Automatic Generation Control in Renewables-Integrated Multi-Area Power Systems: A Comparative Control Analysis

Abstract

Electrical load dynamics result in system instability if not met with adequate power generation. Therefore, monitoring and control plans are necessary to avoid potential consequences. Tie-line-bias control has facilitated power exchange between interconnected areas to cope with load dynamics. However, this approach presents a challenge, as load variation in either area leads to frequency deviations and power irregularities in each of the interconnected areas, which is undesirable. The load frequency control loop method is used to address this issue, which utilizes area control errors. This study focuses on the control of inter-area oscillations in a six-area power system under the effect of renewable energy sources. It evaluates the area control errors in response to changes in load and the penetration of renewable energy into the system. To mitigate these errors efficiently, an adaptive-PID controller is proposed, and its results are compared with PI and PID controllers optimized with heuristic and meta-heuristic algorithms. The findings demonstrate the superiority of the proposed controller over traditional controllers in mitigating tie-line power errors and frequency deviations in each area of the interconnected power system, thus helping to mitigate inter-area oscillations and restore system stability.

1. Introduction

The interconnection of different areas within electrical power systems has become a necessity due to the increasing demand for reliable power and its economic viability. This interconnection is expanding in both reach and scope, encompassing a range of connections from the integration of multiple grids to the establishment of regional, provincial, and even global energy connections [1,2]. The drawback of these vast interconnected areas is the increased stress on the power system, which can lead to a mismatch in the power balance equation if not managed diligently [3]. The integration of renewable energy sources (RESs) with the power grid is also viewed as a promising solution to meet the rising demand for clean and sustainable energy [4]. However, due to their intermittent nature and lack of redundancy, RESs can introduce instability to the power system, making it challenging to control and stabilize [5,6]. Therefore, several challenges need to be addressed to ensure the stability and reliability of the power system. Both of these challenges can result in an imbalance between power generation and load, leading to stress on the power system. This, in turn, can lead to deviations in frequency and voltage, ultimately leading to a reduction in the quality of the power supply.

To address the challenge of power system stresses, the deregulated environment of the power system has been widely adopted [6,7,8]. This involves the use of the tie-line power flow method, which facilitates contractual power flows in and out of a specific area of the power system. Nonetheless, areas that are interconnected via tie-lines can prove problematic during instances of faults, such as power outages or sudden load changes, wherein the resultant load deviation in one area can lead to frequency deviations and power irregularities in every other interconnected area, which is far from ideal [3,6]. To resolve this issue, a secondary control loop modification, commonly known as Load Frequency Control (LFC), is used in power systems. The LFC employs area control errors (ACEs) to restore the electrical power flow on the tie-line and maintain the frequency of an area at nominal values [9].

Several studies have been conducted on how to improve the stability of power systems by swiftly mitigating ACE from system dynamics. These investigations have utilized various techniques, including controller organizations such as proportional–integral (PI), proportional–integral–derivative (PID), proportional–integral–derivative with notch filter (PID-N), fuzzy logic control (FLC), and fractional-order-PID (FOPID), as well as refining techniques for classical controllers such as particle swarm optimization (PSO), Ziegler–Nichols (Z-N), genetic algorithms (GAs), firefly algorithms (FAs), artificial bee colony optimization (ABCO), ant colony optimization (ACO), grasshopper optimization (GOA), grey wolf optimization (GWO), differential evolution (DE), acteria forging (BF), teacher learning-based optimization (TLBO), and Levy flight, and fitness distance balance (FDB)-based coyote optimization algorithm (LRFDBCOA) [10,11,12,13,14,15,16,17,18,19,20]. Some of these works have considered wind and photovoltaic (PV) solar power, along with the stochastic load demand, in order to imitate a realistic power system.

Although these techniques are beneficial in controlling dynamics in a power system, there are a few shortcomings when it comes to effectively controlling complex and dynamic systems, such as the increasing penetration of renewable energy into the systems. Moreover, these methods also require tuning efforts as these techniques require a priori knowledge of system dynamics and a well-tuned set of parameters to ensure optimal performance and system stability. If these controllers are not finely tuned, their performance may deteriorate or even destabilize the system; therefore, the system may not perform efficiently.

