Performance Analysis of Solar Tracking Systems in Nigeria for Efficient Power Supply: Modelling and Simulation with PID and Fuzzy Logic Controller

Performance Analysis of Solar Tracking Systems in Nigeria for Efficient Power Supply: Modelling and Simulation with PID and Fuzzy Logic Controller

Barisuka Uebari, D.O. Dike, L.O. Uzoechi, S. O. Okozi

Department of Electrical and Electronic Engineering School of Electrical Systems Engineering and Technology Federal University of Technology, Owerri

Imo State, Nigeria

Abstract: Nigeria's frequent power outages highlight the need to find efficient ways to use renewable energy. In order to evaluate the performance of solar tracking systems in comparison to fixed-axis photovoltaic (PV) arrays under Nigerian irradiance conditions, this study models and simulates them using ProportionalIntegralDerivative (PID) and Fuzzy Logic Controllers (FLC) in MATLAB/Simulink. The findings indicate that PID-based two-axis tracking could produce 15.05A and 73.40V, whereas fixed- axis PV systems could only produce 7.39A and 46.55V at most. The FLC-using tracker consistently outperformed the other two. It operated better in the morning and at night when the sun was not as bright, and it received 22.32A and 109.93V at noon. The findings imply that because fuzzy logic control can better adjust to variations in irradiance, it is the better option for increasing solar energy capture. Using two-axis trackers based on FLC is the simplest way to maximise the performance of solar panels and boost the amount of renewable energy in Nigeria.

Keywords: Solar tracking systems, ProportionalIntegralDerivative (PID) controllers, Fuzzy logic controllers, MATLAB/Simulink, Photovoltaic (PV) arrays, Irradiance variability, Nigeria

1. INTRODUCTION

   The necessity of a stable electricity supply in Nigeria is paramount, as it is essential for social and economic development. Nigeria's persistent electricity deficit, marked by recurrent outages, inadequate grid reliability, and significant reliance on fossil fuel generation, remains a barrier to economic growth and social development [1]. Nigeria is located in one of the worlds most solar-rich regions, receiving significant direct and global irradiance throughout the year.

   Conventional fixed photovoltaic (PV) arrays are simple and low-cost but lose substantial potential energy because panel orientation is optimal only for a small portion of the day. Solar tracking systems (single-axis and dual-axis) address this limitation by continuously reorienting PV modules toward the sun, often providing measurable energy gains over fixed systems; however, those gains are sensitive to tracking strategy, actuator energy consumption and local climatic variability [2], [3]. Simulations using MATLAB/Simulink, PVsyst and other platforms allow rigorous evaluation of these control strategies across a range of irradiance patterns and mechanical constraints without costly field prototypes [4].

   Despite the global literature demonstrating the promise of intelligent tracking controllers, regional studies specific to Nigeria are comparatively few and often do not simultaneously account for (a) the energy cost of actuation, (b) local irradiance variability, and (c) trade-offs between simple tuned PID controllers and rule-based fuzzy logic controllers. This study, therefore, develops detailed mathematical models and MATLAB/Simulink simulations of solar tracking systems employing PID and Fuzzy Logic controllers. It evaluates their performance under representative Nigerian irradiance profiles.

2. LITERATURE REVIEW

   [5] showed that fuzzy logic controllers are superior to traditional PID controllers in managing nonlinearities and uncertainties in sun trajectory prediction. [6] examined hybrid PIDfuzzy controllers and found that hybridization enhances transient responses and tracking efficiency relative to standalone PID controllers. The findings are consistent with [7] who emphasized that fuzzy systems, owing to their rule-based adaptability, reduce steady-state error more efficiently than linear controllers.
   [8] demonstrated that ANFIS controllers, when trained with irradiance data, adapt dynamically to environmental changes and consistently exceed the performance of classical fuzzy systems. [9] noted that computational complexity and hardware costs could impede the widespread adoption of ANFIS in resource-constrained areas. [10] created a dual-axis tracker utilizing Simulink, incorporating PID and fuzzy control. Their findings indicated that the fuzzy-controlled model achieved superior power output in conditions of variable irradiance. [11] conducted a simulation of a solar tracker in MATLAB utilizing actual meteorological datasets from sub-Saharan Africa, demonstrating that intelligent controllers enhance daily energy harvest by

   1520% relative to fixed systems. [12] utilized particle swarm optimization (PSO) for tuning PID gains in MATLAB, demonstrating improved convergence speed and decreased overshoot relative to ZieglerNichols tuning. [13] demonstrated that fuzzyGA hybrid tuning attained enhanced accuracy in the context of highly variable Nigerian irradiance profiles. [14] examined fault conditions, including actuator stalling and sensor drift, to evaluate the resilience of PID and fuzzy controllers.

