Multi-Criteria Evaluation of MPPT Algorithms under Time-Varying Partial Shading Using a Physics-Based PV Model

2.1 PV electrical model with bypass diodes

We model a PV module comprising ${{N}_{\text{sub}}}$ substrings in series; substring $j\in \left\{ 1,\ldots ,{{N}_{\text{sub}}} \right\}$ has ${{N}_{s,j}}$ cells and an antiparallel bypass diode. The module KVL in Eq. (1) relates the terminal voltage $V$ to the substring voltages ${{V}_{j}}\left( \cdot  \right)$ evaluated at the array current $I$, plus a small series interconnect ${{R}_{s}}>0$:

$V\ =\ \underset{j=1}{\overset{{{N}_{\text{sub}}}}{\mathop \sum }}\,{{V}_{j}}\left( I \right)\ +\ {{R}_{s}}I$        (1)

For each substring $j$, Eq. (2) states the one-diode current balance: ${{\text{ }\!\!\Xi\!\!\text{ }}_{j}}=0$ defines the implicit I–V relation, where ${{I}_{ph,j}}\left( T,{{G}_{j}} \right)$ is the photo-current under temperature $T$ and irradiance ${{G}_{j}}$; ${{I}_{0,j}}\left( T \right)$ is the saturation current; ${{R}_{s,j}},{{R}_{p,j}}$ are substring series/shunt resistances; ${{n}_{j}}>0$ is the effective ideality; and ${{V}_{T}}\left( T \right)={{k}_{B}}T/q$ is the thermal voltage with ${{k}_{B}}$ Boltzmann’s constant and $q$ the elementary charge:

${{\Xi }_{j}}\left( I,{{V}_{j}};T,{{G}_{j}} \right)\ =\ {{I}_{ph,j}}\left( T,{{G}_{j}} \right)-{{I}_{0,j}}\left( T \right)\left( {{e}^{\frac{{{V}_{j}}+I{{R}_{s,j}}}{{{n}_{j}}{{N}_{s,j}}{{V}_{T}}}}}-1 \right)-\frac{{{V}_{j}}+I{{R}_{s,j}}}{{{R}_{p,j}}}-I$       (2)

Bypass conduction is enforced in Eq. (3): once the bypass diode forward-drops to ${{V}_{bd,j}}>0$, the substring clamps to ${{V}_{j}}\left( I \right)\equiv -{{V}_{bd,j}}$ with zero incremental slope $\text{d}{{V}_{j}}/\text{d}I=0$:

${{V}_{j}}\left( I \right)\equiv -{{V}_{bd,j}},\frac{d{{V}_{j}}}{dI}=0$        (3)

Temperature/irradiance scaling in Eq. (4) specify a smooth data-sheet–style law, introducing the photonic reference ${{I}_{ph,ref}}$ at ${{G}_{ref}}$, the temperature coefficient ${{\alpha }_{I}}$, the saturation-current reference ${{I}_{0,ref}}$ at ${{T}_{ref}}$, and its slope ${{\beta }_{T}}$:

${{I}_{ph,j}}\left( T,{{G}_{j}} \right)={{I}_{ph,ref}}\left( \frac{{{G}_{j}}}{{{G}_{ref}}} \right)\left( 1+{{\alpha }_{I}}\left( T-{{T}_{ref}} \right) \right),{{I}_{0,j}}\left( T \right)={{I}_{0,ref}}\,\text{exp}\left( {{\beta }_{T}}\left( T-{{T}_{ref}} \right) \right)$       (4)

Eliminating $\left\{ {{V}_{j}} \right\}$ from Eqs. (2) and (3) yields the module residual $g\left( \cdot  \right)$ in Eq. (5), where ${{\hat{V}}_{j}}\left( I;T,{{G}_{j}} \right)$ denotes the substring voltage obtained by solving ${{\text{ }\!\!\Xi\!\!\text{ }}_{j}}=0$ under the non-bypass inequality ${{V}_{j}}\ge -{{V}_{bd,j}}$; the operating current is the root in Eq. (6):

$g\left( I;V,T,G \right)\ =\ \underset{j=1}{\overset{{{N}_{sub}}}{\mathop \sum }}\,{{\hat{V}}_{j}}\left( I;T,{{G}_{j}} \right)\ +\ {{R}_{s}}I\ -\ V$         (5)

$g\left( {{I}^{\star }};V,T,G \right)=0,{{I}^{\star }}\in \left[ {{I}_{min}},\,{{I}_{max}} \right]$        (6)

with the practical bracket Eq. (7) using substring short-circuit currents ${{I}_{sc,j}}\left( T,{{G}_{j}} \right)$; ${{I}_{\text{min}}}<0$ allows slight back-bias:

${{I}_{\text{max}}}=1.05\,\underset{j}{\mathop{\text{max}}}\,{{I}_{sc,j}}\left( T,{{G}_{j}} \right),{{I}_{\text{min}}}<0$         (7)

For Newton updates, Eq. (8) gives the substring slope by implicit differentiation of Eq. (2); here ${{\zeta }_{j}}=\left( {{V}_{j}}+I{{R}_{s,j}} \right)/\left( {{n}_{j}}{{N}_{s,j}}{{V}_{T}} \right)$. Aggregating substring slopes and the module series ${{R}_{s}}$ produces the Jacobian ${g}'\left( I \right)$ in Eq. (9), with bypassed substrings contributing zero:

$\frac{d{{V}_{j}}}{dI}\ =\ -\,\frac{\partial {{\Xi }_{j}}/\partial I}{\partial {{\Xi }_{j}}/\partial {{V}_{j}}}=-\,\frac{-{{I}_{0,j}}{{e}^{{{\zeta }_{j}}}}\frac{{{R}_{s,j}}}{{{n}_{j}}{{N}_{s,j}}{{V}_{T}}}-\frac{{{R}_{s,j}}}{{{R}_{p,j}}}-1}{-{{I}_{0,j}}{{e}^{{{\zeta }_{j}}}}\frac{1}{{{n}_{j}}{{N}_{s,j}}{{V}_{T}}}-\frac{1}{{{R}_{p,j}}}},{{\zeta }_{j}}=\frac{{{V}_{j}}+I{{R}_{s,j}}}{{{n}_{j}}{{N}_{s,j}}{{V}_{T}}},$          (8)

$\begin{gathered}g^{\prime}(I)=\sum_{j=1}^{N_{\text {sub }}} \frac{\mathrm{d} \hat{V}_j}{\mathrm{~d} I}+R_s, \quad \text { with } \quad \frac{\mathrm{d} \hat{V}_j}{\mathrm{~d} I}=0 \text { if } V_j= -V_{b d, j}\end{gathered}$     (9)

To avoid divergence near small $\left| {g}'\left( I \right) \right|$, the damped Newton step in Eq. (10) adds a slope-contingent regularizer ${{\lambda }_{k}}={{\eta }_{k}}\text{max}\left\{ 0,\epsilon -\left| {g}'\left( {{I}_{k}} \right) \right| \right\}$ (with ${{\eta }_{k}}\ge 0$, $\epsilon >0$); the projected, line-searched update and step clamp appear in Eq. (11), using ${{\gamma }_{k}}\in \left( 0,1 \right]$, the saturation $\text{sa}{{\text{t}}_{\text{ }\!\!\Delta\!\!\text{ }}}\left( \cdot  \right)$ with bound ${{\text{ }\!\!\Delta\!\!\text{ }}_{\text{max}}}$, and the projection $\text{pro}{{\text{j}}_{\left[ {{I}_{\text{min}}},{{I}_{\text{max}}} \right]}}$:

$\Delta I_{k}^{\text{N}}=-\frac{g\left( {{I}_{k}} \right)}{{g}'\left( {{I}_{k}} \right)+{{\lambda }_{k}}},{{\lambda }_{k}}={{\eta }_{k}}\,max\left\{ 0,\ \epsilon -\left| {g}'\left( {{I}_{k}} \right) \right| \right\}$        (10)

${{I}_{k+1}}=\text{pro}{{\text{j}}_{\left[ {{I}_{\text{min}}},{{I}_{\text{max}}} \right]}}\left( {{I}_{k}}+{{\gamma }_{k}}\,\text{sa}{{\text{t}}_{\text{ }\!\!\Delta\!\!\text{ }}}\left( \text{ }\!\!\Delta\!\!\text{ }I_{k}^{\text{N}} \right) \right)$         (11)

When curvature is unreliable, Eq. (12) restores bracketing by a safeguarded bisection using the current interval endpoints $\left( {{I}_{\text{L}}},{{I}_{\text{U}}} \right)$:

${{I}_{k+1}}\leftarrow 1/2\left( {{I}_{L}}+{{I}_{U}} \right),\left\{ {{I}_{L}},{{I}_{U}} \right\}\text{bracket}g\left( I \right)=0$         (12)

For linearized diagnostics, Eq. (13) is the local affine surrogate in $\delta I$; its module slope equals ${g}'\left( I \right)$:

${{\hat{V}}_{j}}\left( I+\delta I \right)\approx {{\hat{V}}_{j}}\left( I \right)+\frac{d{{{\hat{V}}}_{j}}}{dI}\delta I,\quad V+\delta V\approx V+\left( \mathop{\sum }_{j}\frac{d{{{\hat{V}}}_{j}}}{dI}+{{R}_{s}} \right)\delta I$           (13)

Bypass action can also be written as the complementarity system Eq. (14), introducing the nonnegative multiplier ${{\mu }_{j}}$; for smooth Jacobians in fast solvers, Eq. (15) replaces the hard switch by a softplus ${{\phi }_{\tau }}\left( x \right)=\tau \text{ln}\left( 1+{{e}^{x/\tau }} \right)$ with smoothing $\tau >0$ and ${{\text{ }\!\!\Upsilon\!\!\text{ }}_{j}}$ the solution variable inside (2):

$\begin{array}{r}V_j+V_{b d, j} \geq 0, \quad \mu_j \geq 0, \quad \mu_j\left(V_j+V_{b d, j}\right)=  0, \quad \Xi_j\left(I, V_j\right)=0 \text { when } \mu_j=0\end{array}$       (14)

${{V}_{j}}\approx -{{V}_{bd,j}}+{{\phi }_{\tau }}\left( {{\text{ }\!\!\Upsilon\!\!\text{ }}_{j}} \right),{{\text{ }\!\!\Upsilon\!\!\text{ }}_{j}}\text{solves }\!\!~\!\!\text{ (2)},{{\phi }_{\tau }}\left( x \right)=\tau \text{ln}\left( 1+{{e}^{x/\tau }} \right),\tau \downarrow 0$       (15)

Parametric sensitivities follow from implicit differentiation; Eq. (16) collects the temperature and irradiance derivatives, using the generic placeholder “$\bullet $” for any parameter of interest:

$\frac{\partial g}{\partial T}=\underset{j}{\mathop \sum }\,\frac{\partial {{{\hat{V}}}_{j}}}{\partial T},\frac{\partial g}{\partial {{G}_{j}}}=\frac{\partial {{{\hat{V}}}_{j}}}{\partial {{G}_{j}}},\frac{\partial {{{\hat{V}}}_{j}}}{\partial \bullet }=-\frac{\partial {{\text{ }\!\!\Xi\!\!\text{ }}_{j}}/\partial \bullet }{\partial {{\text{ }\!\!\Xi\!\!\text{ }}_{j}}/\partial {{V}_{j}}}$          (16)

Finally, Eq. (17) states the acceptance test with residual and step tolerances ${{\varepsilon }_{V}},{{\varepsilon }_{I}}>0$ and a bypass-consistency slack ${{\varepsilon }_{bd}}>0$:

$\begin{gathered}\left|g\left(I^{\star}\right)\right| \leq \varepsilon_V, \quad\left|\Delta I_k\right| \leq \varepsilon_I, \quad \text { and } \quad V_j \geq-V_{b d, j}-  \varepsilon_{b d} \forall j\end{gathered}$          (17)

2.2 Parameter set and physical constants

For clarity, the complete parameterization used by the PV model is collected in the parameter vector in Eq. (18).