Adaptive PID (APID) control has numerous benefits for AGC because of its inherent nature of adaptability to changing system dynamics, its better performance in non-linear systems, its ability to handle uncertainties and model variations, its improved stability and reliability, and the reduction in manual tuning efforts. With online tuning of the control parameters, it offers a more robust and effective approach to control as it swiftly adapts to the dynamics of the power system and updates its parameters for optimal control. This makes it more suited for load balancing, and renewable-energy-integrated power systems, which are variable and unpredictable.

Adaptive controllers can adapt to system changes, yet they cannot ensure optimal performance. Therefore, the intelligent data-driven approach has led to widespread interest in developing controllers for unknown and complex systems. It is an effective approach that makes use of data to optimize control techniques to enhance the system performance [21,22]. Adaptive dynamic programming (ADP) is a data-driven control technique that works efficiently to achieve the optimal controller performance by keeping the feedback-controlled system stable. It is an effective method that makes use of data to identify optimal control rules, enhancing system stability and performance [23]. ADP and reinforcement learning (RL) are useful strategies for resolving the aforementioned issues with optimal control and adaptive control [24]. In the absence of accurate system dynamics knowledge, online measured data can be used by ADP and RL algorithms to solve the optimal control problem [25].

An RL-based adaptive PID controller for non-linear systems is suggested and presented in [26]. The initial settings of PID are set to zero, meaning that knowledge of the system is not required. Furthermore, it can tune the controller parameters without the use of heuristic methods like PSO, genetic algorithms, etc. As such, the suggested control scheme reduces the amount of calculation, the time, and the complexity. The parameters of this intelligent control can be changed online based on the needs of the system. The authors in [27] use the adaptive PID controller to adjust the load frequency of a two-area power system. However, there is a noticeable lack of typical AGC practices in this investigation, such as an evaluation of the controller’s ability to handle uncertainties resulting from variable parameters or renewable energy sources. Moreover, the evaluation solely focuses on the controller’s performance in disrupting one area while the other remains unaltered. The contributions of this article are as follows:

• A gradient descent algorithm-based adaptive-PID (APID) control is implemented for the automatic generation control of an interconnected multi-area power system.
• The robustness of the proposed controller against an intermittent supply of PV and wind energy sources is demonstrated by comparing the results with the PSO-based as well as ZN-based PI control and PID control.
• The number of interconnected areas is increased from two to six. Since a fault in one area can propagate to other areas, this is useful in assessing the performance of proposed controller regarding its overall stability under large-scale power system dynamics.
• This paper illustrates simultaneous load deviations in more than one area. This demonstration was missing in [27].

2. Modelling of Power System Dynamics

In a power system, a load reference set point (LRSP) is a pre-set target value for maintaining the desired level of electrical load. It is often determined by controllers or system operators based on system restrictions, resource availability, and demand estimates. The governor of a power plant is responsible for keeping track of the turbine speed, active power, and frequency. It detects and reduces the frequency bias brought on by load fluctuations by modifying the inputs to the turbine. The prime mover drives a rotating mass that is linked to a rotating mass or generator, which ultimately governs the load. The transfer function-based model presented in this section represents several components that form part of the primary control loop (PCL), as depicted in Figure 1. This control system is responsible for enhancing the frequency stability of a power system, following the change in load [28].

Thus, a composite transfer function that characterizes the correspondence between the perturbation in load, denoted as $\mathsf{\Delta }{P}_L$, and the ensuing change in frequency, represented by $\mathsf{\Delta }\omega$, can be stated as follows:

$$\frac{\mathsf{\Delta }\omega \left(s\right)}{\mathsf{\Delta }{P}_L\left(s\right)}=\frac{\frac{-1}{Ms+D}}{1+\frac{1}{R}\left[\frac{1}{1+s{T}_G}\right]\left[\frac{1}{1+s{T}_{ch}}\right]\left[\frac{1}{Ms+D}\right]}$$

(1)

Equation (1) shows that an alteration in the load, either through addition or removal, will result in a variation in frequency within the interconnected areas; thus, a power exchange occurs through the tie lines connecting them. This results in the emergence of power differences, denoted as $\mathsf{\Delta }{P}^{tie}$, between the coupled area and a tie-line synchronizing torque coefficient, $P_S$.

