【状态估计】基于UKF法、AUKF法的电力系统三相状态估计研究（Matlab代码实现）

**基于UKF法、AUKF法的电力系统三相状态估计研究**

**一、前言**

电力系统状态估计是电力系统监测和控制的一个重要组成部分。通过估计电力系统的状态，可以实现实时监测和预测，提高电力系统的可靠性和效率。在近年来，基于非线性滤波（UKF）法和自适应扩展卡尔曼滤波（AUKF）法的电力系统状态估计研究得到了广泛关注。这些方法通过利用系统模型和测量数据，可以实现高精度的状态估计。

**二、基于UKF法的电力系统三相状态估计**

###2.1 UKF算法原理UKF是一种用于非线性系统状态估计的滤波器。它通过使用卡尔曼滤波的思想和扩展卡尔曼滤波（EKF）的方法，来实现高精度的状态估计。

###2.2 UKF算法步骤1. **预测**:根据系统模型预测下一时刻的状态。
2. **计算sigma点**:根据系统模型和噪声分布计算sigma点。
3. **评估**:使用sigma点来评估系统状态的误差。
4. **更新**:根据评估结果更新系统状态。

###2.3 Matlab代码实现

matlabfunction [xhat, P] = ukf(x0, P0, f, h, Q, R, tspan)
 % UKF算法参数 alpha =1;
 beta =2;
 kappa =0;
 % sigma点计算 lambda = (alpha^2 * kappa -1) / (alpha^2 + kappa);
 N = size(x0,1);
 n = length(tspan);
 xhat = zeros(N, n);
 P = zeros(N, N, n);
 for i =1:n % 预测 x_pred = f(x0, tspan(i));
 % 计算sigma点 sigma_points = compute_sigma_points(x0, lambda, alpha, beta, kappa);
 %评估 x_eval = zeros(N, n);
 for j =1:n x_eval(:, j) = h(sigma_points(j, :), tspan(i));
 end % 更新 [xhat(:, i), P(:,:,i)] = update(x0, P0, x_pred, sigma_points, x_eval, Q, R);
 % 更新状态和误差矩阵 x0 = xhat(:, i);
 P0 = P(:,:,i);
 end function [sigma_points] = compute_sigma_points(x0, lambda, alpha, beta, kappa)
 N = size(x0,1);
 n = length(tspan);
 sigma_points = zeros(n, N);
 for i =1:n % 计算sigma点 x_sigma = zeros(N,2*n+1);
 for j =1:2*n+1 if j ==1 x_sigma(:, j) = x0;
 elseif j <= n x_sigma(:, j) = x0 + sqrt(lambda + kappa) * [sqrt(chol(P(:,:,i)))]';
 else x_sigma(:, j) = x0 - sqrt(lambda + kappa) * [sqrt(chol(P(:,:,i)))]';
 end end % 计算sigma点的平均值 x_mean = mean(x_sigma,2);
 % 计算sigma点的标准差 sigma_points(i, :) = x_mean;
 end function [xhat, P] = update(x0, P0, x_pred, sigma_points, x_eval, Q, R)
 N = size(x0,1);
 n = length(tspan);
 % 计算权重 weights = zeros(n,1);
 for i =1:n if i ==1 weights(i) =2 * lambda / (lambda + kappa) +1;
 elseif i <= n weights(i) =2 * lambda / (lambda + kappa);
 else weights(i) =2 * lambda / (lambda + kappa);
 end end % 计算状态估计值 xhat = zeros(N,1);
 for i =1:n xhat = xhat + weights(i) * x_eval(i, :)';
 end % 计算误差矩阵 P = zeros(N, N);
 for i =1:n P = P + weights(i) * (sigma_points(i, :) - xhat) * (sigma_points(i, :) - xhat)';
 end % 添加噪声项 P = P + Q;
 % 计算状态估计值的标准差 std_P = sqrt(diag(P));
 % 更新状态和误差矩阵 x0 = xhat;
 P0 = P;
 endend

matlabfunction [xhat, P] = aukf(x0, P0, f, h, Q, R, tspan)
 % AUKF算法参数 alpha =1;
 beta =2;
 kappa =0;
 % sigma点计算 lambda = (alpha^2 * kappa -1) / (alpha^2 + kappa);
 N = size(x0,1);
 n = length(tspan);
 xhat = zeros(N, n);
 P = zeros(N, N, n);
 for i =1:n % 预测 x_pred = f(x0, tspan(i));
 % 计算sigma点 sigma_points = compute_sigma_points(x0, lambda, alpha, beta, kappa);
 %评估 x_eval = zeros(N, n);
 for j =1:n x_eval(:, j) = h(sigma_points(j, :), tspan(i));
 end % 更新 [xhat(:, i), P(:,:,i)] = update(x0, P0, x_pred, sigma_points, x_eval, Q, R);
 % 更新状态和误差矩阵 x0 = xhat(:, i);
 P0 = P(:,:,i);
 end function [sigma_points] = compute_sigma_points(x0, lambda, alpha, beta, kappa)
 N = size(x0,1);
 n = length(tspan);
 sigma_points = zeros(n, N);
 for i =1:n % 计算sigma点 x_sigma = zeros(N,2*n+1);
 for j =1:2*n+1 if j ==1 x_sigma(:, j) = x0;
 elseif j <= n x_sigma(:, j) = x0 + sqrt(lambda + kappa) * [sqrt(chol(P(:,:,i)))]';
 else x_sigma(:, j) = x0 - sqrt(lambda + kappa) * [sqrt(chol(P(:,:,i)))]';
 end end % 计算sigma点的平均值 x_mean = mean(x_sigma,2);
 % 计算sigma点的标准差 sigma_points(i, :) = x_mean;
 end function [xhat, P] = update(x0, P0, x_pred, sigma_points, x_eval, Q, R)
 N = size(x0,1);
 n = length(tspan);
 % 计算权重 weights = zeros(n,1);
 for i =1:n if i ==1 weights(i) =2 * lambda / (lambda + kappa) +1;
 elseif i <= n weights(i) =2 * lambda / (lambda + kappa);
 else weights(i) =2 * lambda / (lambda + kappa);
 end end % 计算状态估计值 xhat = zeros(N,1);
 for i =1:n xhat = xhat + weights(i)