Optimizing Energy Management of Hybrid Battery-Supercapacitor Energy Storage System by Using PSO-Based Fractional Order Controller for Photovoltaic Off-Grid Installation

The off-grid photovoltaic system under investigation is depicted in Figure 1. It comprises a solar PV system connected to the DC bus through a DC-DC boost converter. The hybrid energy storage system (HESS) consists of a combination of batteries and supercapacitors. Each ESS is linked to the DC bus through a DC-DC buck-boost converter.

A DC load is able to be supplied by any of the three sources: the photovoltaic generator is employed as the primary source of power, the batteries are utilized in situations where there is an excess in PV production or a shortage to supply and the SCs are used to minimize variations in either the PV production or the load. A PI controller is utilized to regulate each DC/DC buck-boost converter, the MPPT is used through the DC/DC in order to extract the maximum power from the PV source, and an efficient energy management strategy (EMS) is employed to control the entire system.

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Figure 1. Schematic diagram of off-grid PV systems with HESS

2.1 Modeling of the photovoltaic system

The conventional solar PV model is shown in Figure 2 and consists of a photocurrent, a diode, a parallel resistance (Rp) that represents leakage current, and a series resistance (Rs) that represents internal resistance to current flow. The voltage-current characteristic of a solar cell is given by Eq. (1).

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Figure 2. Photovoltaic equivalent circuit model

Eq. (1) gives the voltage-current characteristic of a solar cell [20].

$I=I p h-I s\left(\exp \left(\frac{q V+I R s}{k T c A}\right) 1\right)-\left(\frac{V+I R s}{R p}\right)$               (1)

• I: Current through the solar cell.
• Iph: Photocurrent, representing the light-generated current.
• Is: Reverse saturation current, a characteristic of the solar cell materials.
• q: Charge of an electron.
• V: Voltage across the solar cell.
• Rs: Series resistance, accounting for internal resistance.
• k: Boltzmann constant.
• Tc: Temperature in Kelvin.
• A: Diode ideality factor.
• Rp: Parallel resistance

2.2 Modeling of DC-DC converter

Figure 3 describes the power stage of a buck-boost converter. It consists of the actual load R, the smoothing inductance L, the output smoothing capacitance C, and the two switching transistors Q1 and Q2.

When employing the buck converter mode, Q1 is always OFF and current flows from the DC bus to the EES source (batteries or SCs). Reducing Vdc voltage to charge the EES is possible for the converter by controlling Q2.

The switch Q2 is turned off and the diode in Q2 permits current to flow only in one direction, from the storage source to the DC bus, when the converter is operating in the boost mode. Power for the DC bus can be supplied by the converter by raising the voltage VEES of the ESS by regulating the duty cycle of Q1 [21, 22].

The EES voltage when operating in buck converter mode is:

$V_{E E S}=\frac{V d c}{D}$                   (2)

$V_{E E S}$  Can be adjusted by controlling the duty cycle D of the converter.

DC voltage when operating in a boost converter is:

$V d c=\frac{V_{E E S}}{1-D}$                (3)

By adjusting the duty cycle D, the gain of the boost converter can be changed.

• V_ESS: Voltage of the energy storage system
• V_dc: DC output voltage of the DC-DC converter.
• D: Duty cycle of the DC-DC converter, representing the fraction of time the switch is turned on.

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Figure 3. Buck-BOOST DC/DC converter circuit model

This section shows the structure of the suggested control strategy, which aims to minimize the battery's peak current demand and dynamic stress. The LPF and FOPID controller are the two components that make up the control strategy. The Fractional Order Proportional Integral Controller (FOPID) parameters are optimized through the implementation of the PSO algorithm to attain optimal performance. The following sections provide an explanation of the proposed control strategy's structure.

4.1 Low pass filter (LPF)

Photovoltaic power generation and load demand actually fluctuate a lot during operation [28]. The batteries are stressed in the conventional system in order to meet the highly fluctuating. A large amount of heat would be produced inside the battery by the highly fluctuating battery current, which would increase internal resistance and reduce efficiency. To avoid this, we could split the power drift between the generation and consumption sides into two parts: the transient high-frequency component (HFC) and the steady low-frequency component (LFC). The battery is going to manage the LFC, and the SC will deal with the HFC [29].

By doing this, the battery's dynamic stress is decreased, and the high-frequency components are prevented from being supplied by the battery. Figure 4 depicts the diagram of the low-pass filter.

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Figure 4. Low pass filter diagram

4.2 Fractional order PID controller (FOPID)

Researchers have been curious about a new PID controller design in recent years because of its advantages over classical PID controllers, which include improving the performance of dynamic, non-linear systems and being less sensitive to changes in system parameters. This new design can be applied in many domains, particularly control theories.