To better understand the behavior of the system, we modify steady-state behaviour in Equation (2) by introducing perturbations. Specifically, we determine the deviations in the steady-state power flow of a tie-line from its nominal value by considering the deviations in phase angle $\mathsf{\Delta }\delta$ [28]. This analysis allows us to gain insight into the behaviour of the system in response to changes in phase angle $\mathsf{\Delta }\omega$,

$$\begin{array}{c}\hfill P^{tie}=\frac{1}{X^{tie}}({\delta }_1-{\delta }_2)\end{array}$$

(2)

$$P^{tie}+\mathsf{\Delta }{P}^{tie}=\frac{1}{X^{tie}}[({\delta }_1+\mathsf{\Delta }{\delta }_1)-({\delta }_2+\mathsf{\Delta }{\delta }_2)]$$

$$\mathsf{\Delta }{P}^{tie}=\frac{1}{X^{tie}}(\mathsf{\Delta }{\delta }_1-\mathsf{\Delta }{\delta }_2)$$

As changes in electrical frequency are the rate of the change in the rotor angle, $\mathsf{\Delta }\omega =\frac{d}{dt}\left(\mathsf{\Delta }\delta \right)$, therefore:

$$\mathsf{\Delta }{P}^{tie}=\frac{P}{s}(\mathsf{\Delta }{\omega }_1-\mathsf{\Delta }{\omega }_2)$$

(3)

3. Generation Control

3.1. Supplementary Action Control

For a shift in frequency followed by a change in electrical load, the PCL can react quickly to modify the system conditions to regain approximate stability, but it is unable to completely mitigate steady-state errors in frequency deviations, which can be sustained for considerable a length of time within a power system [28,29]. Therefore, an additional loop is required, as well as a control action, to achieve zero steady-state response, which is referred to as a supplementary control loop (SCL), as shown in Figure 2.

The supplementary control should ensure that the load demand $\mathsf{\Delta }{P}_{L_i}$ within a specific area is fulfilled by the generation of that same area.

$$\mathsf{\Delta }{P}_{ge{n}_i}=\mathsf{\Delta }{P}_{L_i}$$

(4)

$$\mathsf{\Delta }{P}_{ge{n}_j}=0$$

(5)

3.2. Tie-Line Control

The objective of tie-line control is to limit the power flow from the neighboring areas when a load change occurs in either area.

$$\mathsf{\Delta }{P}^{tie}=P^{tie}-P^{tie,sch}$$

(6)

$P^{tie}$ is total actual net interchange and $P^{tie,sch}$ is the scheduled or desired interchange value. This control system must also have the ability to recognize the following:

• A load increase has taken place outside the area if the frequency dropped and the net interchange power leaving the area increased.
• A load increase has occurred inside the area if the frequency dropped and the net interchange power exiting the area decreased.
• When a change in frequency is negative, this indicates that the area’s frequency is reduced because it experienced an increase in load, and when the change in frequency is positive, this indicates that area’s frequency increased because it experienced a decrease in load.
• The location of the increase in load is determined by looking at $\mathsf{\Delta }{P}^{tie}$ of the tie-line.
• Positive (+) and negative (−) signs in $\mathsf{\Delta }{P}^{tie}$, respectively, represent power leaving and entering the system.

The implementation of the rules outlined in

Table 1. Rule set for a control mechanism.

$\mathit{\Delta }\mathit{\omega }$	$\mathit{\Delta }{\mathit{P}}^{\mathbf{tie}}$	$\mathit{\Delta }{\mathit{P}}_{\mathit{L}}$	Resulting Control Action
$-ive$	−	$\mathsf{\Delta }{P}_{L_i}=increased$; $\mathsf{\Delta }{P}_{L_j}=0$	Increase $P_{gen}$ in area i
$+ive$	+	$\mathsf{\Delta }{P}_{L_i}=decreased$; $\mathsf{\Delta }{P}_{L_j}=0$	Decrease $P_{gen}$ in area i
$-ive$	+	$\mathsf{\Delta }{P}_{L_i}=0$; $\mathsf{\Delta }{P}_{L_j}=increased$	Increase $P_{gen}$ in area j
$+ive$	−	$\mathsf{\Delta }{P}_{L_i}=0$; $\mathsf{\Delta }{P}_{L_j}=increased$	Decrease $P_{gen}$ in area j

This table indicates a control mechanism which mitigates ACE in order to achieve the desired frequency and net interchange value.

