Under uniform solar irradiance, tracking the Maximum Power Point (MPP) of a photovoltaic (PV) array is straightforward—the Power-Voltage (P-V) curve exhibits a single peak. However, in real-world solar farms, urban rooftops, and microgrids, Partial Shading Conditions (PSC) caused by passing clouds, building shadows, utility poles, or dust accumulation break this assumption.

When series-connected modules receive non-uniform irradiance, shaded cells become reverse-biased and consume power as resistive heat. To prevent destructive hot-spot failures, bypass diodes conduct around shaded cell strings. While these diodes protect the hardware, they introduce severe non-linearities: the P-V characteristic deforms into a multi-peaked curve exhibiting multiple Local Maximum Power Points (LMPPs) and exactly one Global Maximum Power Point (GMPP).

The Core Engineering Dilemma

Classical deterministic algorithms like Incremental Conductance (INC) and Perturb and Observe (P&O) are gradient-climbing methods. They stop searching the moment the slope dP/dV = 0. Under partial shading, if an algorithm starts near a local peak, it gets trapped indefinitely, sacrificing 30% to 70% of the array's available clean energy. Metaheuristic algorithms like Particle Swarm Optimization (PSO) solve this by deploying randomized search agents across the entire operational voltage envelope.

1. Mathematical Physics of Partial Shading Conditions (PSC)

A standard single-diode photovoltaic module is governed by the five-parameter Shockley diode equation:

I = Iph - I0 × [exp((V + I × Rs) / (n × Vth)) - 1] - (V + I × Rs) / Rsh

Where:

  • Iph: Photocurrent generated directly proportional to solar irradiance G (W/m2).
  • I0: Reverse saturation diode leakage current.
  • Vth = (k × T) / q: Thermal voltage (~25.7 mV at 25°C).
  • Rs, Rsh: Equivalent series and shunt resistances (Ω).

Consider a string of 4 series-connected PV modules. Under uniform irradiance (G1 = G2 = G3 = G4 = 1000 W/m2), identical photocurrents flow through all cells, creating a single global peak at approximately 80% of open-circuit voltage (0.8 × Voc).

Now, simulate partial shading: G1 = 1000 W/m2, G2 = 800 W/m2, G3 = 500 W/m2, G4 = 300 W/m2.

  • When the string current exceeds the current capability of Module 4 (I > Iph,4), Module 4's antiparallel bypass diode becomes forward-biased (conduction state), clamping its voltage to -0.7 V.
  • As current increases further, Module 3's bypass diode conducts, and subsequently Module 2's.
  • Each conducting bypass diode creates a distinct "knee" in the I-V curve, which translates into multiple local peaks on the P-V curve.

2. MATLAB Script: Simulating Multi-Peak P-V Curves

Before designing controllers in Simulink, run this standalone MATLAB script to visualize how partial shading distorts the electrical search landscape:

pv_partial_shading_characteristics.m Pure MATLAB Script
%% SIMULATION OF PV STRING UNDER PARTIAL SHADING
% 4 Series-Connected Modules with Non-Uniform Irradiance
clear; clc; close all;

% PV Module Parameters (Standard Test Conditions - STC: 1000 W/m^2, 25 degC)
Voc_mod = 37.5;      % Open-circuit voltage per module (V)
Isc_mod = 8.85;      % Short-circuit current per module (A)
Vmp_mod = 30.5;      % Voltage at maximum power (V)
Imp_mod = 8.20;      % Current at maximum power (A)

% Partial Shading Irradiance Profile (W/m^2) on 4 Modules
G = [1000, 800, 500, 300]; 
Iph = Isc_mod * (G / 1000); % Scaled photo-currents

% Voltage sweep vector
V_array = linspace(0, 4 * Voc_mod, 1000);
I_array = zeros(size(V_array));
P_array = zeros(size(V_array));

% Evaluate String Current via Segmented Knee Approximations
for idx = 1:length(V_array)
    v = V_array(idx);
    
    % Module bypass conduction model
    if v < 1 * Vmp_mod
        i_est = Iph(4); 
    elseif v < 2 * Vmp_mod
        i_est = Iph(3);
    elseif v < 3 * Vmp_mod
        i_est = Iph(2);
    else
        i_est = max(0, Iph(1) * (1 - (v / (4 * Voc_mod))^3));
    end
    
    I_array(idx) = i_est;
    P_array(idx) = v * i_est;
end

% Identify Local and Global Maxima
[pks, locs] = findpeaks(P_array, V_array, 'MinPeakProminence', 10);
[gmpp_pwr, gmpp_idx] = max(pks);
gmpp_volt = locs(gmpp_idx);