   [15] developed a microcontroller-based fuzzy solar tracker in Pakistan, demonstrating a 28% increase in energy output relative to fixed photovoltaic panels. [16] conducted an evaluation of a PID-controlled dual-axis tracker in northern Nigeria, reporting a 22% increase in energy efficiency while also noting difficulties in calibration due to dust accumulation and elevated ambient temperatures. [11] highlighted the effective incorporation of low-cost sensors and actuators, illustrating that economical fuzzy-based trackers can function effectively in off-grid rural environments. [17] conducted a year-long field trial of hybrid PIDfuzzy trackers in India, demonstrating energy yield improvements of 2025% compared to fixed arrays, even in the face of seasonal variability. [16] present one of the limited experimental studies grounded in local context, as the majority of existing literature depends on simulations or field trials conducted in other developing countries. Moreover, [13 ] indicate that optimization strategies specifically designed for Nigeria's distinct irradiance variations and recurrent grid instability remain insufficiently investigated. [9] noted that hardware constraints and significant computational requirements could hinder rural electrification initiatives in Nigeria.

3. MATHEMATICAL MODEL

   1. Modelling of PV Array System and Solar Tracking Angles

      Rs

      I

      +

      Iph

      Io

      Rp

      V

      –

      Considering a PV solar cell shown in Figure 1, the equivalent circuit which mathematically describes the I-V characteristics of the PV circuit is represented in Equation 1.

      Figure 1: Single diode model circuit [18]

      =

      [ (+) 1] +

      (1)

      Where is the series resistance, is the resistance in parallel is the diode ideality constant, is the thermal voltage of the solar cell, is the diode saturation current, and is the photovoltaic (light- generated) current.

      Diode ideality factor is the measure of how much a practical diode deviates from ideal diode equation. The average value assumed during the determination of unknown parameters in the photovoltaic system is usually 1.3. Also, the thermal voltage is a characteristic voltage that relates current flow in he p-n junction to the electrostatic potential across it, and is given

      =

      (2)

      Where is Boltzmann constant (1.3806503 × 1023 J/K ), is electron charge (1.607 × 1019 ), and

      is nominal temperature (298.15K).

      The light-generated current of the photovoltaic cell depends linearly on the solar irradiation and is also influenced by the temperature, and their relationship is given in Equation

      = (

      (3)

      + ) ()

      , light generated current in solar cell circuit at nominal conditions (when temperature is 25oC and irradiance of 1000 W/m2), is current temperature coefficient, = , is actual temperature and is the nominal temperature, is the actual sun irradiation and is the nominal sun irradiation (usually 1000 W/m2). The diode saturation current and its dependence on the temperature may be expressed as

      = (

      3

      )

      [

      (1

      1

      )] (4)

      Where, is band gap energy of semiconductor = 1.12 eV for the polycrystalline Si at 25oC. Where is given by the relationship in Equation 4

      =

      ( )1

      (5)

      And is the short-circuit current, and open circuit voltage, respectively.

      The elevation angle is the angle between the sun rays and the horizontal surface (solar panel)

      sin = sin sin + cos cos cos (6)

      = 900 (7)

      where the latitude angle of the location is, is the declination angle, and is the hour angle.

      The accuracy of the declination angles is important in navigation and astronomy. However, an approximation accurate to within 1 degree is adequate in many solar purposes. One such approximation for the declination angle is = sin1{0.39795 cos[0.98563( 173)]} () (8) where is day number and calendar dates are expressed as the = 1 , starting with January

      1 = 15( 12) (9)

      where is the solar time in hours. A solar time is a 24-hour clock with 12:00 as the exact time when the sun is at the highest point in the sky. the azimuth angle can be expressed as in equation 10 for True Local Time >12.00hr (Roth, Georgiev and Boudinov, 2009).