Podlunby has been proposing this control device, known as fractional order PID (or FOPID) [30]. A classical PID controller additionally includes the fractional components of the integral and derivative parts, indicated by λ and μ, in addition to the control parameters Kp, Ki, and Kd.

The equation below gives the control law of the fractional order PID controller:

$u(t)=\left(K p+K i D^{-\lambda}+K d D^\mu\right) e(t)$              (4)

• $u(t)$ is the controller input
• e(t) is the error

By using the Laplace transform on Eq. (4), Eq. (5) defines the transfer function of this controller under zero initial conditions.

$G(s)=K_P+K_i s^{-\lambda}+K_d s^\mu$                (5)

The fractional orders of the integral and derivative terms are denoted by λ and μ, respectively. The proportional, integral, and derivative gain constants are represented by Kp, Ki, and Kd. The values (λ and μ) for a classical PID controller are equal to 1. Figure 5 illustrates the concept of the FOPID (Fractional Order Proportional Integral Derivative) controller and Figure 6 displays the control domains for FOPID and classical PID.

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Figure 5. FOPID controller concept

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Figure 6. PID controller versus FOPID controller

While the classical PID controller only needs to optimize three parameters, the FOPID controller design requires five parameters. The control system's dynamics can be realized with greater flexibility thanks to this extension [31].

In the context of our work, opting for a FOPI structure with Kd set to zero in the controller is justified by the need to simplify the design, prevent unstable responses, reduce implementation costs, and enhance robustness against variations in system parameters. This choice aligns with our specific objectives.

4.3 Particle swarm optimization algorithm (PSO)

The social behavior of fish and bird flocks searching for food serves as an inspiration for the PSO algorithm, a population-based optimization technique [29, 32, 33], PSO is introduced and discussed. It demonstrates how birds could determine their directions by using the collective knowledge of their group. Therefore, during each flight, birds work as particles that update their positions and velocities based on their own and the group's best experiences.

One of the most popular techniques for resolving optimization issues nowadays is the Particle Swarm Optimization algorithm. Its simplicity, high performance, and cheap computational cost have made it heralded as a promising and effective optimization method. PSO algorithms are widely applied in science and are effective at solving the majority of optimization issues [34, 35].

In order to attain the ideal controller parameters and, consequently, the best system output, the PSO technique is advised in this study to tune the controller gains.

The position and velocity updates for each particle within the population are computed using the following mathematical expressions at each cycle:

$\begin{gathered}V i(k+1)=w \cdot V i(k)+C 1 \cdot r 1[P i(k)-x i(k)] \\ +C 2 \cdot r 2[G i(k)-x i(k)]\end{gathered}$                 (6)

$x i(k+1)=x i(k)+V i$              (7)

• $w$: Adaptive inertia factor
• $V i(k)$: Velocity of particles at iteration k
• $x i(k)$: Position of particles in the search space at iteration k
• $C 1$: Cognitive factor controlling the individual behavior of each particle
• $C 2$: Social factor controlling the collective behavior of each particle
• $r 1$: Random number uniformly distributed in the interval [0, 1]
• $r 2$: Random number uniformly distributed in the interval [0, 1]

Figure 7 displays the flowchart outlining the step-by-step process of Particle Swarm Optimization (PSO). This visual guide illustrates how particles update their positions and velocities iteratively, providing a straightforward representation of the optimization algorithm's dynamics.

The control parameters for the PSO technique used to optimize the FOPI controller are listed in Table 1.

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Figure 7. Sequential steps of the Particle Swarm Optimization (PSO) process

Table 1. Parameters of the PSO algorithm

The values of the PSO parameters were selected after a thorough analysis of their effects on algorithm performance, combined with empirical adjustments to optimize results within the specific framework of our optimization problem.

The optimization process employs various suitability criteria, such as Integral Absolute Error (IAE), Integral Square Error (ISE), and Integral Time-Weighted Absolute Error (ITAE), to assess system performance. In the current context, the fitness function utilized to evaluate the performance of the system's output response is the Integral Absolute Error (IAE). The IAE is defined as follows:

$\mathrm{IAE}=\int_0^{\mathrm{t}}(\mid$ Isc_err $\mid) \mathrm{dt}$            (8)

$I s c_{e r r}=I s c_{r e f}-I s c$              (9)

The use of IAE as the fitness function in the optimization process is justified because it provides an intuitive, robust, and easily interpretable measure of system performance, making it suitable for evaluating and enhancing the efficiency of the proposed control algorithm.

Figure 8 shows the block diagram illustrating the optimization process for the FOPI-PSO controller.

Figure 9 shows the proposed FOPI-PSO HESS control scheme. As was previously mentioned, the main purpose of the SC is to reduce the stress associated with battery charging by absorbing the transient peak energy that arises from sudden changes in load or weather. This will be accomplished by splitting the power drift between the generation and consumption sides into two parts: the transient high-frequency component (HFC) and the steady low-frequency component (LFC).