This grouping follows common practice in PV modeling surveys and state-space treatments of MPPT test procedures [1, 43, 65].

$\boldsymbol{\Theta}=\left\{\begin{array}{c}N_{s u b},N_{s, j}, n_j, R_s, R_{s, j}, R_{p, j}, \\ V_{b d, j}, I_{p h, \text { ref}}, I_{0, \text { ref}}, \alpha_{I s c}, \beta_{V o c}, \\ E_g, G_{\text {ref }}, T_{\text {ref }}, V_{\min }, V_{\max }\end{array}\right\}$          (18)

Temperature $T$ is in kelvin with ${{V}_{T}}\left( T \right)={{k}_{B}}T/q$. The photocurrent scaling in Eq. (19) reflects irradiance proportionality and a first-order temperature coefficient consistent with data-sheet conventions and modeling references [1, 43, 65].

${{I}_{ph,j}}\left( T,{{G}_{j}} \right)={{I}_{ph,ref}}\left( \frac{{{G}_{j}}}{{{G}_{ref}}} \right)\left[ 1+{{\alpha }_{Isc}}\left( T-{{T}_{ref}} \right) \right]$        (19)

The saturation current law in Eq. (20) adopts a bandgap-aware Arrhenius form with a polynomial prefactor $\kappa \in \left[ 2,4 \right]$, widely used for c-Si devices over moderate temperature spans [43, 65].

${{I}_{0,j}}\left( T \right)={{I}_{0,\text{ref}}}{{\left( \frac{T}{{{T}_{\text{ref}}}} \right)}^{\kappa }}\text{exp}\left[ -\frac{{{E}_{g}}}{{{k}_{B}}}\left( \frac{1}{T}-\frac{1}{{{T}_{\text{ref}}}} \right) \right],\kappa \in \left[ 2,4 \right]$           (20)

Eq. (21) combines a first-order open-circuit temperature drift with the standard diode-equation approximation at ${{T}_{\text{ref}}}$, which links ${{V}_{oc}}$ to the ${{I}_{ph}}/{{I}_{0}}$ ratio for a substring with ideality ${{n}_{j}}$ and ${{N}_{s,j}}$ series cells [43, 65].

${{V}_{oc,j}}\left( T \right)\approx {{V}_{oc,ref}}\left[ 1+{{\beta }_{Voc}}\left( T-{{T}_{ref}} \right) \right],{{V}_{oc,ref}}\simeq {{n}_{j}}{{N}_{s,j}}{{V}_{T}}\left( {{T}_{ref}} \right)ln\left( 1+\frac{{{I}_{ph,ref}}}{{{I}_{0,ref}}} \right)$           (21)

The nominal values in the in-text table are representative of c-Si modules and consistent with modeling choices used in comparative MPPT evaluations [43, 52].

The admissible operating sets in Eq. (22) provide explicit solver/controller bounds; the current bracket is chosen from substring short-circuit conditions as in standard EN50530-style setups [43, 52].

$\begin{gathered}\mathcal{V}=\left\{V \in \mathbb{R}: V_{\min } \leq V \leq V_{\max }\right\}, \quad \mathcal{J}= {\left[I_{\min }, I_{\max }\right], I_{\max } \approx 1.05 \max _j I_{s c, j}}\end{gathered}$          (22)

This compact Table 1 fixes the symbols, nominal values, and temperature/irradiance scalings enforced with series/parallel resistances and ideal bypass diodes used throughout the benchmark.

Table 1. Parameterization of the substring-resolved single-diode PV model

2.3 Partial-shading scenario generator

Time discretization ${{t}_{k}}=k\text{ }\!\!\Delta\!\!\text{ }t$ with a fixed seed $s$ ensures repeatable irradiance sequences; the four primitives below mirror patterns recommended for dynamic MPPT evaluation and reported in PSC characterization studies [1, 43, 52]. Uniform forcing is constant in time as in Eq. (23):

${{G}_{U}}\left( t \right)\equiv {{g}_{0}}$          (23)

The step profile in Eq. (24) captures abrupt regime changes used in EN50530-type dynamic tests and many PSC case studies [43, 52].

$\boldsymbol{G}_S(t)=\boldsymbol{g}_0+\sum_{\ell=1}^m\left(\boldsymbol{h}_{\ell}-\boldsymbol{h}_{\ell-1}\right) \mathbb{1}_{\left\{t \geq t_{\ell}\right\}}, \quad \boldsymbol{h}_0=\mathbf{0}$          (24)

Cloud events in Eq. (25) model localized, stochastic irradiance deficits per substring, consistent with reports on PSC morphology and its impact on multi-peak $P-V$ characteristics [1, 20, 47].

