Metaheuristic-Based Optimization of Nonlinear PI Controllers for Maximum Power Extraction in PV Systems

Photovoltaic energy has received particular attention in recent years due to its sustainability and wide range of applications, including stand-alone systems, grid-connected installations and small-scale uses such as street lighting [1, 2]. However, the efficiency of photovoltaic systems is very sensitive and highly dependent on environmental factors, including irradiation and temperature, which can significantly reduce energy production and lead to energy losses [3].

In such circumstances, traditional MPPT methods such as perturbation and observation (P&O) and incremental conductance (InC) often offer limited performance and numerous ripples [4]. Consequently, achieving reliable performance of photovoltaic systems under these specific conditions is still a critical challenge in ongoing research.

To overcome these limitations in photovoltaic systems, the integration of a robust control loop is essential for regulating output and enhancing overall system performance. Among the most commonly used techniques is the PID controller, widely recognized for its effectiveness, simplicity, and ease of implementation [5].

However, one of the main drawbacks of PID controllers is to properly choose a suitable and correct control gains to get the best possible performance [6], which is even trickier in non-linear systems such as photovoltaics. Various techniques are available for this purpose, such as manual tuning, Ziegler and Nichols, and more recent ones introducing artificial intelligent computing [7].

While recent strategies offer notable advantages, they are also associated with certain trade-offs. Consequently, there is a growing interest on artificial-intelligence-based methods and soft computing techniques along with optimization algorithms in enhancing solar PV panels and to cope with the system's inherent nonlinear behavior. However, despite the effectiveness of AI-based techniques such as the Fuzzy Logic Controller (FLC) and Artificial Neural Networks (ANN) in achieving high tracking performance, they often struggle under conditions of strong nonlinearities, particularly during partial shading. As a result, these methods are prone to stagnation at local power points, exhibit prolonged settling times, and suffer from reduced tracking accuracy.

Moreover, it is worth mentioning that the combination of these methods is also considered a go-to solution to mitigate the limitations of both strategies, offering very efficient tracking of MPP and GMPP in partial shading scenarios. Many maximum power point tracking algorithms (MPPT) techniques was discussed and summarized in the study [8], evaluated under both uniform and partial shading scenarios.

To meet this challenge, several techniques have been used, based on well-known classical techniques and on more recent intelligence-based techniques. An intelligent control technique combining an SPSOA MPPT algorithm with a non-linear NPID controller and compared to a conventional technique such as a conventional PID controller was proposed in the study [9], the results showed a reduction in power losses and an increase in PV system reliability as the proposed control outperformed other techniques. In the study [10], an intelligent self-adaptive MPPT strategy based on the integration of Adaptive Neuro-Fuzzy Inference System (ANFIS) with PSO optimization to enhance energy extraction efficiency. The proposed approach was evaluated under a variety environmental condition, demonstrating MPPT efficiencies of 99.2% under uniform irradiation and 98.7% under dynamically changing conditions, outperforming conventional MPPT algorithms Notably, the proposed ANFIS-PSO technique can avoid the local power maxima under partial shading scenarios by accurately identifying the point of maximum overall power. Overall, the approach significantly improved energy tacking efficiency with minimal grid disturbances, and enhanced the robustness of the PV system. In addition, for parameter optimization, a hybrid GA-PSO algorithm was used for both controllers, resulting in improved system stability. In the study conducted by Malarvili and Mageshwari [11] a modified nonlinear PID controller was formulated in compliance with the Popov criterion, ensuring system hyperstability for both MPPT control and the regulation of a single-stage single-phase (SSSP) grid-connected inverter in a photovoltaic system. The proposed approach demonstrated enhanced MPPT performance, characterized by rapid convergence and reduced oscillations around the maximum power point. Furthermore, the nonlinear PID controller exhibited improved effectiveness in inverter control, achieving a total harmonic distortion (THD) of less than 0.5%, minimizing energy losses through accurate MPP tracking, and maintaining a unity power factor on the grid side. A nonlinear control strategy was proposed in the study [12] for regulating current and voltage in a photovoltaic (PV) system integrated into a microgrid. The study presented a comparative analysis between the nonlinear PI (NPI) controller and a conventional PI controller. The proposed method employs a variable nonlinear gain function that dynamically adjusts according to the system error. when the error is large, the gain increases to accelerate the convergence toward the reference, while near the desired value, it decreases to ensure smooth tracking. This adaptive mechanism effectively suppresses high-frequency oscillations and reduces the settling time by stabilizing the system more rapidly. Furthermore, simulation results demonstrated a notable improvement in performance, with a consistent reduction in the Integral of Absolute Error (IAE) across all tested scenarios. In the study [13], the PID controller gains are optimized using the Particle Swarm Optimization (PSO) algorithm and compared to those obtained through the Cohen-Coon method. The results demonstrate that the PSO-optimized PID controller significantly enhances both transient and steady-state performance, achieving a settling time of 2.37 s and an overshoot of 0.78%, compared to 6.12 s and 20.14% with the Cohen-Coon method. These findings underscore the advantages of incorporating artificial intelligence techniques in control system design. Overall, PSO-tuned controllers were shown to outperform the Cohen-Coon method in all areas. In the study [6], the Gorilla Troops Optimizer (GTO) algorithm was used to fine-tune the PID controller gains. Four types of error evaluation indicators were used: integral absolute error (IAE), integral time absolute error (ITAE), integral square error (ISE) and integral time square error (ITSE). The results showed that soft computing tools provide better results when tuning the gain of PID controllers than more traditional methods such as Z-N tuning. In addition, GTO showed promising results in terms of rise time, stability and settling time, as well as minimum steady-state error. In the study [14], introduced a teaching-learning-based optimization TLBO to optimize the parameter of a fractional order PID (FOPID) controller in a grid connected PV system, simulation results showed the effectiveness of the proposed control strategy with better dynamic response and outperforming other techniques. Finally, in the study [15] GWO was used to adjust the converter duty cycle to maximize output power, resulting in greater stability and rapid convergence with less oscillation. A 43.6% improvement in output power was observed compared to the situation without MPPT.