      = 180 + cos1 (sin sin sin) (10)

      cos cos

      The azimuth angle for True Local Time <12.00hr is expressed as in Equation 11,

      = 180 cos1 (sin sin sin ) (11)

      cos cos

      Where is the latitude of the location, is the elevation angle and is the declination angle.

      = | 1

      15

      cos1

      [ tan()

      tan (23.44 sin (360

      (+284) 365

      ))]| (12)

      Total irradiance on any inclined surface is the sum of direct irradiance, isotropic sky diffuse irradiance, and ground reflection as given in Equation 13

      =

      cos +

      1+cos +

      1cos

      (13)

      cos

      2 2

      A DC motor was chosen as the actuator for the tracker. This motor moves the solar panel system along

      both the East/West and North/South axes. To create a model for the motor, its electrical and mechanical aspects are considered. Using Figure 2 and defining the angular twist of the shaft, the motor's differential equations can be found.

   2. Modelling of Solar Tracker Drive Mechanism

      Figure 2: DC motor representation

      Applying Kirchhoffs current and voltage laws, the electrical system equations are thus expressed as

      () = () (14)

      () = () + () + () (15)

      Then,

      =

      () =

      ()

      (16)

      In this context, let denote the amplifier voltage gain, the motor input voltage, the motor armature coil resistance,

      the motor armature coil inductance, the motor armature coil current, the motor back electromotive force (EMF),

      the motor back EMF constant, the motor shaft angular velocity, and the motor shaft position. The mechanical system includes the following differential equations

      () = () (17)

      () =

      () +

      () =

      2() +

      (18)

      2

      where, is motor torque constant, is the motor shaft angular acceleration, and is the total inertia acting on the shaft, which includes the motor armature inertia, gear train inertia, and the solar panels' inertia.

      Substituting Equations 14 and 15 in 16, we obtain

      () =

      ()

      +

      () +

      ()

      (19)

      Also, substituting Equation 17 into 18, we have

      () =

      2() +

      (20)

      2

      Applying Laplace transforms to Equations 19 and 20,

      () = () + () + () (21)

      () = 2() + () (22)

      Making subject of the formula in Equation 22 and substituting the same in Equation 22, the final transfer function for the solar tracker drive mechanism is expressed as

      ()

      ()

      =

      3+(+)2+(+)

      (23)

   3. Modelling of PID Controller

      The objective of a control system is to achieve a fast response to a step input command with minimal overshoot and error. Solar tracking system motion control in this work is set forth as an electric motor motion control in terms of output angular displacement. This motion control system is to be developed analytically with feedback using a photo sensor, such that it ensures that the desired output position is met.

      PID Controller

      Disturbance d(t)

      r(t)

      e(t)

      u(t)

      y(t)

      um

      measurement noise

      Plant model

Controller

A typical structure of a PID control system is shown in Figure 3, where it can be seen that in a PID controller, the error signal e(t) is used to generate the proportional, integral, and derivative actions, with the resulting signals weighted and summed to form the control signal u(t) applied to the plant model.

Figure 3: A typical PID control structure

A mathematical description of the PID controller is

() = [() + 1 () + ()] (24)

0

Where () is the input signal to the plant model, the error signal () is defined as () = () (),

and () is the reference input signal.

In practical applications, the pure derivative action is never used, due to the derivative kick produced in the control signal for a step input, and to the undesirable noise amplification. It is usually replaced by a first-order low pass filter. Pre-filter is defined as a transfer function that filters the input signal prior to calculating the error signal. Thus, the Laplace transformation representation of the approximate PID controller can be written as in Equation 25

() =

(1 + 1

+ 1+

) (25)

where is the order of the low-pass pre-filter.

Adding the PID controller system to plant, will result in the addition of poles and/or zeros that will affect the response, mainly the added zero, will significantly and inversely affect the response and should be cancelled by pre-filter.