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Figure 8. Proposed FOPI-PSO controller bloc diagram

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Figure 9. Proposed FOPI-PSO hybrid energy storage system control scheme

The reference voltage and the DC output voltage are compared in this algorithm, and the error is fed to the proportional-integral controller. The hybrid energy storage system's total current required, or I_ref is generated by the PI controller.

$I_{\text {bat_ref }}=$ lowpassfilter $\left(I_{\text {ref }}\right)$              (10)

The reference current to the battery represents the low-frequency component of the current. The error (I_{bat_err}) is sent to the PI controller after I_{bat_ref} and I_bat, the actual battery current, are compared. The duty ratios are produced by the PI controller. The PWM generator receives these duty ratios and outputs switching pulses that correspond to battery switches.

The following formula gives the power (P_{bat_uncomp}) that must be supplied by the SC and is not compensated by the battery.

$P_{\text {Bat }_{\text {uncomp }}}=V_{\text {Bat }}\left(I_{\text {Bat }_{\text {err }}}+I_{\text {ref-HF }}\right)$        (11)

$I_{\text {ref-HF }}=I_{\text {ref }}-I_{\text {ref-LF }}$                (12)

SC is responsible for making up for this uncompensated battery power. As a result, the SC reference current is taken as:

$I_{S C_{-} \text {ref }}=\frac{P_{\text {Bat }}{ }_{\text {uncomp }}}{V_{S C}}$               （13）

The battery and SC voltages are denoted by V_bat and V_SC respectively.

After comparing I_{sc_ref} with the actual SC current (I_sc), the FOPI controller receives the error. The necessary duty ratios are produced by the FOPI controller. The PWM generator receives these duty ratios and uses them to produce switching pulses that correspond to SC witches [36].

The synergy between FOPI regulator and PSO optimization lies in their complementary strengths. While FOPI provides a robust and adaptive control framework capable of handling system uncertainties and nonlinearities, PSO facilitates automatic tuning of regulator parameters to achieve optimal performance. This combination leverages the benefits of both methodologies, resulting in a powerful and versatile control strategy.

In this study, the exclusive use of FOPI-PSO on the supercapacitor control loop (Figure 8) was motivated by the necessity for a specialized optimization approach customized to the complex dynamics and diverse parameters associated with supercapacitor control. This targeted application enables a more precise adjustment and adaptation of FOPI controller parameters to enhance the overall performance of the supercapacitor within the system.

By introducing the derivative term, the controller may become more sensitive to noise and disturbances in the system, requiring more precise parameter tuning. Additionally, adjusting five variables simultaneously for a FOPID controller, instead of three variables for a FOPI controller, will be necessary to avoid undesirable behaviors such as overshoot. Therefore, in certain situations, such as with the FOPI controller mentioned earlier, it may be preferable to opt for controllers without a derivative term [17, 19, 30] when implementation complexity needs to be reduced without impacting the overall system performance.

In conclusion, this paper proposes and simulates a standalone PV system incorporating a hybrid battery/supercapacitor HESS controlled by an EMS that uses a Fractional Order Proportional Integral (FOPI) controller. The introduced FOPI-PSO controller, distinguished by its increased parameter flexibility, has demonstrated superior performance compared to the conventional PI controller. Exploiting the PSO optimization technique for simultaneous adjustment of multiple parameters, the proposed FOPI-PSO controller was fine-tuned to achieve optimal settings.

Through various simulation scenarios with load fluctuations and irradiance conditions, the efficacy and robustness of both the PI controller and the novel FOPI-PSO controller for synchronizing batteries and supercapacitors were thoroughly evaluated. Results indicate that the use of PSO-based FOPI in the control strategy consistently outperformed the PI controller, demonstrating superior robustness and overall system performance.

This research contributes to the enhanced adaptability and effectiveness of the FOPI-PSO control approach, showcasing its potential for optimizing standalone PV systems with hybrid energy storage.

The study and proposed approach are subject to some limitations. These include:

• Initial conditions dependency: The PSO algorithm used for controller parameter optimization may be sensitive to initial conditions and configuration parameters. Additional trials with different initial setups may be necessary to ensure result reliability.

• Impact of real environmental variations: Our study may not comprehensively account for real-time environmental variations such as sudden climate changes or shading, which can affect the system performance under real conditions.

We recommend exploring the following ideas in future research:

• Experimental validation: Conducting experimental tests on a real system to validate the effectiveness of the approach under real-world conditions.
• Incorporation of machine learning: Exploring the use of machine learning techniques to enhance controller performance by dynamically adapting to system and environmental variations.
• Multi-objective optimization: Extending the strategy to take into account several goals at once, including reducing operating costs.