${{G}_{C,j}}\left( t \right)={{g}_{0,j}}-\underset{k:{{j}_{k}}=j}{\mathop \sum }\,{{d}_{k}}\,exp\left( -\frac{{{(t-{{c}_{k}})}^{2}}}{2w_{k}^{2}} \right),j=1,2,3$             (25)

In the proposed shading scenario generator, all stochastic and deterministic parameters are explicitly defined to ensure full reproducibility. The irradiance is constrained within the physically admissible range of ${{G}_{\text{min}}}=200\,\text{W}/{{\text{m}}^{2}}$and ${{G}_{\text{max}}}=1000\,\text{W}/{{\text{m}}^{2}}$, with ${{G}_{\text{STC}}}=1000\,\text{W}/{{\text{m}}^{2}}$ and ${{T}_{\text{STC}}}=298.15\,\text{K}$. Each transient cloud is modelled by a Gaussian attenuation applied to the background irradiance ${{G}_{0}}\left( t \right)$, with a fixed amplitude ${{A}_{c}}=0.6~$(dimensionless) and temporal spread ${{\sigma }_{c}}=0.35\,\text{s}$, centred at the event time ${{t}_{c}}$. For burst-type events, three overlapping Gaussian components $\left( {{N}_{c}}=3 \right)$are superimposed within the interval ${{t}_{i}}\in \left[ 3,6 \right]\,\text{s}$, with individual amplitudes sampled from a uniform distribution ${{A}_{i}}\sim \mathcal{U}\left( 0.4,~0.8 \right)$ to emulate variable cloud density. A subsequent linear ramp is applied during the interval $t\in \left[ 6,10 \right]\,\text{s}$with a slope of ${{k}_{r}}=80\,\text{W}/{{\text{m}}^{2}}/\text{s}$, reflecting a realistic large-scale irradiance transition. All stochastic processes are generated using a fixed random seed of 2025, thereby guaranteeing deterministic and repeatable shading profiles across independent simulation runs.

The ramp forcing in Eq. (26) emulates slow drifts superimposed on PSC to test tracker persistence under nonstationary envelopes [43, 52].

${{G}_{R}}\left( t \right)={{g}_{0}}+r\,max\left\{ 0,t-{{t}_{\star }} \right\}$           (26)

The convex mixture and clipping in Eq. (27) generate mixed, bounded sequences while preserving reproducibility; this construction aligns with systematic, code-driven evaluation pipelines [43, 52].

${{G}_{M}}\left( t \right)={{\Pi }_{\left[ {{G}_{min}},{{G}_{max}} \right]}}\left( \alpha \,{{G}_{S}}\left( t \right)+\beta \,{{G}_{C}}\left( t \right)+\gamma \,{{G}_{R}}\left( t \right)+\left( 1-\alpha -\beta -\gamma  \right){{G}_{U}}\left( t \right) \right)$          (27)

Seed determinism in Eq. (28) is standard good practice to enable exact reproducibility of PSC realizations across runs and algorithms [43].

$\text{RNG}\_\text{Init}\left( s \right)\Rightarrow \{{{c}_{k}},{{w}_{k}},{{j}_{k}},{{d}_{k}}\}_{k=1}^{K}\text{reproducible}$          (28)

To certify multi-peak $P-V$ landscapes, we compute instantaneous power $P\left( V;t \right)=V\,{{I}^{\star }}\left( V;t \right)$ using the module solution from Eqs. (1)-(6) and track bypass states $\mathbf{\sigma }\left( t,V \right)$, the bound in Eq. (29) reflects that each clamped substring can introduce at most one additional local extremum. Empirical and analytical discussions of such multi-peak behavior under PSC appear widely in the literature [1, 23, 43].

$LM\left( P\left( \cdot ;t \right) \right)\le 1+\underset{j=1}{\overset{{{N}_{sub}}}{\mathop \sum }}\,\sigma _{j}^{max}\left( t \right)\le {{N}_{sub}}$          (29)

Condition Eq. (30) records the existence of at least two stationary points with distinct voltages when one substring is heavily shaded, typical in step-cloud mixtures, and selects the global one by power comparison [1, 23]:

$\frac{dP}{dV}\left( V_{1}^{\circ };t \right)=0,\frac{dP}{dV}\left( V_{2}^{\circ };t \right)=0,V_{1}^{\circ }\ne V_{2}^{\circ },P\left( V_{2}^{\circ };t \right)>P\left( V_{1}^{\circ };t \right)$            (30)

Finally, the peak-count functional $\mathcal{M}\left( t \right)$ in Eq. (31) implements a simple, voltage-sweep diagnostic consistent with state-space and dynamic-test analyses of PV trackers [43, 52].

$\begin{gathered}\mathcal{M}(t)=\sum_i \mathbb{1}\left(\mathrm{~d} P / \mathrm{d} V\left(V_i^{-} ; t\right) \cdot \mathrm{d} P / \mathrm{d} V\left(V_i^{+} ; t\right)<0\right),  \left\{V_i\right\} \text { critical points along } \mathcal{V}\end{gathered}$       (31)

2.4 MPPT algorithms

This subsection formalizes four MPPT controllers, perturb-and-observe (P&O), incremental conductance (INC), particle swarm optimization (PSO), and firefly/flower-pollination (FP), on the PV model in Eqs. (1)-(17) and the admissible voltage set in Eq. (22), using EN50530-style dynamics and recent refinements as context [51, 53, 64].

P&O (directional perturbation with safeguarded step).

The ascent direction is defined in Eq. (32), where $\mathcal{P}\left( V \right)=V\,{{I}^{\star }}\left( V;{{t}_{k}} \right)$ is the instantaneous power and ${{\sigma }_{k}}\in \left\{ -1,0,+1 \right\}$ aligns the next move with the last power change [51, 62, 69].