can be achieved through the use of a control mechanism that takes into account both the frequency deviation $\mathsf{\Delta }\omega$ and the net interchange power $\mathsf{\Delta }{P}^{tie}$. These rules can be utilized to compute the frequency deviations and tie flows that arise due to a load change $\mathsf{\Delta }{P}_{L_i}$ using Equations (7) and (8).

$$\mathsf{\Delta }\omega =\frac{\mathsf{\Delta }{P}_{L_i}}{{\sum }_i\frac{1}{R_i+D_i}+{\sum }_j\frac{1}{R_j+D_j}}$$

(7)

$$\mathsf{\Delta }{P}_{ij}^{tie}=\frac{\mathsf{\Delta }{P}_{L_i}\times \left[{\sum }_j\frac{1}{R_j+D_j}\right]}{{\sum }_i\frac{1}{R_i+D_i}+{\sum }_j\frac{1}{R_j+D_j}}$$

(8)

3.3. Area Control Error

ACE is the amount of adjustment in the generation of an area that is necessary to restore the frequency and net interchange to their desired values [28]. LFC aims to permit the scheduled power exchange owing to the tie-line as well as to mitigate frequency deviations in every area, since all system steady-state frequency deviations lead to tie-line power errors. The ACEs in each area can be reduced to zero to eliminate both the frequency errors and tie-line power errors. If $P^{tie}$ is the tie-line power and $\beta =\frac{1}{R}+D$ is the frequency bias, then ACE can be calculated as follows [29]:

$$AC{E}_i=\sum _i^m\left({\beta }_i\mathsf{\Delta }{f}_i+\sum _j^n\mathsf{\Delta }{P}_{ij}^{tie}\right); i\ne j$$

(9)

The control signal from the LFC loop in any area is applied in Equation (9) as $\mathsf{\Delta }{P}_i^C={\varphi }_i\left(AC{E}_i\right)$. Where ${\varphi }_i\left(\right)$ indicates the use of the LFC controller as a function. The detailed flowchart of the problem is provided in Figure 3.

4. Renewable Energy Sources

Each control area exclusively comprises conventional non-reheat generation sources. However, due to growing environmental concerns, RESs are being integrated into the global energy mix in significant amounts [4]. Their impact on frequency regulation has recently become a significant point of interest since their power outputs are primarily influenced by uncontrollable weather conditions. In addition to their intermittent nature, RESs are linked to the power system grid through power converters, which allow them to operate independently of the grid [30]. The inclusion of RESs like wind and/or solar introduces an additional instability factor to power networks, which are already inherently fluctuating [5,6].

A first-order transfer function is utilized to simulate the effects of RESs on the system, with the penetration level serving as the gain of the transfer function [31]. Each transfer function is presented below.

$$\frac{\mathsf{\Delta }{P}_{WTG}\left(S\right)}{\mathsf{\Delta }{P}_{WT}\left(S\right)}=\frac{K_{WTG}}{1+s{T}_{WTG}}$$

(10)

$$\frac{\mathsf{\Delta }{P}_{PV}\left(S\right)}{\mathsf{\Delta }\psi \left(S\right)}=\frac{K_{PV}}{1+s{T}_{PV}}$$

(11)

where $\mathsf{\Delta }{P}_{WTG}$ is the change in wind turbine generation, $\mathsf{\Delta }\psi$ is the change in solar irradiance, and $\mathsf{\Delta }{P}_{PV}$ is the change in solar power output [32]. The frequency variations caused by RESs depend on how much of the total electrical power is produced by them [6]. The impact of RESs on the LFC loop is limited in power networks with high inertia, where the majority of the power is generated by traditional resources, as the system’s inherent stability and damping properties can help absorb fluctuations caused by an intermittent RES output; however, where there is large penetration of RESs, the LFC strategy has to be changed to consider the dynamics of the power variation caused by the RESs [33]. Thus, the modified equations for the net interchange of tie-line flow and ACEs under the influence of RESs are represented as follows:

$$\mathsf{\Delta }{P}_i^{tie,TOTAL}=\mathsf{\Delta }{P}_i^{tie,RES}+\mathsf{\Delta }{P}_i^{tie}$$

(12)

$$AC{E}_i={\beta }_i\mathsf{\Delta }{f}_i+P_i^{tie,TOTAL}$$

(13)

The dynamic behaviour of wind and solar energy, shown in Figure 4a,b is introduced in area 2 and area 3 respectively, of the test system, as seen in Figure 5:

5. Multi-Area Interconnected System

As the inter-machine oscillations inside each area are not primarily of concern, an analogous generation unit connected to a single bus can represent a control area. Figure 6 displays the six areas of equally rated capacities with a block diagram of the complete transfer function. Each area has multiple neighbors; hence ${\mathbb{A}}_1^{Ne}=[2,4,5],$ ${\mathbb{A}}_2^{Ne}=[1,3,4,5,6],$ ${\mathbb{A}}_3^{Ne}=[2,5,6],$ ${\mathbb{A}}_4^{Ne}=[1,2,5],$ ${\mathbb{A}}_5^{Ne}=[1,2,3,4,6],$ ${\mathbb{A}}_6^{Ne}=[2,3,5]$, as seen on the single-line-diagram in Figure 5.

The parametric description of Figure 6 can be found in Appendix A.

5.1. Tie-Line Model for Six-Area Systems

The increase in frequency due to an increase in area load is a representation of the incremental power $[\mathsf{\Delta }{P}_G-\mathsf{\Delta }{P}_D]$, where $\mathsf{\Delta }{P}_G$ and $\mathsf{\Delta }{P}_D$ are the change in the generated power and the change in the demanded power, respectively. The equations for tie-line power are distinct for each area and are provided by the following equations. The tie-line power for each area is represented by $\mathsf{\Delta }{P}_i$, where $i=1,2,\dots ,6$.

$$\mathsf{\Delta }{P}_1=\mathsf{\Delta }{P}_{12}+{\alpha }_{14}\mathsf{\Delta }{P}_{14}+{\alpha }_{15}\mathsf{\Delta }{P}_{15}$$

(14)

$$\mathsf{\Delta }{P}_2=\mathsf{\Delta }{P}_{21}+{\alpha }_{23}\mathsf{\Delta }{P}_{23}+{\alpha }_{24}\mathsf{\Delta }{P}_{24}+{\alpha }_{25}\mathsf{\Delta }{P}_{25}+{\alpha }_{26}\mathsf{\Delta }{P}_{26}$$

(15)

$$\mathsf{\Delta }{P}_3=\mathsf{\Delta }{P}_{32}+{\alpha }_{35}\mathsf{\Delta }{P}_{35}+{\alpha }_{36}\mathsf{\Delta }{P}_{36}$$

(16)

$$\mathsf{\Delta }{P}_4=\mathsf{\Delta }{P}_{45}+{\alpha }_{41}\mathsf{\Delta }{P}_{41}+{\alpha }_{42}\mathsf{\Delta }{P}_{42}$$

(17)

$$\mathsf{\Delta }{P}_5=\mathsf{\Delta }{P}_{51}+{\alpha }_{52}\mathsf{\Delta }{P}_{52}+{\alpha }_{53}\mathsf{\Delta }{P}_{53}+{\alpha }_{54}\mathsf{\Delta }{P}_{54}+{\alpha }_{56}\mathsf{\Delta }{P}_{56}$$

(18)

$$\mathsf{\Delta }{P}_6=\mathsf{\Delta }{P}_{62}+{\alpha }_{63}\mathsf{\Delta }{P}_{63}+{\alpha }_{65}\mathsf{\Delta }{P}_{65}$$

(19)

$\mathsf{\Delta }{P}_i$ is the change in power in the $i\mathrm{th}$ area and $\mathsf{\Delta }{P}_{ij}$ is the change in power from area i to j. Since we are considering equal areas, $\alpha$ can be $\pm 1$ depending upon the power entering or leaving the area.

5.2. ACEs for Six-Area Systems

Bias control, which mandates that each control area is in charge of power exchange and frequency control, is utilized to remove frequency errors and is calculated as follows:

$$AC{E}_1=(\mathsf{\Delta }{P}_{12}+\mathsf{\Delta }{P}_{14}+\mathsf{\Delta }{P}_{15})+{\beta }_1\mathsf{\Delta }{f}_1$$

(20)

$$AC{E}_2=(\mathsf{\Delta }{P}_{21}+\mathsf{\Delta }{P}_{23}+\mathsf{\Delta }{P}_{24}+\mathsf{\Delta }{P}_{25}+\mathsf{\Delta }{P}_{26})+{\beta }_2\mathsf{\Delta }{f}_2$$

