line;

2.5) the mathematical optimization model of the power system environmental economic dispatching problem considering the power transmission line loss is as follows:

<math> <mrow> <mi>Minimize</mi> <mo>:</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>f</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mtable> </mtable> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <munderover> <mi>Σ</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>n</mi> <mi>g</mi> </msub> </munderover> <msub> <mi>f</mi> <mi>fuel</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <munderover> <mi>Σ</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>n</mi> <mi>g</mi> </msub> </munderover> <msub> <mi>f</mi> <mi>emi</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow> </math>

s.t. (3)(4)(5)

wherein f is_fuel(x_i) And f_emi(x_i) Are represented by formula (1) and formula (2), respectively.

3. The power system scheduling method based on the hybrid multi-objective lambda iterative method and the Newton method according to claim 1 or 2, wherein: step 3), solving the power system environmental economic dispatching problem without considering the loss of the power transmission line by adopting a multi-target lambda iteration method to obtain a pareto optimal solution, which is as follows:

3.1) giving a force output value of the fixed force output unit;

3.2) for the lambda value with the size of-0.00001, obtaining the active output corresponding to each unit, and calculating the unbalance amount of the power generation load;

3.3) for the lambda value of-6000, calculating the corresponding active output of each unit, and calculating the unbalance amount of the power generation load;

3.4) if the amount of the generated load unbalance calculated in the step 3.2) and the step 3.3) is the same, the potential pareto optimal solution does not exist, and the step 3.1) is returned; and if the number of the generated load unbalance calculated in the step 3.2) and the number of the generated load unbalance calculated in the step 3.3) are different, modifying the lambda value by adopting a dichotomy until one lambda value is found, the active output of each unit corresponding to the lambda value meets the load balance, storing the lambda value and the active output of each unit, and obtaining the active output of each unit as the pareto optimal solution.

4. The power system scheduling method based on the hybrid multi-objective lambda iterative method and the Newton method according to claim 3, wherein: the generating load unbalance is the difference between the total generating power of each unit and the total load of the system.

5. The power system scheduling method based on the hybrid multi-objective lambda iterative method and the Newton method according to claim 3, wherein: the active power output corresponding to each unit enables every two units to meet the equal difference ratio micro-increment rate rule.

6. The power system scheduling method based on the hybrid multi-objective lambda iterative method and the Newton method according to claim 1, wherein: step 5) the multi-target decision method adopts a sequencing method approaching to an ideal value, and the method specifically comprises the following steps:

5.1) first, a per-unit weighted decision matrix v is calculated_ij：

Wherein,

f_ijthe ith objective function value, N, for the jth solution in the optimal solution set_objThe number of targets in the multi-target optimization is J, and the number of solutions in the optimal solution set is J;

5.2) calculating the optimal points A respectively⁺And the least ideal point A^-：

A^{+} = {v_{1}^{+}, v_{2}^{+}, . . ., v_{N}^{+}}

A^{-} = {v_{1}^{-}, v_{2}^{-}, . . ., v_{N}^{-}}

Wherein,

v_{i}^{+} = \max_{J} v_{ij}, v_{j}^{-} = \min_{j} v_{ij};

5.3) respectively calculating the distance D from each optimal solution to the optimal point⁺And a distance D to the least ideal point^-：

Wherein J is 1, 2, 3, … J;

5.4) calculating the distance ratio R of each optimal solution_j：

R_{j} = D_{j}^{-} / (D_{j}^{-} + D_{j}^{+})

Wherein J is 1, 2, 3, … J;

5.5) reacting R_jAnd selecting the maximum optimal solution as a final unit overhaul and output scheme.