$\text{ }\!\!\Delta\!\!\text{ }{{\mathcal{P}}_{k}}\triangleq \mathcal{P}\left( {{V}_{k}} \right)-\mathcal{P}\left( {{V}_{k-1}} \right),{{\sigma }_{k}}\triangleq \text{sgn}\left( \text{ }\!\!\Delta\!\!\text{ }{{\mathcal{P}}_{k}} \right)\text{sgn}\left( {{V}_{k}}-{{V}_{k-1}} \right)$           (32)

The step selection and projection law in Eq. (33) clips ${{\alpha }_{k}}$ to $\left[ {{\alpha }_{\text{min}}},{{\alpha }_{\text{max}}} \right]$, scales by a local slope surrogate $\left| {{\nabla }_{V}}\mathcal{P} \right|$, and enforces feasibility via ${{\text{ }\!\!\Pi\!\!\text{ }}_{\mathcal{V}}}\left[ \cdot  \right]$; $\varepsilon >0$ regularizes small gradients [51, 66, 69].

${{\alpha }_{k}}\triangleq \text{clip}\left( {{\alpha }_{\text{min}}},\frac{\left| \,\text{ }\!\!\Delta\!\!\text{ }{{\mathcal{P}}_{k}}\, \right|}{\left| \,{{\nabla }_{V}}\mathcal{P}\left( {{V}_{k-1}} \right)\, \right|+\varepsilon },{{\alpha }_{\text{max}}} \right),{{V}_{k+1}}={{\text{ }\!\!\Pi\!\!\text{ }}_{\mathcal{V}}}\left[ {{V}_{k}}+{{\sigma }_{k}}{{\alpha }_{k}} \right]$        (33)

Near a stationary point ${{V}^{\star }}$ with ${{\nabla }_{V}}\mathcal{P}\left( {{V}^{\star }} \right)=0$, the curvature-based bound in Eq. (34) tempers overshoot using a local Hessian envelope on $\mathcal{N}\left( {{V}^{\star }} \right)$; $\epsilon >0$ prevents division by small denominators [53, 64].

$\left|\alpha_k\right|>\frac{2\left|\nabla_V \mathcal{P}\left(V^{\star}\right)\right|}{\sup _{V \in \mathcal{N}\left(V^{\star}\right)}\left|\nabla_V^2 \mathcal{P}(V)\right|+\epsilon}, \quad \nabla_V \mathcal{P}\left(V^{\star}\right)=0$           (34)

INC (incremental conductance with signed regulation).

The MPP condition follows from ${{\nabla }_{V}}\left( VI \right)=0$, yielding the conductance relation in Eq. (35) with $\mathcal{J}(V)$ the I–V [56, 64].

$\nabla_V \mathcal{P}(V)=\mathcal{J}(V)+V \frac{d \jmath}{d V} \Rightarrow \frac{d \jmath}{d V}=-\frac{\jmath}{V} \quad(M P P)$          (35)

The implementable control in Eq. (36) uses the error surrogate ${{\hat{\delta }}_{k}}=\frac{\text{ }\!\!\Delta\!\!\text{ }{{I}_{k}}}{\text{ }\!\!\Delta\!\!\text{ }{{V}_{k}}}+\frac{{{I}_{k}}}{{{V}_{k}}+\epsilon }$, a positive gain ${{\mu }_{k}}$, a symmetric limiter $\text{sa}{{\text{t}}_{\eta }}\left( \cdot  \right)$ of width $\eta $, and projection ${{\text{ }\!\!\Pi\!\!\text{ }}_{\mathcal{V}}}$ to respect bounds [56, 66].

${{V}_{k+1}}={{\Pi }_{\mathcal{V}}}\left[ {{V}_{k}}-{{\mu }_{k}}\,sa{{t}_{\eta }}\left( {{{\hat{\delta }}}_{k}} \right) \right],{{\hat{\delta }}_{k}}\triangleq \frac{\Delta {{I}_{k}}}{\Delta {{V}_{k}}}+\frac{{{I}_{k}}}{{{V}_{k}}+\epsilon }$          (36)

PSO (population search maximizing power).

The velocity update in Eq. (37) combines inertia $\omega $, cognitive/social gains ${{c}_{1}},{{c}_{2}}$, and i.i.d. uniform vectors ${{\mathbf{r}}_{1}},{{\mathbf{r}}_{2}}$ to pull particle $i$ toward its personal best ${{\mathbf{p}}_{i}}$ and the swarm best ${{\mathbf{g}}^{t}}$; $\odot $ denotes elementwise multiplication [52, 57, 66].

$v_{i}^{t+1}=\omega \,v_{i}^{t}+{{c}_{1}}{{r}_{1}}\odot \left( {{p}_{i}}-x_{i}^{t} \right)+{{c}_{2}}{{r}_{2}}\odot \left( {{g}^{t}}-x_{i}^{t} \right)$            (37)

The position/slew constraint in Eq. (38) advances ${{\mathbf{x}}_{i}}$ with limiter $\text{sa}{{\text{t}}_{\delta V}}$, projects into $\mathcal{V}$, and reselects ${{\mathbf{g}}^{t+1}}$ as the power-maximizing candidate among $\left\{ {{\mathbf{p}}_{i}},{{\mathbf{g}}^{t}} \right\}$ [52, 57, 66].

$x_{i}^{t+1}={{\Pi }_{\mathcal{V}}}\left[ x_{i}^{t}+sa{{t}_{\delta V}}\left( v_{i}^{t+1} \right) \right],{{g}^{t+1}}=arg\underset{\left\{ {{p}_{i}},{{g}^{t}} \right\}}{\mathop{min}}\,\left( -\mathcal{P}\left( \cdot  \right) \right)$           (38)

Firefly / Flower-Pollination (intensity–attractiveness with Lévy jumps).