(21)

$$AC{E}_3=(\mathsf{\Delta }{P}_{32}+\mathsf{\Delta }{P}_{35}+\mathsf{\Delta }{P}_{36})+{\beta }_3\mathsf{\Delta }{f}_3$$

(22)

$$AC{E}_4=(\mathsf{\Delta }{P}_{41}+\mathsf{\Delta }{P}_{42}+\mathsf{\Delta }{P}_{45})+{\beta }_4\mathsf{\Delta }{f}_4$$

(23)

$$AC{E}_5=(\mathsf{\Delta }{P}_{51}+\mathsf{\Delta }{P}_{52}+\mathsf{\Delta }{P}_{53}+\mathsf{\Delta }{P}_{54}+\mathsf{\Delta }{P}_{56})+{\beta }_5\mathsf{\Delta }{f}_5$$

(24)

$$AC{E}_6=(\mathsf{\Delta }{P}_{62}+\mathsf{\Delta }{P}_{64}+\mathsf{\Delta }{P}_{65})+{\beta }_6\mathsf{\Delta }{f}_6$$

(25)

where ${\beta }_{1-6}$ refers to the frequency bias of the areas $1,2\dots 6$.

6. Controller Design

This study aims to investigate the effectiveness of using different tuning methods for the LFC application. By comparing the results of the benchmark methods with APID, the aim is to identify the most suitable technique for improving the performance of LFC systems. For this purpose, commonly used PI and PID controllers are employed, along with benchmark tuning techniques, such as the Z-N and PSO algorithms, to optimize their performance for LFC. The results of these benchmark methods are compared with an APID controller.

6.1. Ziegler–Nichols Optimization

This method starts by making the integral and differential gains zero initially and then raising the proportional gain until the system is marginally stable. The value of $K_p$ at this point of stability is referred to as $K_u$. The period of oscillation at $K_u$ is; therefore, $T_u$ [34]. After successfully finding these parameters, the gains for PI and PID can be found by using their respective formula in

Table 2. ZN-based gain parameters.

Controller Type	${\mathit{K}}_{\mathit{P}}$	${\mathit{K}}_{\mathit{I}}$	${\mathit{K}}_{\mathit{D}}$
PI	$0.45\times K_u$	$0.54\times \frac{K_u}{T_u}$	−
PID	$0.6\times K_u$	$1.2\times \frac{K_u}{T_u}$	$0.075\times K_u\times T_u$

The values of $K_u=0.992$, $T_u=2.729$ for PI, and $K_u=0.6775$, $T_u=2.790$ for PID are obtained by visualising the bode-plot in the MATLAB R2021A version 9.10.0 environment.

.

6.2. Particle Swarm Optimization

The goal of PSO is to find an optimal set of values for the gains of the PID controller. This method involves a group of agents or particles exploring a search space with multiple dimensions. These particles adjust their position, velocity, and direction based on their own experience and the experience of their neighbours. By considering the optimal position found by its neighbours, each particle moves towards a better solution. This approach facilitates systematic population-based exploration in the search space [35]. Mathematically:

$$v_i^{k+1}=w\times v_i^k+c_1\times rand\times (P_{bes{t}_i}^k-x_i^k)+c_2\times rand\times (G_{bes{t}_i}^k-x_i^k)$$

(26)

$$x_i^{k+1}=x_i^k+v_i^{k+1}$$

(27)

Here,

• w is the inertia of the particle.
• $v_i^k$ is the $i\mathrm{th}$ particle’s velocity at iteration k.
• c is the learning rate.
• $rand$ is any numeral value between 0 and 1.
• $x_i^k$ is the current location of the $i\mathrm{th}$ particle at iteration k.
• $P_{bes{t}_i}^k$ is the personal best of $i\mathrm{th}$ particle at iteration k, and $G_{bes{t}_i}^k$ is the global best at iteration k.

For $i=16$ particles after $k=100$ iterations with weight $w=1$ and a relative tolerance of ${10}^{-6}$, the effectiveness of the method is assessed by the objective function, which is a minimization of the integral-time-absolute-error (ITAE) of ACE. The gain parameters for PI and PID controllers using PSO are shown in

Table 3. Gain parameters with PSO.

