基于遗传算法的经济负荷分配

基于遗传算法的经济负荷分配是电力系统中优化发电资源分配的经典问题。

问题概述

经济负荷分配的目标是在满足电力需求和各发电机约束条件下，最小化总发电成本，同时实现负荷在各发电机间的优化分配。

数学模型

1. 目标函数

总发电成本最小化：

Minimize F_total = Σ F_i(P_i)  (i=1,2,...,N)

其中发电机成本函数通常为二次函数：

F_i(P_i) = a_i × P_i² + b_i × P_i + c_i

2. 约束条件

• 功率平衡约束：Σ P_i = P_load + P_loss
• 发电机出力限制：P_i_min ≤ P_i ≤ P_i_max
• 爬坡率约束（如考虑）：|P_i(t) - P_i(t-1)| ≤ R_i

遗传算法实现步骤

1. 染色体编码

% 实数编码 - 每个基因代表一台发电机的出力
function population = initialize_population(pop_size, n_gen, P_min, P_max)
    population = zeros(pop_size, n_gen);
    for i = 1:pop_size
        % 在约束范围内随机生成个体
        population(i,:) = P_min + rand(1, n_gen) .* (P_max - P_min);
    end
end

2. 适应度函数

function fitness = calculate_fitness(individual, a, b, c, P_load, penalty_factor)
    % 计算总成本
    total_cost = 0;
    for i = 1:length(individual)
        P_i = individual(i);
        total_cost = total_cost + a(i)*P_i^2 + b(i)*P_i + c(i);
    end
    
    % 功率平衡约束惩罚
    total_power = sum(individual);
    power_imbalance = abs(total_power - P_load);
    
    % 适应度值（最小化问题取倒数）
    fitness = 1 / (total_cost + penalty_factor * power_imbalance^2);
end

3. 选择操作

function parents = selection(population, fitness, method='roulette')
    pop_size = size(population, 1);
    
    switch method
        case 'roulette'
            % 轮盘赌选择
            prob = fitness / sum(fitness);
            cum_prob = cumsum(prob);
            parents = zeros(size(population));
            
            for i = 1:pop_size
                r = rand();
                idx = find(cum_prob >= r, 1);
                parents(i,:) = population(idx,:);
            end
            
        case 'tournament'
            % 锦标赛选择
            tournament_size = 2;
            parents = zeros(size(population));
            
            for i = 1:pop_size
                contestants = randperm(pop_size, tournament_size);
                [~, best_idx] = max(fitness(contestants));
                parents(i,:) = population(contestants(best_idx),:);
            end
    end
end

4. 交叉操作

function offspring = crossover(parents, crossover_rate)
    pop_size = size(parents, 1);
    n_gen = size(parents, 2);
    offspring = parents;
    
    for i = 1:2:pop_size-1
        if rand() < crossover_rate
            % 算术交叉
            alpha = rand();
            offspring(i,:) = alpha * parents(i,:) + (1-alpha) * parents(i+1,:);
            offspring(i+1,:) = alpha * parents(i+1,:) + (1-alpha) * parents(i,:);
        end
    end
end

5. 变异操作

function mutated_pop = mutation(population, mutation_rate, P_min, P_max)
    pop_size = size(population, 1);
    n_gen = size(population, 2);
    mutated_pop = population;
    
    for i = 1:pop_size
        for j = 1:n_gen
            if rand() < mutation_rate
                % 高斯变异
                mutated_pop(i,j) = population(i,j) + randn() * 0.1 * (P_max(j)-P_min(j));
                % 边界处理
                mutated_pop(i,j) = max(P_min(j), min(P_max(j), mutated_pop(i,j)));
            end
        end
    end
end

完整MATLAB实现

function [best_solution, best_cost, convergence] = GA_economic_dispatch()
    % 参数设置
    pop_size = 50;          % 种群大小
    max_gen = 200;          % 最大迭代次数
    crossover_rate = 0.8;   % 交叉概率
    mutation_rate = 0.1;    % 变异概率
    penalty_factor = 1000;  % 惩罚因子
    
    % 发电机参数 (a, b, c, P_min, P_max)
    % 示例：3台发电机
    a = [0.0056, 0.0050, 0.0048];
    b = [4.5, 4.0, 3.8];
    c = [100, 120, 110];
    P_min = [50, 50, 50];
    P_max = [200, 200, 200];
    
    P_load = 400;  % 总负荷需求
    n_gen = length(a);
    
    % 初始化种群
    population = initialize_population(pop_size, n_gen, P_min, P_max);
    
    % 记录收敛过程
    convergence = zeros(max_gen, 1);
    
    for gen = 1:max_gen
        % 计算适应度
        fitness = zeros(pop_size, 1);
        for i = 1:pop_size
            fitness(i) = calculate_fitness(population(i,:), a, b, c, P_load, penalty_factor);
        end
        
        % 记录最佳解
        [best_fit, best_idx] = max(fitness);
        convergence(gen) = 1/best_fit;  % 转换为成本
        
        % 选择
        parents = selection(population, fitness, 'tournament');
        
        % 交叉
        offspring = crossover(parents, crossover_rate);
        
        % 变异
        mutated_pop = mutation(offspring, mutation_rate, P_min, P_max);
        
        % 精英保留
        population = mutated_pop;
        population(1,:) = parents(best_idx,:);
    end
    
    % 提取最优解
    best_solution = population(best_idx,:);
    best_cost = convergence(end);
    
    % 显示结果
    display_results(best_solution, best_cost, a, b, c);
end

function display_results(solution, cost, a, b, c)
    fprintf('最优负荷分配结果:\n');
    for i = 1:length(solution)
        P_i = solution(i);
        cost_i = a(i)*P_i^2 + b(i)*P_i + c(i);
        fprintf('发电机%d: %.2f MW, 成本: %.2f $\n', i, P_i, cost_i);
    end
    fprintf('总成本: %.2f $\n', cost);
    fprintf('总发电量: %.2f MW\n', sum(solution));
end

参考代码 基于遗传算法的经济负荷分配 www.youwenfan.com/contentcno/82428.html

高级改进策略

1. 考虑网损的经济调度

% B系数法计算网损
function P_loss = calculate_losses(P, B_coeff)
    P_loss = P * B_coeff * P';
end

% 修改适应度函数考虑网损
function fitness = calculate_fitness_with_loss(individual, a, b, c, P_load, B_coeff, penalty_factor)
    total_cost = sum(a.*individual.^2 + b.*individual + c);
    total_power = sum(individual);
    P_loss = calculate_losses(individual, B_coeff);
    power_imbalance = abs(total_power - P_loss - P_load);
    
    fitness = 1 / (total_cost + penalty_factor * power_imbalance^2);
end

2. 阀点效应考虑

% 考虑阀点效应的成本函数
function cost = valve_point_cost(P, a, b, c, e, f)
    % e, f 为阀点效应系数
    cost = a*P^2 + b*P + c + abs(e * sin(f * (P_min - P)));
end

性能优化建议

1. 参数调优：

   • 种群大小：50-100
   • 交叉概率：0.7-0.9
   • 变异概率：0.01-0.1

2. 收敛准则：

   • 最大迭代次数
   • 适应度改善阈值
   • 早停策略

3. 混合算法：

   • GA + 局部搜索
   • GA + 模拟退火
   • 多目标优化扩展

实际应用考虑

• 备用容量约束
• 禁止运行区间
• 最小启停时间
• 燃料约束
• 排放约束（多目标优化）