The attractiveness kernel $\beta \left( {{r}_{ij}} \right)={{\beta }_{0}}{{e}^{-\gamma r_{ij}^{2}}}$ and move rule in Eq. (39) pull agent $i$ toward a brighter neighbor $j$ with baseline attractiveness ${{\beta }_{0}}$, absorption factor $\gamma $, distance ${{r}_{ij}}=\left| x_{i}^{t}-x_{j}^{t} \right|$, and a Lévy jump $\varsigma \,{{\mathcal{L}}_{\lambda }}$ scaled by $\varsigma $; feasibility is enforced by ${{\text{ }\!\!\Pi\!\!\text{ }}_{\mathcal{V}}}$ [60, 63, 68].

$\beta \left( {{r}_{ij}} \right)={{\beta }_{0}}{{e}^{-\gamma r_{ij}^{2}}},x_{i}^{t+1}={{\Pi }_{\mathcal{V}}}\left[ x_{i}^{t}+\beta \left( {{r}_{ij}} \right)\left( x_{j}^{t}-x_{i}^{t} \right)+\varsigma \,{{\mathcal{L}}_{\lambda }} \right],{{r}_{ij}}=\left| x_{i}^{t}-x_{j}^{t} \right|.$          (39)

The Lévy increment in Eq. (40) follows the Mantegna sampler with tail index $1<\lambda <2$, where $\text{ }\!\!\Gamma\!\!\text{ }\left( \cdot  \right)$ is the gamma function and $u,v\sim \mathcal{N}\left( 0,1 \right)$ are standard normals, enabling heavy-tailed exploration on multi-peak P–V landscapes [60, 63, 68].

${{\mathcal{L}}_{\lambda }}\sim \frac{\Gamma \left( 1+\lambda  \right)sin\left( \pi \lambda /2 \right)}{\Gamma \left( 1+\lambda /2 \right)\lambda \,{{2}^{\left( \lambda -1 \right)/2}}}\frac{u}{|v{{|}^{1/\lambda }}},u,v\sim \mathcal{N}\left( 0,1 \right)$             (40)

The acceptance/relaxation rule in Eq. (41) uses $\rho \in \left( 0,1 \right]$ to convex-combine rejected proposals with the incumbent, mitigating destabilizing drops in $\mathcal{P}$ during aggressive exploratory jumps [61, 63, 68].

$\operatorname{accept}\left(x_i^{t+1}\right)$ iff $\mathcal{P}\left(x_i^{t+1} \geq \mathcal{P}\left(x_i^t\right)\right.$ else $x_i^{t+1} \leftarrow(1-$ρ)$x_i^t+\rho x_i^{t+1}$           (41)

We evaluate four trackers within a single, reproducible pipeline: a three-substring PV model with bypass diodes is driven by a 10-s mixed shading profile at 1 ms resolution; identical voltage bounds/slew limits and a fixed per-step compute budget apply to all controllers; an internal oracle yields the instantaneous global optimum solely for scoring. From the resulting power–voltage trajectories we first reveal how each method migrates across the moving multi-peak surface, then compress behavior into the core metrics defined earlier, energy normalized to the oracle integral, event-window time-to-sustain, steady-window ripple, and an ε-dwell hit ratio. Accuracy–speed trade-offs are exposed on a common event definition, followed by error-series and a spectral proxy on steady segments to connect oscillations with power quality. Instantaneous deficits define a loss process whose histograms/CDF/boxplots quantify tails and outliers; cumulative-time curves show when energy advantages accrue. A weight-transparent composite score is paired with the empirical Pareto set to separate near-dominance from dominated regions, while median/IQR and nonparametric inference establish robustness. The sensitivity sweeps (step sizes, swarm/coefficients, attractiveness/Lévy, weights) report stability vs. performance, per-cycle cost and memory footprints assess embedded feasibility, and a brief validity analysis covers oracle use, solver safeguards, and modeling assumptions.

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Figure 1. Dynamic MPPT performance under time-varying partial shading conditions

Figure 1 summarizes dynamic MPPT behavior under four irradiance regimes (uniform 0–2 s, step change 2–4 s, cloud burst 4–6 s, dynamic ramp 6–10 s) by juxtaposing instantaneous power and operating voltage against the global MPP reference. The overlaid traces correspond to P&O, INC, PSO, and FP, with the dashed line indicating the time-varying global maximum power point, enabling direct visual comparison of tracking stability and transient behavior across all irradiance regimes. Incremental Conductance (INC) snaps to the GMPP almost immediately in each regime (consistent with t₉₅ ≈ 3 ms) and preserves near-coincident voltage tracking during rapid peak relocations around 2.0, 4.0, and 6.0 s, limiting energy loss when the optimum shifts from the ≈200-210 W plateau to intermediate levels near110-150 W. PSO initially aligns well under uniform irradiance but exhibits intermittent zero-power gaps at regime boundaries (swarm re-exploration) that elongate effective convergence (t₉₅ ≈ 206 ms) and create visible voltage dropouts. P&O lags after each transition, remaining pinned near a sub-optimal ≈18-20 V shelf during intervals where the optimal voltage resides≈27-32 V, illustrating local peak capture and finite perturbation step inertia. The firefly (FP) controller, despite very fast initial response (t₉₅ ≈ 2.6 ms), becomes highly volatile after 6s, oscillating and collapsing to zero power for extended spans; its voltage trace fragments into wide excursions that rarely dwell at the optimal plateau, explaining its low cumulative energy and hit ratio. The joint trajectories thus expose distinct failure modes, exploration latency (PSO), local entrapment (P&O), and instability (FP), while highlighting INC’s superior balance of rapid reacquisition and voltage steadiness across nonstationary multi-peak conditions.

