AI-Based Optimal Power Flow

🎯 Lab Objectives

• Understand the difference between Economic Dispatch and OPF.
• Formulate a simplified Optimal Power Flow problem.
• Include generation cost and network losses in the objective.
• Handle voltage and generation constraints.
• Use penalty functions for infeasible operating points.
• Implement a simplified evolutionary optimizer in MATLAB.
• Interpret the optimal solution physically.

1. From Economic Dispatch to OPF

Economic Dispatch mainly determines generator active powers. Optimal Power Flow also considers the electrical network.

2. Simplified Two-Bus Network

Consider a simple educational two-bus system.

The generator supplies:

The load consumes:

3. Simplified Network-Loss Model

For educational purposes, approximate active-power losses as:

where:

This is not a full AC power-flow model, but it introduces the idea that larger power transfers produce larger losses.

4. Power-Balance Constraint

Generation must supply both the load and losses:

Therefore:

5. Generator Cost

Assume:

The first component of the objective is operating cost.

6. Simplified Voltage Model

To introduce voltage constraints, use the educational approximation:

with:

Qc represents a controllable reactive-power support variable.

7. Voltage Constraint

Operating points outside this range are penalized.

8. Decision Variables

We optimize:

with limits:

9. Multi-Objective Cost

Define:

where:

The weight wL determines how strongly losses are penalized relative to cost.

10. Power-Balance Penalty

Use:

11. Voltage Penalty

If:

then:

and if:

then:

12. Complete Objective Function

13. Population Representation

Each chromosome contains:

Example:

14. Optimization Parameters

popSize = 50;
maxGen  = 120;

Pc = 0.8;
Pm = 0.12;

sigmaPG = 5;
sigmaQ  = 0.05;

15. Initial Population

PGmin = 80;
PGmax = 140;

Qmin = -0.5;
Qmax = 0.5;

population = zeros(popSize,2);

population(:,1) = ...
    PGmin + rand(popSize,1)*(PGmax-PGmin);

population(:,2) = ...
    Qmin + rand(popSize,1)*(Qmax-Qmin);

16. Evaluate One Operating Point

For one chromosome:

calculate:

17. MATLAB Objective Evaluation

PG = population(i,1);
Qc = population(i,2);

Ploss = k*PG^2;

V2 = 1 - r*PG/1000 + q*Qc;

Cgen = 0.005*PG^2 + 5*PG + 300;

mismatch = PG - PL - Ploss;

penP = lambdaP*mismatch^2;

penV = 0;

if V2 < Vmin
    penV = lambdaV*(Vmin-V2)^2;
elseif V2 > Vmax
    penV = lambdaV*(V2-Vmax)^2;
end

J = Cgen + wLoss*Ploss + penP + penV;

18. Selection, Crossover, Mutation

We use the same evolutionary logic introduced in Lab 05.

19. Complete MATLAB Program

clear; clc; close all;

% ------------------------------------------------
% AI-Based Optimal Power Flow
% Simplified educational example
% ------------------------------------------------

PL = 100;

% Simplified loss coefficient
k = 0.0005;

% Simplified voltage model coefficients
r = 0.08;
q = 0.04;

% Voltage limits
Vmin = 0.95;
Vmax = 1.05;

% Decision-variable limits
PGmin = 80;
PGmax = 140;

Qmin = -0.5;
Qmax = 0.5;

% Objective weights
wLoss = 100;

lambdaP = 10000;
lambdaV = 10000;

% GA parameters
popSize = 50;
maxGen = 120;

Pc = 0.8;
Pm = 0.12;

sigmaPG = 5;
sigmaQ  = 0.05;

% ------------------------------------------------
% Initial population
% ------------------------------------------------

population = zeros(popSize,2);

population(:,1) = ...
    PGmin + rand(popSize,1)*(PGmax-PGmin);

population(:,2) = ...
    Qmin + rand(popSize,1)*(Qmax-Qmin);

bestHist = zeros(maxGen,1);

% ------------------------------------------------
% Evolutionary optimization
% ------------------------------------------------

for gen = 1:maxGen

    Jvec = zeros(popSize,1);

    for i = 1:popSize

        PG = population(i,1);
        Qc = population(i,2);

        % Network losses
        Ploss = k*PG^2;

        % Simplified voltage
        V2 = 1 - r*PG/1000 + q*Qc;

        % Generation cost
        Cgen = 0.005*PG^2 + 5*PG + 300;

        % Power mismatch
        mismatch = PG - PL - Ploss;

        % Power-balance penalty
        penP = lambdaP*mismatch^2;

        % Voltage penalty
        penV = 0;

        if V2 < Vmin

            penV = ...
                lambdaV*(Vmin-V2)^2;

        elseif V2 > Vmax

            penV = ...
                lambdaV*(V2-Vmax)^2;

        end

        % Penalized objective
        Jvec(i) = ...
            Cgen + ...
            wLoss*Ploss + ...
            penP + ...
            penV;

    end

    % ------------------------------------------------
    % Elitism
    % ------------------------------------------------

    [bestJ,bestIdx] = min(Jvec);

    elite = population(bestIdx,:);

    bestHist(gen) = bestJ;

    % ------------------------------------------------
    % New population
    % ------------------------------------------------

    newPopulation = zeros(size(population));

    newPopulation(1,:) = elite;

    kpop = 2;

    while kpop <= popSize

        % --------------------------------------------
        % Tournament Parent 1
        % --------------------------------------------

        a = randi(popSize);
        b = randi(popSize);

        if Jvec(a) < Jvec(b)
            parent1 = population(a,:);
        else
            parent1 = population(b,:);
        end