Controller	PI		PID
	${\mathit{K}}_{\mathit{P}}$	${\mathit{K}}_{\mathit{I}}$	${\mathit{K}}_{\mathit{P}}$	${\mathit{K}}_{\mathit{I}}$	${\mathit{K}}_{\mathit{D}}$
Area 1	0.07997	0.21875	1	1	1
Area 2	−0.19399	0.44547	1	1	0.44732
Area 3	−0.20879	0.47944	1	1	0.82825
Area 4	0.08521	0.19439	1	1	0.90532
Area 5	−0.42646	0.21091	1	0.99813	0.89595
Area 6	−0.02965	0.09085	1	1	0.87521

The limit values for both controllers are set as $\pm 1$.

. The weight of the error per iteration is shown in Figure 7a,b for PI and PID controllers, respectively.

6.3. Adaptive PID

A PID controller with the automated tuning of controller gains is known as an APID. It functions as a learning mechanism and updates the parameters throughout system operation to take the process of dynamic variations into consideration [36]. On the basis of the error signal, the gains are modified in real-time using the gradient descent approach. The error between the desired value and the actual value of a certain power system parameter is determined; in our case, the system is the ACE.

Mathematically, the APID controller is shown as follows:

$$u_{Adap}^k=k_{p_{Adap}}{e}^t\left(k\right)+k_{d_{Adap}}\frac{d}{dt}{e}^t\left(k\right)+k_{i_{Adap}}\int e^t\left(k\right)dt$$

(28)

Here, the adaptive proportional, derivative, and integralgain parameters are denoted by $k_{p_{Adap}}>0$, $k_{d_{Adap}}>0$, and $k_{i_{Adap}}>0$, respectively.

• $k_{p_{Adap}}$ yields an output which is directly proportional to the current error value. For a given error $e\left(t\right)$, a higher $k_{p_{Adap}}$ yields a more aggressive response. A system that sets $k_{p_{Adap}}$ too high may become unstable and oscillate, although raising $k_{p_{Adap}}$ normally decreases the rise time and steady-state inaccuracies.
• The integral gain $k_{i_{Adap}}$ determines the response based on the accumulated preceding errors, and the integral term accumulates errors over time and incorporates them into the control output. In doing so, the residual steady-state errors that cannot be addressed using only $k_{p_{Adap}}$ are significantly reduced. While raising $k_{i_{Adap}}$ lowers the steady-state errors, if it is not tuned appropriately, it can also slow down the response and result in overshooting; conversely, raising $k_{i_{Adap}}$ too high induces oscillations or instability in the system.
• The derivative gain $k_{d_{Adap}}$ provides a damping effect that improves system stability by counteracting the error’s rate of change, while the derivative term predicts future errors based on its rate of change. Raising $k_{d_{Adap}}$ decreases overshoot and enhances stability, but if $k_{d_{Adap}}$ is set too high, it may lead to increased sensitivity to error signal noise and unpredictable control behavior.

Hence, a finetuning of controller gains is inevitable so that the objective of reducing the error function is achievable, as in $e\left(t\right)=\frac{1}{2}{(y-r)}^2$, where y and r are the desired and actual values of the system. The control signal $u_{Adap}^k$ uses the error signal $e^k\left(t\right)$ at any point in time t for a specific iteration k and updates the gain parameters of APID using the gradient descent approach, such that the error is minimized.

Gradient Descent Method

Gradient descent is an optimization process that moves in an iterative manner in the direction of the steepest descent, which is indicated by a negative gradient, in order to minimize a function [37]. Its application steps are as follows:

• Parameters’ initialization: The weights are provided, along with their initial values. These can be random values.
• Gradient determination: The gradient of cost function is determined in relation to each parameter. The gradient shows the cost function’s steepest rise in terms of both direction and rate.
• Parameters update: The parameters are adjusted by a certain step size, ak, learning rate, in the gradient’s opposite direction. In our case, the learning rate was $0.25$. This was to guarantee that the parameters tended to approach the cost function’s minimum.
• Reiteration: The process is repeated until convergence occurs, i.e, there is almost no variation in the parameters between iterations.

The update equation is given as follows:

$$K_{Adap}^k=K_{Adap}^{k-1}+{\alpha }^{Adap}·\nabla ·e^k\left(t\right)$$

(29)

where ${\alpha }^{Adap}>0$ is a step size whose values are typically measured in the range $[0,1]$, ∇ is the gradient, and $e^k\left(t\right)$ is the error function. The updated APID parameters reduce error and hasten convergence in a closed-loop system [38].