Improving the performance of photovoltaic systems remains a persistent challenge despite academic research, and the various existing conventional strategies have known limitations. To fill this persistent gap, nonlinear PID controllers comes as a viable answer. As an extended version of conventional PID controllers, they are characterized by additional non-linear gain cascaded with the conventional structure, they offer more flexibility and better alignment with the inherently nonlinear nature of PV systems, thereby making them ideally suited for such applications. Furthermore, tuning the parameters of such controllers is essential to achieve maximum system efficiency. Among the many existing algorithms, Genetic Algorithm (GA), Particle Swarm Optimization (PSO) and Grey Wolf Optimizer (GWO) have enjoyed remarkable success in various applications.

This paper is organized as follows: Section 2 presents the description of the photovoltaic system and behavior analysis. Section 3 deals with the design of the NPID controller. Optimization algorithms and the fitness function used in this work are carried out in Sections 4 and 5 respectively. The results, along with performance evaluations, are discussed in Section 6. The concluding remarks are provided in Section 7.

Despite the widespread use and popularity of PID controllers for their robustness, they lack adaptability in non-linear systems such as photovoltaics due to abrupt changes in environmental operating conditions [16]. Non-linear PID controllers have been developed to overcome the shortcomings of conventional PID controllers. The typical PID controller is often expressed in its standard form, as employed in references [17, 18], acting on the error by means of three actions: proportional, integral and derivative, each weighted by a specific gain, K_p, K_i and K_d respectively, and is defined as follows:

$U_c(t)=\left(K_p E(t)+K_i \int E(t) d t+K_d \frac{d E(t)}{d t}\right)$                (1)

3.1 Design of non-linear gain – Popov’s hyper stability criterion

Non-linear PID controller is derived by introducing a non-linear hyperbolic cosine gain function in cascade with the conventional PID structure [11, 19]. Block diagrams are displayed in Figures 5 and 6.

