考虑不确定性的电力系统蒙特卡洛负荷模拟与潮流计算

整体框架概述

% 主程序框架
clear; clc; close all;

%% 系统参数初始化
n_buses = 30;           % 节点数量
n_lines = 41;           % 支路数量
n_samples = 1000;       % 蒙特卡洛样本数
confidence_level = 0.95; % 置信水平

%% 蒙特卡洛负荷模拟
load_scenarios = generate_load_scenarios(n_buses, n_samples);

%% 概率潮流计算
[voltage_results, line_loading_results] = probabilistic_power_flow(load_scenarios);

%% 结果分析与可视化
analyze_results(voltage_results, line_loading_results, confidence_level);

不确定性负荷建模

1. 基于概率分布的负荷模型

function load_scenarios = generate_load_scenarios(n_buses, n_samples)
    % 生成蒙特卡洛负荷场景

    load_scenarios = zeros(n_buses, n_samples);

    % 基准负荷 (MW)
    base_load = [125, 90, 100, 120, 60, 60, 200, 200, 60, 60, ...
                 45, 60, 60, 120, 60, 60, 60, 90, 90, 90, ...
                 90, 90, 90, 420, 420, 60, 60, 60, 120, 120]';

    for i = 1:n_buses
        % 正态分布：均值=基准负荷，标准差=10%均值
        load_mean = base_load(i);
        load_std = 0.1 * load_mean;  % 10%变异系数

        % 生成随机负荷样本
        load_scenarios(i, :) = normrnd(load_mean, load_std, 1, n_samples);

        % 确保负荷非负
        load_scenarios(i, :) = max(load_scenarios(i, :), 0.1 * load_mean);
    end

    fprintf('生成了 %d 个负荷场景，每个场景 %d 个节点\n', n_samples, n_buses);
end

2. 考虑时间相关性的负荷模型

function [load_time_series] = generate_correlated_loads(n_buses, n_timesteps, n_samples)
    % 生成具有时空相关性的负荷序列

    % 负荷相关性矩阵 (简化模型)
    correlation_matrix = eye(n_buses);
    for i = 1:n_buses
        for j = i+1:n_buses
            % 相邻节点相关性更高
            if abs(i-j) <= 2
                correlation_matrix(i,j) = 0.7;
                correlation_matrix(j,i) = 0.7;
            else
                correlation_matrix(i,j) = 0.3;
                correlation_matrix(j,i) = 0.3;
            end
        end
    end

    % Cholesky分解生成相关随机数
    L = chol(correlation_matrix, 'lower');

    load_time_series = zeros(n_buses, n_timesteps, n_samples);

    for sample = 1:n_samples
        for t = 1:n_timesteps
            % 基础负荷模式 (日负荷曲线)
            base_pattern = 0.7 + 0.3 * sin(2*pi*(t-6)/24);

            % 随机扰动
            random_component = L * randn(n_buses, 1);

            % 组合生成负荷
            base_load = [125, 90, 100, 120, 60, 60, 200, 200, 60, 60, ...
                         45, 60, 60, 120, 60, 60, 60, 90, 90, 90, ...
                         90, 90, 90, 420, 420, 60, 60, 60, 120, 120]';

            load_time_series(:, t, sample) = base_load .* base_pattern .* ...
                (1 + 0.1 * random_component);
        end
    end
end

概率潮流计算核心

1. 基于蒙特卡洛的潮流计算

function [voltage_results, line_loading_results] = probabilistic_power_flow(load_scenarios)
    % 概率潮流计算主函数

    [n_buses, n_samples] = size(load_scenarios);

    % 初始化结果存储
    voltage_results = zeros(n_buses, n_samples);
    line_loading_results = zeros(n_lines, n_samples);

    % 系统基准数据
    [bus_data, line_data, gen_data] = initialize_system_data();

    % 并行计算所有场景
    parfor i = 1:n_samples
        if mod(i, 100) == 0
            fprintf('正在计算第 %d/%d 个场景...\n', i, n_samples);
        end

        % 更新负荷数据
        current_load = load_scenarios(:, i);

        % 执行潮流计算
        [V, line_flows] = run_power_flow(bus_data, line_data, gen_data, current_load);