For the metaheuristic trackers, the reported behaviour is closely tied to the chosen parameter sets. The PSO controller uses a moderate swarm size and iteration budget (12 particles, 6 inner updates per control step with w = 0.6, c₁ = c₂ = 1.4), which biases the search toward cautious, consensus-driven exploration of the P–V surface. This configuration lifts the harvested energy from 0.143 Wh (P&O) to 0.231 Wh and improves the hit ratio from 37.9% to48.9%, but it also stretches the average convergence time to t₉₅ = 206.2 ms and introduces zero-power gaps at regime boundaries, because the swarm is periodically re-aligned when the GMPP moves. Within this budget, more aggressive gains or larger swarms tend to reduce local entrapment at the cost of longer computation per step and larger excursions in voltage, whereas more conservative settings would shorten t₉₅ but push PSO closer to the behaviour of a local hill-climber.

The FP controller is tuned in the opposite direction: a small population (five agents) with a high global-pollination probability and Lévy jumps ($\lambda \approx 1.5$, relaxation factor $0.3$) produces very fast initial moves toward bright regions of the P–V curve, which explains the short event-wise convergence time t₉₅ = 2.6 ms. However, the same aggressive global exploration prevents the agents from settling near the GMPP under mixed shading. This is visible in the low energy yield (0.094 Wh), the very poor hit ratio (10.8%), and the large steady-state ripple (92.35 W), as the operating point repeatedly leaves the optimal plateau to probe distant candidates. In this sense, the PSO and FP configurations in this benchmark instantiate complementary ends of a speed–stability trade-off: PSO sacrifices reacquisition speed for smoother trajectories and higher energy, while FP attains near-instantaneous response but loses robustness and power quality under strongly nonstationary, multi-peak conditions.

Table 2. Comprehensive performance metrics of MPPT algorithms under partial shading conditions

Table 2 presents quantitative performance indicators revealing distinct operational trade-offs among the four evaluated algorithms. INC achieves the highest energy harvest (0.281 Wh) with the second-fastest convergence ($t_{95}=3 \mathrm{~ms}$), optimal GMPP hit ratio (62.8%), and minimal power ripple (41.38 W), establishing clear multi-metric superiority. PSO delivers moderate energy yield (0.231 Wh) but suffers from excessive convergence delay (206.2 ms), approximately 69-fold slower than INC, limiting its practical deployment in rapidly changing conditions. FP exhibits the fastest response (2.6 ms) yet produces the lowest energy (0.094 Wh) and poorest tracking accuracy (10.8% hit ratio), demonstrating that raw speed without sustained GMPP residence yields inferior cumulative performance. P&O maintains baseline energy extraction (0.143 Wh) with intermediate convergence (46.5 ms) but generates the highest ripple (100.12 W), reflecting inherent oscillatory behavior around the operating point.

Figure 2 illustrates the multi-dimensional performance landscape across four critical metrics, revealing distinct operational trade-offs among the algorithms. Panels (a)–(d) report the aggregated performance of P&O, INC, PSO, and FP, respectively highlighting energy yield, dynamic response, tracking accuracy, and steady-state power variability for a consistent cross-algorithm comparison. Panel (a) demonstrates that INC achieves maximum energy harvest (0.281 Wh), exceeding PSO by 21.6% and baseline P&O by 96.5%, while FP yields only 0.094 Wh despite its superior speed characteristics. Panel (b) exposes a bimodal convergence distribution with fast responders (FP at 2.6 ms, INC at 3.0 ms) contrasting sharply against slower methods (P&O at 46.5ms, PSO at 206.2 ms), where PSO’s 79-fold penalty relative to FP reflects iterative swarm optimization overhead. Panel (c) quantifies tracking accuracy, with INC maintaining the highest GMPP residence (62.8%) followed by PSO (48.9%), while FP’s minimal hit ratio (10.8%) confirms that rapid response without stability yields poor steady-state performance. Panel (d) reveals power quality hierarchy where INC produces minimal ripple (41.4 W), PSO shows moderate fluctuation (51.7 W), and both P&O (100.1 W) and FP (92.3 W) exhibit substantial oscillations that compromise grid integration quality. The composite view establishes INC as the sole algorithm achieving concurrent superiority in energy yield, acceptable speed, maximum accuracy, and optimal power quality, while other methods sacrifice at least one critical dimension.

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Figure 2. Comparative analysis of four key MPPT performance metrics: (a) Total energy harvested, (b) Convergence time $t_{95}$, (c) GMPP hit ratio, and (d) Power ripple

Table 3 reveals critical power quality differentiators where INC achieves the lowest ripple (41.38 W) and mean tracking error (62.26 W) while maintaining the highest stability index (9.00), demonstrating superior steady-state regulation. FP exhibits the worst performance with maximum mean error (117.76 W) and poor stability (index 6.00), confirming volatile operation despite fast initial response. P&O generates the highest ripple (100.12 W) characteristic of fixed-step perturbation oscillations, while PSO shows intermediate values across all metrics. The maximum error remains uniformly high (220.54-223.24 W) for all algorithms during irradiance transitions, indicating universal vulnerability to abrupt environmental changes. The 2.4-fold ripple variation and 1.9-fold mean error spread between best and worst performers underscore the substantial impact of control strategy on power quality.