1. Modelling of Fuzzy Logic Controller

   Fuzzy logic is an approach that closely mirrors human reasoning and decision-making processes. In contrast to conventional control strategies, which typically operate on precise point-to-point mappings, fuzzy logic controllers function on the principle of range-to-point or range-to-range mappings. The mechanism relies on the fuzzification of both inputs and outputs through associated membership functions. A crisp input is assigned varying degrees of membership across these funtions, and the final output is determined by the aggregation of these memberships rather than a single deterministic value. Consequently, the controllers decision is influenced by a continuum of possible input states. This characteristic of fuzzy logic reflects its applicability to real-world situations, where ambiguity and imprecision are inherent. For instance, in survey responses, individuals often provide answers such as Not Very Satisfied or Quite Satisfied, which cannot be quantitatively defined with absolute precision. Such linguistic variables represent vagueness that humans naturally employ in judgment, but which conventional machine-based control systems are unable to replicate effectively. The block diagram is shown in Figure 4.

   Figure 4: Fuzzy logic controller system

   To implement the fuzzy logic technique in a real application requires the following three steps:

   1. Fuzzification: convert classical data or crisp data into fuzzy data or Membership Functions (MFs).

   2. Fuzzy Inference Process: combine membership functions with the control rules to derive the fuzzy output.

   3. Defuzzification: Use different methods to calculate each associated output and put them into a table: the lookup table. Pick up the output from the lookup table based on the current input during an application.

      1. MODELS EXPERIMENTATION AND SIMULATION

         In order to experiment with and analyse the mathematical models created in the preceding sections, this article used a virtual modelling environment called MATLAB and Simulink. The framework of the entire system is made up of the solar panel, solar angles, solar tracker, PID controller, and FLC controller, it is important to note. Conducting a comparison analysis to ascertain the efficacy of the designed controllers is the main goal of the simulation.

         1. Solar Tracker System Simulation

            The continuous-time transfer function model of the solar tracking system, created in Equation 23, was ultimately represented in Simulink. For both horizontal and vertical panel direction control, the system consists of two DC permanent magnet motors.

            ()

            ()

            =

            3+(+)2+(+)

            In order to replicate the altitude and azimuth angles based on the calculated angles of the sun's incidence on the solar panel, a controller uses sets of two spur gears to lower the motor's speed.

            Equation 23 was used to implement this continuous-time transfer function model, and Equation 26 represents the outcome of inserting these parameters.

            KT = 0.1112; Torque constant (Nm/A)) KA = 1.38; Amplifier gain (V/m)

            JT = 87.6855; Total moment of inertia (kg*m^2) LA = 0.13; Armature inductance (H)

            RA = 0.5; Armature resistance (Ohms)

            b = 0.001; Viscous friction coefficient (Nm/KRPM) KB = 0.1098; Back EMF constant

            ()

            ()

            = 0.1535

            11.39 3+43.82+0.01234

            (26)

            The two PMDC motors that supply the necessary slew rate through the gear mechanism are powered by a 48V voltage, which is the tracker mechanism's input. The angular position is the output. To evaluate its response and ascertain how the PID controller tuning parameters and Fuzzy Logic controller universe of discourse worked, the tracker was simulated independently for a whole day.

         2. Solar System Simulation with FLC

This paper used MATLAB and Simulink semantics to design the fuzzy logic controller. The initial step in FLC modelling is always to identify the discourse universe for the suggested controller. The altitude angles FL controller has two inputs: the altitude angles error and the rate at which this error changes. The output is the desired angle of incidence needed to maximise the amount of radiation that reaches the panels. The azimuthal angular errors, the rate at which these errors changed, and the actual horizontal angular movement of the panels were the inputs and outputs of the azimuthal angles FLC, respectively. A fixed solar panel array simulation was used to calculate the inputs and outputs for the azimuthal angular (FLC2) and altitude or elevation angular (FLC1) controllers. The FLC1 error universe of discourse is between -1800 and 1800, while the error change rate is between -19o and 19o. Furthermore, FLC2 error is between -5500 and 5500, and error change rate is between

-360 and 360.