The non-linear gain is designed based on the hyperstability criterion, which guarantees robust and global system stability [20]. Consequently, the output of the NPID controller is defined as follows:

$\begin{gathered}U_{-} c(t)=(K(e))\left(K_p E(t)+K_i \int E(t) d t\right.  \left.+K_d \frac{d E(t)}{d t}\right)\end{gathered}$              (2)

$U_c(t)=(\psi(e))\left(K_p+K_i \int d t+K_d \frac{d}{d t}\right)$               (3)

5.png

Figure 5. Linear PID control block

6.png

Figure 6. Non-linear PID block diagram

Here, $K(e)$ denotes the nonlinear gain, and $\Psi(e)$ is a nonlinear transformation of the error signal, defined as:

$\psi(e)=(K(e) \times E(t))$              (4)

7.png

Figure 7. Hyper stable nonlinear PID block diagram

Illustration of a modified hyper stable nonlinear PID controller is shown in Figure 7.

The non-linear gain is designed based on Popov’s criterion, which states global asymptotic stability if both of the following conditions are satisfied:

For the nonlinear part:

$\eta\left(t_0, t_1\right)=\int_{t_0}^{t_1} E(t) \psi(e) d t \geq-\gamma_0^2 \forall t_1 \geq t_0$                (5)

where, $\gamma_0^2$ is a positive finite constant, and for the linear part:

The transfer function $\left(K_p+K_i \int d t+K_d \frac{d}{d t}\right)\left(\frac{C(t)}{U_c(t)}\right)$ must be strictly positive real by Kalman-Yakubovich-Lemma.

By inserting (4) in (5), we derive:

$\begin{array}{r}\eta\left(t_0, t_1\right)=\int_{t_0}^{t_1} E(t) K(e) E(t) d t  \geq-\gamma_0^2 \quad \forall t_1 \geq t_0\end{array}$               (6)

To satisfy the inequality (6), $K(e)$ could be in any form of nonlinear function of error $E(t)$ that is bounded in the range $0 \leq K(e) \leq K_{\max }$.

To efficiently design the nonlinear gain function, it is essential to analyze the closed-loop control behavior. During the transient phase, the error $E(t)$ is relatively large; therefore, $K(e)$ should also be large to accelerate the system's response. Conversely, in the steady-state phase, $E(t)$ becomes small, and $K(e)$ should decrease accordingly to minimize steadystate oscillations while maintaining accuracy. This adaptive adjustment ensures rapid convergence when the system experiences disturbances while preserving stability once it reaches the desired operating point.

As a result, the nonlinear gain $K(e)$ is defined as in (7), exhibiting a nonlinear dependence on the error and being proportional to an exponential function with base $\beta$.

$K(e)=K_0(\beta)^e$ where: $K_0, \beta \geq 0$               (7)

For $0<\beta<1$: the error is defined as

$e= \begin{cases}e, & e \geq-e_{\max } \\ -e_{\max }, & e<-e_{\max }\end{cases}$

For $\beta>1$

$e= \begin{cases}e, & e \leq e_{\max } \\ +e_{\max }, & e>e_{\max }\end{cases}$

$e_{\max }$  can be determined according to the dynamics of the system. Thus, by substituting Eq. (6) in Eq. (7):

$\begin{aligned} & \eta\left(t_0, t_1\right)  & =\int_{t_0}^{t_1} E(t)\left[K_0(\beta)^e\right] E(t) d t  & =\int_{t_0}^{t_1} E(t) K_0 \exp (e \ln (\beta)) E(t) d t \geq 0 \geq-\gamma_0^2\end{aligned}$                (8)

$K(e)$ is defined in $[0, \infty]$. From Eq. (8), we get:

$K(e)=K_0 \exp (e \ln (\beta))$                (9)

Inserting $\alpha=\ln (\beta)$ in Eq. (9), we obtain: $K(e)= K_0 \exp (\alpha e)$, with $K_0, \beta$ and $\alpha \geq 0$.

Hence, the nonlinear gain is specified separately for the two regions $\beta<1$ and $\beta>1$. As follows:

For $\beta=\exp (-\alpha)<1 ; K(e)$ is defined as:

$\begin{aligned} & K(e)=K_0 \exp (-\alpha e) \\ & =\left\{\begin{array}{cl}K_0 & e=0 \\ 0 & e \rightarrow+\infty: \text { lower bound } \\ K_0 \exp \left(\alpha e_{\max }\right) & e<-e_{\max }: \text { upper bound }\end{array}\right.\end{aligned}$

For $\beta=\exp (\alpha)>1$:

$\begin{aligned} & K(e)=K_0 \exp (\alpha e) \\ & =\left\{\begin{array}{cl}K_0 & e=0 \\ 0 & e \rightarrow-\infty: \text { lower bound } \\ K_0 \exp \left(\alpha e_{\max }\right) & e>+e_{\max }: \text { upper bound }\end{array}\right.\end{aligned}$

A limitation common to both expressions of $K(e)$ for regions $\beta<1$ and $\beta>1$ arises at their lower boundary, where the gain becomes inefficient. Specifically, as the error approaches large magnitudes ($\mathrm{e} \rightarrow \pm \infty$) the minimum gain reaches $K_{\min }=0$, causing the control action to lose the desired responsiveness. Therefore, an enhanced nonlinear gain formulation can be introduced by combining both of the two previous cases giving in hyperbolic cosine function to enhance performance while preserving the stability properties of the closed-loop system.

$K(e)=K_0 \frac{\exp (\alpha e)+\exp (-\alpha e)}{2}=K_0 \cosh (\alpha e)$                  (10)

where,

$e=\left\{\begin{array}{cl}e, & |e| \leq e_{\max } \\ e_{\max } \operatorname{sign}(e), & |e|>e_{\max }\end{array}\right.$

$K(e)=\left\{\begin{array}{cl}K_0, & e=0 \\ K_0 \cosh \left(\alpha e_{\max }\right), & |e|>e_{\max }\end{array}\right.$

To evaluate system performance, there are various processing criteria, whose aim is to minimize the error between the desired setpoint and the system output. They play a crucial role in guiding the algorithm towards the best solution. In the context of this research, the mean square error (MSE) cost function was chosen, along with metrics like: Maximum Overshoot (M_p), Steady State Error (e_ss), and Settling Time (t_s).

The MSE calculates the mean square error and is defined as follows:

$M S E=\frac{1}{n} \sum_{i=1}^n(r(i)-y(i))^2$                (18)

$y(i)$ is the output voltage of PV array and $r(i)$ is the desired output voltage generated from MPPT algorithm.

Maximum Overshoot (M_p): maximum peak of the output $\left(x_{\text {peak}}\right)$ from the desired setpoint ($x_{\text {setpoint}}$) during a transient response. It is expressed in percentage (%) and calculated as follows:

$M_p=\frac{x_{\text {peak }}-x_{\text {setpoint }}}{x_{\text {setpoint }}} \times 100 \%$

Steady State Error (e_ss): difference between the output and the desired output (reference) during steady state.

Settling Time (t_s): Time taken for the system to stay within a certain band (±5% in this work) of the setpoint.

To capture the trade-offs among various dynamic performance metrics, a weighted sum objective function is employed. This allows simultaneous minimization of error, overshoot, steady state error, and settling time:

$J=\omega_1 \times M S E+\omega_2 \times M_p+\omega_3 \times e_{s s}+\omega_4 \times t_s$

where, $\omega_1, \omega_2, \omega_3, \omega_4$ are the weighting coefficients assigned to each metric. In this study, the weights are set as follows: $\omega_1=0.4, \omega_2=0.2, \omega_3=0.2$, and $\omega_4=0.2$. They were selected to emphasize tracking accuracy through MSE, and balancing out the other metrics.

In this study, the performance of advanced MPPT control strategies for a PV system was evaluated under both Standard Test Condition (STC) and dynamic operating conditions, in our case, varying irradiance levels. A nonlinear PI controller was implemented and optimized using three metaheuristic algorithms: PSO, GA, and GWO. Simulation results demonstrated that the use of nonlinear control significantly improved system performance compared to the conventional P&O method, particularly in terms of reduced overshoot, faster settling time, and minimized ripple, which is primarily due to the non-linear added gain. Among the tested methods, the NPI-GWO controller exhibited superior tracking accuracy, stability, and robustness.

These results highlight the need for a more robust control technique than conventional methods, as well as the use of more modern methods, such as artificial intelligence, to obtain more precise and calculated results, in this case optimal controller gains. In particular of non-linear PI controllers, which have more parameters than conventional PIs, making them more complicated to find. Future work could include experimental validation on real-time hardware, as well as extending the approach to grid-connected systems in partial shading.