        % --------------------------------------------
        % Tournament Parent 2
        % --------------------------------------------

        a = randi(popSize);
        b = randi(popSize);

        if Jvec(a) < Jvec(b)
            parent2 = population(a,:);
        else
            parent2 = population(b,:);
        end

        % --------------------------------------------
        % Crossover
        % --------------------------------------------

        if rand < Pc

            alpha = rand;

            child1 = ...
                alpha*parent1 + ...
                (1-alpha)*parent2;

            child2 = ...
                (1-alpha)*parent1 + ...
                alpha*parent2;

        else

            child1 = parent1;
            child2 = parent2;

        end

        % --------------------------------------------
        % Mutation
        % --------------------------------------------

        if rand < Pm
            child1(1) = child1(1) + sigmaPG*randn;
        end

        if rand < Pm
            child1(2) = child1(2) + sigmaQ*randn;
        end

        if rand < Pm
            child2(1) = child2(1) + sigmaPG*randn;
        end

        if rand < Pm
            child2(2) = child2(2) + sigmaQ*randn;
        end

        % --------------------------------------------
        % Variable limits
        % --------------------------------------------

        child1(1) = ...
            min(max(child1(1),PGmin),PGmax);

        child1(2) = ...
            min(max(child1(2),Qmin),Qmax);

        child2(1) = ...
            min(max(child2(1),PGmin),PGmax);

        child2(2) = ...
            min(max(child2(2),Qmin),Qmax);

        % --------------------------------------------
        % Store offspring
        % --------------------------------------------

        newPopulation(kpop,:) = child1;

        if kpop+1 <= popSize
            newPopulation(kpop+1,:) = child2;
        end

        kpop = kpop + 2;

    end

    population = newPopulation;

end

% ------------------------------------------------
% Final evaluation
% ------------------------------------------------

Jvec = zeros(popSize,1);

for i = 1:popSize

    PG = population(i,1);
    Qc = population(i,2);

    Ploss = k*PG^2;

    V2 = 1 - r*PG/1000 + q*Qc;

    Cgen = 0.005*PG^2 + 5*PG + 300;

    mismatch = PG - PL - Ploss;

    penP = lambdaP*mismatch^2;

    penV = 0;

    if V2 < Vmin

        penV = ...
            lambdaV*(Vmin-V2)^2;

    elseif V2 > Vmax

        penV = ...
            lambdaV*(V2-Vmax)^2;

    end

    Jvec(i) = ...
        Cgen + ...
        wLoss*Ploss + ...
        penP + ...
        penV;

end

[bestJ,bestIdx] = min(Jvec);

best = population(bestIdx,:);

PGbest = best(1);
Qbest  = best(2);

PlossBest = k*PGbest^2;

V2best = ...
    1 - r*PGbest/1000 + q*Qbest;

Cbest = ...
    0.005*PGbest^2 + ...
    5*PGbest + 300;

mismatchBest = ...
    PGbest - PL - PlossBest;

% ------------------------------------------------
% Display results
% ------------------------------------------------

fprintf('AI-Based OPF Result\n\n')

fprintf('PG = %.4f MW\n',PGbest)

fprintf('Qc = %.4f pu\n',Qbest)

fprintf('Load = %.4f MW\n',PL)

fprintf('Losses = %.4f MW\n',PlossBest)

fprintf('Voltage V2 = %.4f pu\n',V2best)

fprintf('Power mismatch = %.6f MW\n', ...
        mismatchBest)

fprintf('Generation cost = %.4f\n', ...
        Cbest)

fprintf('Best objective J = %.4f\n', ...
        bestJ)

% ------------------------------------------------
% Convergence curve
% ------------------------------------------------

figure

plot(1:maxGen,bestHist,'LineWidth',1.6)

grid on

xlabel('Generation')
ylabel('Best Objective Function')

title('AI-Based OPF Convergence')

20. Interpreting the Final Solution

After optimization, check:

If both conditions are satisfied and the objective is low, the candidate represents a good operating point.

21. Experiment 1 — Change Load

PL = 80;
PL = 100;
PL = 120;

Observe PG, losses, voltage, and total objective value.

22. Experiment 2 — Increase Loss Weight

wLoss = 10;
wLoss = 100;
wLoss = 500;

23. Experiment 3 — Tighten Voltage Limits

Vmin = 0.98;
Vmax = 1.02;

Observe how the optimal Qc changes.

24. Experiment 4 — Remove Reactive Support

Force:

Qmin = 0;
Qmax = 0;

Compare voltage and objective value with the controllable-Q case.

25. Toward Real AC-OPF

A real AC Optimal Power Flow problem uses:

with constraints on:

• Generator active power
• Generator reactive power
• Bus voltages
• Line currents
• Transformer taps
• Battery operation

26. AI and OPF

AI can support OPF in several ways:

27. Lab Tasks

1. Implement the simplified loss model.
2. Implement the simplified voltage model.
3. Construct the objective function.
4. Add power-balance penalties.
5. Add voltage-limit penalties.
6. Run the evolutionary optimizer.
7. Record PG, Qc, Ploss, and V2.
8. Verify power balance.
9. Change the load and repeat.
10. Change wLoss and compare results.
11. Remove reactive support and compare voltage.
12. Explain how the simplified model differs from AC-OPF.

📝 Review Questions

1. What is the difference between Economic Dispatch and OPF?
2. Why are network losses included in OPF?
3. What does Qc represent in this laboratory?
4. Why do we need a voltage penalty?
5. Why do we need a power-balance penalty?
6. What is the effect of increasing wLoss?
7. Why must an optimal solution also be physically feasible?
8. What additional equations appear in AC-OPF?
9. How can AI accelerate OPF calculations?