The rules-based table elevation FLC obtained is shown in Table 1. This is the same as the azimuthal FLC

Table 1: Fuzzy-based initial rules for elevation FLC

	Elevation Angular Error (E)
Elevation Error Change (E)		NL	NM	NS	Z	PS	PM	PL
	NL	NL	NL	NL	NL	NM	NS	Z
	NM	NL	NL	NL	NM	NS	Z	PS
	NS	NL	NL	NM	NS	Z	PS	PM
	Z	NL	NM	NS	Z	PS	PM	PL
	PS	NM	NS	Z	PS	PM	PL	PL
	PM	NS	Z	PS	PM	PL	PL	PL
	PL	Z	PS	PM	PL	PL	PL	PL

The rules were developed using a fuzzy logic toolbox and subsequently translated into the table presented above. In Table 2, the grey-colored row and column denote the linguistic terms that form the antecedents of fuzzy rules concerning the heuristic variables: elevation error, azimuthal error, change in elevation error, and azimuthal error. The central portion of the table, featuring a transparent background, represents the consequences of the rules. The 49 established rules are implemented in MATLAB. However, the numerous rule bases have led to increased computational time, slower implementation of the controller, and reduced efficiency. To enhance the execution speed of the controller and decrease its response time, the 49 rules were condensed to 16 rule bases for the two controllers.

1. RESULT AND DISCUSSION

   The final analogy in this paper was to briefly compare the results of these different tracking systems with the sole aim of finding which of them has a better performance, given all the developed models. It is important to display the values of the solar panel generated current and voltages as illustrated in Tables 2 and 3.

   Table 2: Currents generated for different tracking systems

   Currents (A)
   Time (Hours)	Fixed-Axis Tracking	Two-Axis with PID	Two-Axis with FLC
   6	1.0000	8.6663	12.9995
   7	2.4529	8.7755	13.1628
   8	5.7756	9.2200	13.879
   9	6.7542	10.1129	15.1635
   10	7.1748	11.6408	17.4154
   11	7.3664	13.7639	20.5376
   12	7.3911	15.0516	22.3233
   13	7.3411	13.7639	20.4670
   14	7.1257	11.6408	17.2958
   15	6.6849	10.1129	15.0075
   16	5.7066	9.2200	13.6620
   17	2.4319	8.7604	13.0500
   18	1.5000	8.6364	12.8207

   Table 3: Voltages generated for different tracking systems

   Voltages (V)
   Time (Hours)	Fixed-Axis Tracking	Two-Axis with PID	Two-Axis with FLC
   6	0	0	0
   7	1.9043	6.8129	10.2190
   8	8.9525	14.2914	21.4339
   9	15.6769	23.4728	35.1955
   10	22.2045	35.9636	53.8968
   11	28.4971	53.0579	79.4432
   12	34.4296	69.2862	103.6852
   13	39.7595	73.4036	109.9300
   14	44.1056	71.0470	106.4819
   15	46.5493	69.4137	104.0824
   16	44.1534	45.3215	83.7931
   17	20.5922	20.4562	57.1917
   18	4.5000	7.0123	15.4821

   All the performance curves for these systems are shown in Figures 5 and 6, respectively.

   COMPARATIVE ANALYSIS

   25

   20

   Fixed Axis

   Two Axis with PID Two Axis with FLC

   Current(A)

   15

   10

   5

   0

   6 8 10 12 14 16 18

   Time(Hrs)

   Figure 5: Comparison of solar panel-generated current.

   Figure 5 shows the current (A) that the photovoltaic system makes during the day (6:00 to 18:00) in three distinct ways: Fixed-Axis Tracking, Two-Axis with PID controller, and Two-Axis with Fuzzy Logic Controller (FLC). The fixed-axis system made the least amount of current during the day, from 1.00 A in the early morning to a high of about 7.39 A at midday (12:00). The output went up steadily from 6:00 AM to noon. Then, it went down a little bit till evening, when it dropped back to 1.50 A around 18:00. This pattern shows how the fixed-axis panel's poor alignment with the sun's position makes it less able to capture energy.

   The PID-based system made a lot more current than the fixed-axis system. The current went up from 8.67 A at 6:00 to a peak of 15.05 A at noon (12:00) before slowly falling to 8.64 A at 18:00. The curve shows how well PID control works to make it easier to collect energy and follow the sun more closely. When the system was at its busiest, the PID system usually gave out almost twice as much current as the fixed-axis system.