        % 存储结果
        voltage_results(:, i) = abs(V);
        line_loading_results(:, i) = line_flows;
    end

    fprintf('所有 %d 个场景计算完成\n', n_samples);
end

2. 牛顿-拉夫逊潮流计算

function [V, line_flows] = run_power_flow(bus_data, line_data, gen_data, load_vector)
    % 牛顿-拉夫逊潮流计算

    max_iterations = 20;
    tolerance = 1e-8;

    n_buses = length(bus_data);

    % 初始化电压
    V = ones(n_buses, 1);  % 电压幅值
    theta = zeros(n_buses, 1); % 电压相角

    % 构建导纳矩阵
    Y = build_admittance_matrix(bus_data, line_data);

    for iter = 1:max_iterations
        % 计算功率不平衡量
        [P_calc, Q_calc] = calculate_power(V, theta, Y, n_buses);

        % 计算功率偏差
        [delta_P, delta_Q] = calculate_power_mismatch(P_calc, Q_calc, gen_data, load_vector, bus_data);

        % 检查收敛
        max_mismatch = max([abs(delta_P); abs(delta_Q)]);
        if max_mismatch < tolerance
            break;
        end

        % 构建雅可比矩阵
        J = build_jacobian_matrix(V, theta, Y, bus_data, n_buses);

        % 求解修正方程
        mismatch_vector = [delta_P; delta_Q];
        correction = J \ mismatch_vector;

        % 更新电压
        d_theta = correction(1:n_buses);
        d_V = correction(n_buses+1:end);

        theta = theta + d_theta;
        V = V + d_V;
    end

    % 计算线路潮流
    line_flows = calculate_line_flows(V, theta, line_data, Y);
end

结果统计分析与可视化

1. 概率统计与风险指标

function analyze_results(voltage_results, line_loading_results, confidence_level)
    % 结果统计分析与可视化

    [n_buses, n_samples] = size(voltage_results);
    [n_lines, ~] = size(line_loading_results);

    %% 电压统计分析
    voltage_stats = struct();
    for i = 1:n_buses
        voltage_data = voltage_results(i, :);

        voltage_stats.mean(i) = mean(voltage_data);
        voltage_stats.std(i) = std(voltage_data);
        voltage_stats.prob_low(i) = sum(voltage_data < 0.95) / n_samples;  % 低电压概率
        voltage_stats.prob_high(i) = sum(voltage_data > 1.05) / n_samples; % 高电压概率

        % 置信区间
        voltage_stats.ci_lower(i) = quantile(voltage_data, (1-confidence_level)/2);
        voltage_stats.ci_upper(i) = quantile(voltage_data, 1 - (1-confidence_level)/2);
    end

    %% 线路负载率统计分析
    line_stats = struct();
    for i = 1:n_lines
        line_data = line_loading_results(i, :);

        line_stats.mean_loading(i) = mean(line_data);
        line_stats.max_loading(i) = max(line_data);
        line_stats.prob_overload(i) = sum(line_data > 1.0) / n_samples;  % 过载概率
        line_stats.expected_overload(i) = mean(max(line_data - 1.0, 0)); % 期望过载程度
    end

    %% 可视化结果
    create_visualizations(voltage_stats, line_stats, n_buses, n_lines);

    %% 输出关键风险指标
    fprintf('\n=== 系统风险评估结果 ===\n');
    fprintf('节点电压越限概率: %.2f%%\n', 100 * max(voltage_stats.prob_low + voltage_stats.prob_high));
    fprintf('线路过载概率: %.2f%%\n', 100 * max(line_stats.prob_overload));
    fprintf('最危险节点: %d (越限概率: %.2f%%)\n', ...
        find(voltage_stats.prob_low == max(voltage_stats.prob_low), 1), ...
        100 * max(voltage_stats.prob_low));
    fprintf('最危险线路: %d (过载概率: %.2f%%)\n', ...
        find(line_stats.prob_overload == max(line_stats.prob_overload), 1), ...
        100 * max(line_stats.prob_overload));
end

2. 结果可视化函数

function create_visualizations(voltage_stats, line_stats, n_buses, n_lines)
    % 创建综合可视化图表

    figure('Position', [100, 100, 1400, 800]);

    % 子图1: 电压概率分布
    subplot(2, 3, 1);
    errorbar(1:n_buses, voltage_stats.mean, ...
             voltage_stats.mean - voltage_stats.ci_lower, ...
             voltage_stats.ci_upper - voltage_stats.mean, 'o');
    hold on;
    plot([1, n_buses], [0.95, 0.95], 'r--', 'LineWidth', 2);
    plot([1, n_buses], [1.05, 1.05], 'r--', 'LineWidth', 2);
    xlabel('节点编号');
    ylabel('电压 (pu)');
    title('节点电压统计 (95%置信区间)');
    grid on;
    legend('电压均值', '电压限值');