Table 3. Comprehensive performance metrics of MPPT algorithms under partial shading conditions

Figure 3(a) demonstrates that INC achieves the highest mean efficiency $\left( 44.3\pm 49.6 \right)\text{ }\!\!%\!\!\text{ }$ despite substantial variance, followed by P&O $\left( 41.5\pm 48.3 \right)\text{ }\!\!%\!\!\text{ }$, while FP exhibits the poorest performance $\left( 28.4\pm 39.7 \right)\text{ }\!\!%\!\!\text{ }$ with PSO intermediate at $\left( 38.0\pm 48.3 \right)\text{ }\!\!%\!\!\text{ }$. The large standard deviations across all algorithms indicate significant efficiency fluctuations during partial shading transitions, with the coefficient of variation exceeding unity for all methods. Figure 3(b) reveals temporal efficiency patterns where INC maintains near-100% tracking during the step-change period $\left( 2-4\,\text{s} \right)$, contrasting sharply with other algorithms that collapse to zero efficiency at irradiance boundaries. P&O and PSO show delayed recovery after transitions at $4\,\text{s}$ and $6\,\text{s}$, while FP demonstrates erratic behavior with multiple efficiency drops throughout the dynamic ramp phase $\left( 6-10\,\text{s} \right)$. The sustained high-efficiency plateau of INC during 2-4 s, where competing algorithms struggle, accounts for its superior cumulative energy harvest and validates gradient-based tracking superiority under multi-peak conditions.

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Figure 3. Tracking efficiency analysis: (a) Average efficiency with standard deviation error bars, (b) Real-time efficiency dynamics under varying irradiance conditions

Figure 4 captures algorithm behavior across evolving P–V landscapes, revealing distinct tracking strategies under multi-peak conditions. At t = 1.0 s (uniform irradiance), all algorithms converge near the single GMPP ($\approx 220\,\text{W}$ at $28\,\text{V}$), with P&O and FP positioned at near-zero power on the low-voltage shoulder, indicating delayed initialization. During step shading ($t=3.0\,\text{s}$), the P–V curve exhibits dual peaks at $18\,\text{V}$ ($\approx 120\,\text{W}$, global) and $32\,\text{V}$ ($\approx 125\,\text{W}$, local), where INC correctly identifies the higher-voltage peak while P&O remains trapped at the first local maximum, demonstrating hill-climbing limitations. Cloud burst recovery ($t=5.0\,\text{s}$) reinstates a dominant peak at $28\,\text{V}$ ($\approx 220\,\text{W}$), with INC and PSO successfully converging while P&O and FP collapse to zero power at $20\,\text{V}$, revealing poor transient recovery capability. The dynamic conditions snapshot ($t=8.0\,\text{s}$) presents a complex three-peak topology where P&O settles on a suboptimal local peak ($\approx 113\,\text{W}$ at $18\,\text{V}$), PSO occupies a valley near $30\,\text{V}$ with minimal power extraction, and FP approaches zero at $20\,\text{V}$, confirming that only gradient-based tracking (INC) maintains consistent global peak identification across rapidly changing multi-modal surfaces.

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Figure 4. P–V characteristic curves with MPPT operating points at four critical time instances: (a) Uniform irradiance at t = 1.0 s, (b) Step shading at t = 3.0 s, (c) Cloud burst recovery at t = 5.0 s, (d) Dynamic conditions at t = 8.0 s

Figure 5(a) establishes the performance hierarchy through weighted composite scoring (energy 30%, hit ratio 25%, speed 25%, ripple 20%), with INC achieving a dominant score of $90.7$, substantially exceeding PSO ($53.4$), P&O ($41.9$), and FP ($40.3$), demonstrating clear multi-metric superiority. Figure 5(b) reveals loss distribution characteristics where INC exhibits the narrowest interquartile range with median near zero and three high-value outliers (>150 W) during transitions, while FP shows the widest dispersion (IQR: $25-220\,\text{W}$) indicating unstable operation. Figure 5(c) traces temporal error evolution, highlighting INC’s distinctive pattern of near-zero error maintenance except during the $2-4\,\text{s}$ interval where moderate oscillations ($\approx 100\,\text{W}$) occur, contrasting with FP’s persistent high-error baseline ($>200\,\text{W}$) and P&O’s late-stage degradation after $8\,\text{s}$. Figure 5(d) quantifies relative improvements using FP as baseline (0%), showing INC at 200%, PSO at 147%, and P&O at 53%, where the 147 percentage-point gap between INC and P&O validates the substantial performance differential. The composite visualization confirms INC’s Pareto-optimal position, achieving simultaneous excellence in accuracy (minimal loss distribution), stability (lowest error variance), and efficiency (maximum energy gain) across all evaluation dimensions.

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Figure 5. Statistical performance analysis: (a) Overall algorithm ranking with composite scores, (b) Power loss distribution boxplots, (c) Real-time tracking error dynamics, (d) Energy improvement relative to baseline

Overall, Incremental conductance yields the best multi-metric balance, energy $0.281\,\text{Wh}$ (+21.6% vs. PSO; +95.8% vs. P&O), ${{t}_{95}}=3\,\text{ms}$, hit ratio $62.8\text{ }\!\!%\!\!\text{ }$, and the lowest ripple ($\approx 41.4\,\text{W}$), placing it near a Pareto-dominant envelope across speed, accuracy, and power quality. Particle Swarm Optimization attains intermediate energy ($0.231\,\text{Wh}$) but its convergence penalty (${{t}_{95}}=206.2\,\text{ms}$) and re-exploration gaps erode yield during nonstationary intervals despite only moderate ripple. Firefly control is the fastest (${{t}_{95}}=2.6\,\text{ms}$) yet unstable, energy $0.094\,\text{Wh}$, hit ratio $10.8 \%$, ripple $\approx 92\,\text{W}$, while P&O remains simple but trap-prone, with lower energy ($0.143\,\text{Wh}$) and the highest ripple ($\approx 100\,\text{W}$). Distributional evidence aligns with these patterns: INC shows the smallest median/IQR in loss, PSO exhibits heavy tails at regime boundaries, FP the broadest dispersion, and all methods share $\sim 220\,\text{W}$ transition spikes, confirming that speed alone is insufficient without sustained dwell at the global optimum.