Fuzzy Logic Voltage Control

🎯 Lab Objectives

• Implement fuzzy membership functions in MATLAB.
• Calculate voltage error.
• Convert a crisp voltage error into fuzzy memberships.
• Evaluate fuzzy IF–THEN rules.
• Calculate a crisp reactive-power control action.
• Plot membership functions and controller response.
• Understand the physical meaning of fuzzy voltage control.

1. Voltage-Control Problem

Consider a bus in an electrical distribution network.

The desired voltage is:

The measured voltage is:

Define the voltage error:

therefore:

2. Fuzzy Input Sets

We define three fuzzy sets for the voltage error:

3. Membership Functions

Let the error range be:

4. Triangular Membership Function

We use a triangular function:

This is convenient for MATLAB implementation.

5. MATLAB Membership Function

trimf = @(x,a,b,c) ...
    max(min((x-a)/(b-a),(c-x)/(c-b)),0);

This anonymous MATLAB function evaluates a triangular membership function.

6. Define the Error Membership Sets

muN = @(e) trimf(e,-0.10,-0.05,0.00);
muZ = @(e) trimf(e,-0.05, 0.00,0.05);
muP = @(e) trimf(e, 0.00, 0.05,0.10);

where:

• muN = Negative membership
• muZ = Zero membership
• muP = Positive membership

7. Fuzzification

For:

calculate:

mN = muN(e);
mZ = muZ(e);
mP = muP(e);

For the selected membership functions, approximately:

8. Output Variable

Let the fuzzy-controller output be a reactive-power correction:

We define three singleton output values:

9. Fuzzy Rule Base

10. Rule Activation

The activation levels are simply the input membership degrees:

a1 = mN;
a2 = mZ;
a3 = mP;

11. Defuzzification Using Weighted Average

Because we use singleton outputs, a convenient defuzzification equation is:

where:

12. Numerical Example

Assume:

then:

13. MATLAB Controller Calculation

Vref = 1.00;
V    = 0.96;

e = Vref - V;

mN = muN(e);
mZ = muZ(e);
mP = muP(e);

qN = -0.10;
qZ =  0.00;
qP =  0.10;

dQ = (mN*qN + mZ*qZ + mP*qP) / ...
     (mN + mZ + mP);

14. Complete MATLAB Program

clear; clc; close all;

% ------------------------------------------------
% Fuzzy Logic Voltage Controller
% ------------------------------------------------

Vref = 1.00;
V    = 0.96;

% Voltage error
e = Vref - V;

% Triangular membership function
trimf = @(x,a,b,c) ...
    max(min((x-a)/(b-a),(c-x)/(c-b)),0);

% Input fuzzy sets
muN = @(e) trimf(e,-0.10,-0.05,0.00);
muZ = @(e) trimf(e,-0.05, 0.00,0.05);
muP = @(e) trimf(e, 0.00, 0.05,0.10);

% Fuzzification
mN = muN(e);
mZ = muZ(e);
mP = muP(e);

% Singleton output values
qN = -0.10;
qZ =  0.00;
qP =  0.10;

% Defuzzification
den = mN + mZ + mP;

if den > 0
    dQ = (mN*qN + mZ*qZ + mP*qP)/den;
else
    dQ = 0;
end

% Display results
fprintf('Vref = %.3f pu\n',Vref)
fprintf('V    = %.3f pu\n',V)
fprintf('Error e = %.3f pu\n\n',e)

fprintf('muNegative = %.3f\n',mN)
fprintf('muZero     = %.3f\n',mZ)
fprintf('muPositive = %.3f\n\n',mP)

fprintf('Reactive power correction dQ = %.4f pu\n',dQ)

% ------------------------------------------------
% Plot membership functions
% ------------------------------------------------

eVec = linspace(-0.10,0.10,500);

N = zeros(size(eVec));
Z = zeros(size(eVec));
P = zeros(size(eVec));

for k = 1:length(eVec)

    N(k) = muN(eVec(k));
    Z(k) = muZ(eVec(k));
    P(k) = muP(eVec(k));

end

figure

plot(eVec,N,'LineWidth',1.6)
hold on
plot(eVec,Z,'LineWidth',1.6)
plot(eVec,P,'LineWidth',1.6)

xline(e,'--','Current Error')

grid on

xlabel('Voltage Error e [pu]')
ylabel('Membership Degree')

legend('Negative','Zero','Positive')

title('Fuzzy Membership Functions')

% ------------------------------------------------
% Controller characteristic
% ------------------------------------------------

dQVec = zeros(size(eVec));

for k = 1:length(eVec)

    aN = muN(eVec(k));
    aZ = muZ(eVec(k));
    aP = muP(eVec(k));

    den = aN + aZ + aP;

    if den > 0

        dQVec(k) = ...
            (aN*qN + aZ*qZ + aP*qP)/den;

    else

        dQVec(k) = 0;

    end
end

figure

plot(eVec,dQVec,'LineWidth',1.6)

grid on

xlabel('Voltage Error e [pu]')
ylabel('Reactive Power Correction \DeltaQ [pu]')

title('Fuzzy Voltage Controller Characteristic')

15. Understanding the Controller Characteristic

The second figure shows:

16. Physical Interpretation

In many simplified voltage-control situations:

17. Experiment 1 — Change the Measured Voltage

Try:

V = 0.92;
V = 0.98;
V = 1.00;
V = 1.03;
V = 1.08;

Record the resulting:

18. Experiment 2 — Change Membership Width

Compare:

muZ = @(e) trimf(e,-0.05,0,0.05);

with a wider Zero set:

muZ = @(e) trimf(e,-0.08,0,0.08);

19. Experiment 3 — Change Output Strength

Compare:

qN = -0.10;
qP =  0.10;

with:

qN = -0.20;
qP =  0.20;

20. Optional Closed-Loop Simulation

We can create a very simple educational voltage-response model:

where KQ is an assumed voltage sensitivity.

KQ = 0.10;

V = 0.94;

for k = 1:20

    e = Vref - V;

    mN = muN(e);
    mZ = muZ(e);
    mP = muP(e);

    den = mN + mZ + mP;

    if den > 0
        dQ = (mN*qN + mZ*qZ + mP*qP)/den;
    else
        dQ = 0;
    end

    V = V + KQ*dQ;

    Vhist(k) = V;

end

figure
plot(1:20,Vhist,'o-','LineWidth',1.6)
hold on
yline(Vref,'--')
grid on

xlabel('Iteration')
ylabel('Voltage [pu]')
title('Closed-Loop Fuzzy Voltage Control')

21. Lab Tasks

1. Run the complete fuzzy-controller program.
2. Record membership degrees for V=0.96 pu.
3. Calculate ΔQ manually and compare with MATLAB.
4. Test five different voltage values.
5. Change the width of the Zero membership function.
6. Change the output singleton values.
7. Plot the controller characteristic e→ΔQ.
8. Run the optional closed-loop example.
9. Explain the physical meaning of the resulting control action.

📝 Questions

1. Why do fuzzy membership functions overlap?
2. What is fuzzification?
3. Why can several rules be active simultaneously?
4. What does a positive voltage error mean?
5. Why does a positive error produce positive ΔQ in this example?
6. What is the effect of widening the Zero membership function?
7. What is the effect of increasing qP?
8. Why is the fuzzy-control characteristic smooth?