    % 子图2: 电压越限概率
    subplot(2, 3, 2);
    bar(voltage_stats.prob_low + voltage_stats.prob_high);
    xlabel('节点编号');
    ylabel('越限概率');
    title('节点电压越限概率');
    grid on;

    % 子图3: 线路负载率分布
    subplot(2, 3, 3);
    boxplot(line_loading_results');
    hold on;
    plot([0, n_lines+1], [1, 1], 'r--', 'LineWidth', 2);
    xlabel('线路编号');
    ylabel('负载率 (pu)');
    title('线路负载率分布');

    % 子图4: 线路过载概率
    subplot(2, 3, 4);
    [sorted_prob, sorted_idx] = sort(line_stats.prob_overload, 'descend');
    bar(sorted_prob(1:min(10, n_lines)));
    set(gca, 'XTickLabel', sorted_idx(1:min(10, n_lines)));
    xlabel('线路编号');
    ylabel('过载概率');
    title('TOP 10 过载风险线路');

    % 子图5: 概率密度函数
    subplot(2, 3, 5);
    [f, xi] = ksdensity(voltage_results(10, :));  % 示例: 节点10的电压分布
    plot(xi, f, 'LineWidth', 2);
    hold on;
    xline(0.95, 'r--', 'LineWidth', 2);
    xline(1.05, 'r--', 'LineWidth', 2);
    xlabel('电压 (pu)');
    ylabel('概率密度');
    title('节点10电压概率密度');
    grid on;

    % 子图6: 累积分布函数
    subplot(2, 3, 6);
    ecdf(voltage_results(10, :));
    hold on;
    xline(0.95, 'r--', 'LineWidth', 2);
    xline(1.05, 'r--', 'LineWidth', 2);
    xlabel('电压 (pu)');
    ylabel('累积概率');
    title('节点10电压累积分布');
    grid on;
end

高级功能扩展

1. 基于场景削减的快速计算

function [reduced_scenarios, weights] = scenario_reduction(load_scenarios, n_clusters)
    % 使用K-means聚类进行场景削减

    [n_buses, n_samples] = size(load_scenarios);

    % 执行K-means聚类
    [cluster_idx, cluster_centers] = kmeans(load_scenarios', n_clusters);

    % 计算每个场景的权重
    reduced_scenarios = cluster_centers';
    weights = histcounts(cluster_idx, 1:n_clusters+1) / n_samples;

    fprintf('场景从 %d 削减到 %d\n', n_samples, n_clusters);
end

2. 灵敏度分析与关键参数识别

function sensitivity_analysis(load_scenarios, voltage_results)
    % 灵敏度分析：识别对系统状态影响最大的负荷节点

    [n_buses, n_samples] = size(load_scenarios);

    sensitivities = zeros(n_buses, 1);

    for i = 1:n_buses
        % 计算每个节点负荷与系统电压的相关系数
        for j = 1:n_buses
            correlation_matrix = corrcoef(load_scenarios(i, :), voltage_results(j, :));
            sensitivities(i) = sensitivities(i) + abs(correlation_matrix(1, 2));
        end
        sensitivities(i) = sensitivities(i) / n_buses;
    end

    % 排序并显示关键节点
    [~, sorted_idx] = sort(sensitivities, 'descend');

    fprintf('\n=== 灵敏度分析结果 ===\n');
    fprintf('对系统电压影响最大的前5个负荷节点:\n');
    for i = 1:min(5, n_buses)
        fprintf('%d. 节点 %d (灵敏度指数: %.4f)\n', i, sorted_idx(i), sensitivities(sorted_idx(i)));
    end
end

参考代码 蒙特卡洛和概率密度 www.youwenfan.com/contentalh/81314.html

应用场景与优势

优化建议

1. 并行计算：使用parfor循环加速蒙特卡洛模拟
2. 场景削减：对大规模问题使用聚类方法减少计算量
3. 智能采样：采用拉丁超立方采样提高采样效率
4. 简化模型：对非关键区域使用等效模型

这套方法能够有效评估电力系统在不确定性负荷条件下的运行风险，为电网规划和运行提供重要的决策支持。通过概率统计方法，可以更全面地认识系统的安全边界，而不仅仅是依赖确定性分析。