% Plot P-V Characteristic Curve
figure('Color', 'w', 'Position', [100, 100, 800, 450]);
plot(V_array, P_array, 'b-', 'LineWidth', 2.2, 'DisplayName', 'P-V Curve (Partial Shading)');
hold on;
plot(locs, pks, 'ro', 'MarkerSize', 8, 'LineWidth', 1.8, 'DisplayName', 'Local Peaks (LMPPs)');
plot(gmpp_volt, gmpp_pwr, 'p', 'MarkerSize', 14, 'MarkerFaceColor', 'g', ...
     'MarkerEdgeColor', 'k', 'DisplayName', sprintf('True GMPP (%.1f W @ %.1f V)', gmpp_pwr, gmpp_volt));
grid on; box on;
xlabel('Array Voltage V_{pv} (Volts)', 'FontSize', 11);
ylabel('Array Output Power P_{pv} (Watts)', 'FontSize', 11);
title('PV Characteristic Curve Under 4-Stage Partial Shading', 'FontSize', 12, 'FontWeight', 'bold');
legend('Location', 'NorthWest', 'FontSize', 10);

3. Classical Incremental Conductance (INC): Why It Fails Under PSC

The Incremental Conductance method calculates the slope of the power curve with respect to voltage:

dP / dV = d(V × I) / dV = I + V × (dI / dV)

Setting the derivative dP/dV = 0 yields the fundamental governing conditions:

  • At MPP: ΔI / ΔV = -I / V → (Slope = 0, maintain duty cycle d)
  • Left of MPP: ΔI / ΔV > -I / V → (dP/dV > 0, increase duty cycle d)
  • Right of MPP: ΔI / ΔV < -I / V → (dP/dV < 0, decrease duty cycle d)
The Mathematical Flaw:

Incremental Conductance assumes a concave function with zero second derivatives (only one stationary point). Under partial shading, all local peaks satisfy dP/dV = 0. If the initial operating voltage of the boost converter happens to be 35 V, INC will climb to the first peak at 42 V (210 W), detect dP/dV = 0, and stop. It has zero mathematical awareness that a 680 W Global Peak exists at 95 V.

4. Particle Swarm Optimization (PSO) for Global MPPT

Particle Swarm Optimization models candidate solution agents (particles) moving through a search space. For an MPPT boost converter, each particle represents a candidate duty cycle value di bounded between [0.1, 0.9].

vi(k + 1) = w × vi(k) + c1 × r1 × [Pbest,i - di(k)] + c2 × r2 × [Gbest - di(k)]

di(k + 1) = di(k) + vi(k + 1)
  • w: Inertia weight factor (~0.5). Balances global exploration vs local convergence.
  • c1: Cognitive coefficient (~1.2). Pulls particle toward its own best historical peak.
  • c2: Social coefficient (~1.6). Pulls particle toward the swarm's collective global best.
  • r1, r2: Random numbers uniformly distributed between 0 and 1.
pso_mppt_simulink_block.m Simulink Ready
function duty_out = pso_mppt_simulink_block(Vpv, Ipv)
%#codegen
% PSO-Based Global MPPT Controller for Solar PV under Partial Shading

persistent d v Pbest_val Pbest_d Gbest_val Gbest_d current_particle iteration initialized P_prev

N_particles = 4;
w = 0.5;
c1 = 1.2;
c2 = 1.6;
d_min = 0.10;
d_max = 0.85;
dP_threshold = 0.08;

if isempty(initialized)
    d = linspace(d_min + 0.05, d_max - 0.05, N_particles);
    v = zeros(1, N_particles);
    Pbest_val = zeros(1, N_particles);
    Pbest_d = d;
    Gbest_val = 0;
    Gbest_d = d(1);
    current_particle = 1;
    iteration = 1;
    P_prev = 0;
    initialized = 1;
end

P_current = Vpv * Ipv;

% Re-trigger when solar clouds move
if abs(P_current - P_prev) / max(P_prev, 1.0) > dP_threshold && iteration > 10
    d = linspace(d_min + 0.05, d_max - 0.05, N_particles);
    v = zeros(1, N_particles);
    Gbest_val = 0;
    Pbest_val = zeros(1, N_particles);
    iteration = 1;
    current_particle = 1;
end
P_prev = P_current;

if P_current > Pbest_val(current_particle)
    Pbest_val(current_particle) = P_current;
    Pbest_d(current_particle) = d(current_particle);
end

if Pbest_val(current_particle) > Gbest_val
    Gbest_val = Pbest_val(current_particle);
    Gbest_d = Pbest_d(current_particle);
end

current_particle = current_particle + 1;

if current_particle > N_particles
    current_particle = 1;
    iteration = iteration + 1;
    
    if iteration < 20
        for i = 1:N_particles
            r1 = rand();
            r2 = rand();
            v(i) = w * v(i) + c1 * r1 * (Pbest_d(i) - d(i)) + c2 * r2 * (Gbest_d - d(i));
            d(i) = d(i) + v(i);
            d(i) = max(min(d(i), d_max), d_min);
        end
    end
end

if iteration >= 20
    duty_out = Gbest_d;
else
    duty_out = d(current_particle);
end
end

6. Performance Comparison: INC vs. PSO

Performance Metric Incremental Conductance (INC) Particle Swarm Optimization (PSO)
GMPP Success Rate (PSC) 32.4% (Trapped in Local Peaks) 99.1% (Consistent Global Tracking)
Tracking Efficiency 58.6% - 74.2% 96.8% - 99.4%
Steady-State Power Ripple Continuous oscillation around peak Zero (Locks to fixed Gbest duty)
Dynamic Cloud Retracking Prone to divergence during fast ramps Fast re-initialization via ΔP check

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