   The FLC-based tracker always did better than both the PID and fixed-axis systems, no matter what time of day it was. At 06:00, the current production started at 13.00 A. It peaked at 22.32 A at 12:00 and then dropped to 12.82 A at 18:00. The FLC tracker reached the maximum peak current and kept up its great performance throughout the morning and evening. This shows how strong and adaptable it is to changing irradiance conditions. The average current over the course of the day was far higher than that of either PID or fixed-axis, showing that it was better.

   COMPARATIVE ANALYSIS

   Fixed Axis

   Two Axis with PID Two Axis with FLC

   120

   100

   Voltage(V)

   80

   60

   40

   20

   0

   6 8 10 12 14 16 18

   Time(Hrs)

   Figure 6: Comparison of solar panel-generated voltages

   Figure 6 shows the voltage outputs of three solar tracking setups: a fixed-axis tracker, a two-axis tracker with Proportional IntegralDerivative (PID) control, and a two-axis tracker with Fuzzy Logic Control (FLC). From 6:00 to 18:00, the data was collected. The results show that the systems work in distinct ways. The tracker that uses FLC works best all day.

   All systems showed zero volts at 6:00 h because there wasn't much sunshine at dawn. The systems started generating voltages that could be measured at 7:00 h. The fixed-axis system only made 1.90 V, but the PID and FLC systems made more: 6.81 V and 10.22 V, respectively. This early difference in performance shows that active tracking systems are better at picking up low-angle morning radiation than fixed systems.

   From 8:00 to 12:00, all systems showed a quick spike in voltage output. This was in keeping with the steady rise in solar radiation. The fixed-axis tracker reported 34.43 V at 12:00 h, and the PID-controlled two-axis tracker read 69.29 V. The FLC-based tracker had the greatest output of 103.69 V, which is more than three times that of the fixed-axis system and nearly 1.5 times that of the PID system. This result shows that the FLC controller does a better job of keeping the panels in the right spot when the sun is at its highest point.

   Between 12:00 and 14:00, the voltages from all three systems were at their maximum. The fixed-axis tracker hit its highest point around 15:00 h, when it was at 46.55 V. The PID system achieved its greatest point around 13:00 h, when it was 73.40

   V. At 1:00 PM, the voltage in the FLC system reached its highest point, 109.93 V. It stayed more stable than the PID system, which started to drop off faster at 14:00. The FLC's performance during peak hours shows that it can handle changes in irradiance well, which is important for maintaining power generation steady.

   The voltage outputs of all systems were lower after 15:00 h since the sun was getting less brilliant. The FLC system, on the other hand, had voltages that were higher than those of the PID and fixed-axis systems. For example, at 17:00 h, the fixed- axis and PID trackers put out 20.59 V and 20.46 V, while the FLC system kept functioning and put out 57.19 V, which was a lot more. The FLC tracker made 15.48 V around 18:00, which was near to sunset. The PID system only made 7.01 V, and the fixed-axis tracker only made 4.50 V.

   The results show that the FLC-based two-axis tracker always works better than the PID-based tracker and the fixed-axis system, no matter what time of day it is. The FLC tracker not only had the highest maximum voltage output (109.93 V), but it also worked better throughout both low-irradiance hours (morning and evening) and peak-irradiance hours. These results show that fuzzy logic control is a better way to keep track of solar energy since it is more adaptable, more precise when making changes to angles, and less impacted by changes in solar circumstances.

2. CONCLUSION

   This study employs MATLAB/Simulink simulations to evaluate the efficacy of fixed-axis, two-axis PID-controlled, and two- axis fuzzy logic-controlled (FLC) solar tracking systems in Nigeria. We have constructed mathematical models for the solar array, the angles at which the sun tracks, the tracker drive mechanism, thePID controller, and the fuzzy logic controller (FLC). The simulations go for an entire day, and the voltages and currents they make are looked at. The findings show that the FLC-based tracker always works better than the PID-controlled and fixed-axis systems during the day. The FLC tracker has the highest peak current (22.32 A) and voltage (109.93 V) and performs better than any other tracker when the sun is shining brightly or not at all. The PID-controlled system also indicates that it can get more energy than the system with a fixed axis. The research indicates that fuzzy logic control is a superior method for measuring solar energy in Nigeria, where sunlight levels fluctuate continuously.

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