7. Performance Analysis

This section presents the findings of the schemes implemented over a six-area system (represented in Figure 6). The results are categorized into three cases to provide a clear picture of the LFC problem and to identify the most effective control technique.

• Case 1 shows the system with an unequal load distribution.
• Case 2 investigates the system with renewable penetration only.
• Case 3 depicts the test system along with the effect of load change and renewable penetration.

7.1. Case 1

At $t=15$ s, Areas 2, 4, and 6 are subjected to a step increase in load $\mathsf{\Delta }L$ of $0.1$ pu, $0.2$ pu, $0.3$ pu, respectively, while the load in remaining areas remains unchanged. As the electrical load suddenly increased, the LRSP, which determines the turbines’ spinning rate, aka the frequency of subjected areas, was altered. It must be noted that areas 1, 3, and 5 also experienced a decrease in frequency due to the increase in load in other areas, as they are interconnected. To avoid this situation, the APID controller sends the control signal to the governor to increase the turbine’s spinning rate. It can be seen from the performance indicator, ITAE, as shown in

Table 4. ITAE Case 1.

Performance Criterion	ZN_PI	ZN_PID	PSO_PI	PSO_PID	APID
ITAE	29.4	3.516	8.274	0.998	0.4187

and Figure 8, that this decrease in frequency is quickly restored in the case of APID. Figure 8 also shows the smaller overshoot and shorter settling time and rise time in the case of APID, followed by PID-PSO, PID-ZN, PI-PSO, and PI-ZN. The overshoot and settling time are important factors, as they represent the magnitude of error and the time an error remains in the system, respectively. The rise time, however, represents the quickness with a controller responds to a change. As the frequency deviation of each area is brought back to zero, the inter-area frequency oscillations are dampened. Hence, the system becomes more stable.

Electrical power flowing through a tie-line is rapidly reduced to zero in the case of APID, meaning that no access power flows from any area. However, an important factor to observe here is the overshoot and the undershoot. Areas where the load has increased experience an overshoot in tie-line power while the remaining areas experience an undershoot. This happens because, after the load change, the areas with an undershoot started sharing their power with areas with an overshoot to meet that change in load. This is unwanted and quickly resolved since each area should provide its own load. Therefore, with a secondary control loop, a given area will be responsible for its own load change, as seen in Figure 9. In the case of ACE, the APID controller adapts its gain in accordance with system dynamics and helps to swiftly diminish the area control errors in each area, as seen in in Figure 10, which, in turns, helps to restore the tie-line power and system frequency back to their nominal values, thereby reducing unwanted oscillations and improving system stability.

7.2. Case 2

This case represents the system response when subjected to pure RES penetration. when RESs are integrated with a power system, it causes an imbalance in the power balance equation, which introduces inter-area oscillations. Although RESs inject power into the system, the power system sees their integration as variations in its load dynamics. This can be observed by noticing the overshoot in the frequency of the system, especially in area 2 and area 3 of Figure 11. Change in tie-line power $\mathsf{\Delta }P$ for a six-area interconnected system is provided in Figure 12. As the RESs are introduced to the system, this alters the LRSP, which, in turn, increases the prime-mover speed; hence, the frequency of each area in an inter-connected area is increased. Figure 13 shows that as long as RESs are injected into power systems, the ACEs will persist. This means that if there is more renewable penetration, the stability of the system reduces. Nonetheless, in response to the increase in frequency due to the injection of RESs, the SCL effectively minimizes frequency and power flow deviations in the case of APID, as shown in Figure 11 and Figure 12. Figure 13 and

Table 5. ITAE Case 2.

Performance Criterion	ZN_PI	ZN_PID	PSO_PI	PSO_PID	APID
ITAE	17.6	8.514	9.661	3.435	1.617

show that the APID controller quickly minimizes the ITAE and strives to maintain ACEs closer to zero when compared with PID-PSO, PID-ZN, PI-PSO, and PI-ZN. APID also outperforms the rest of controllers in terms of rise time, settling time, and the overshoot of frequency deviation and power flow on tie-lines.

7.3. Case 3

In practical power systems, there are simultaneous changes that occur in both the load and RES power injected into the system. It is, therefore, crucial to observe the system response. In this case too, the change in frequency, change in tie-line power and ACE are provided in Figure 14, Figure 15 and Figure 16, respectively. In this case, area 2 is subjected to a load change of ${\mathsf{\Delta }}_{P_{load}}=0.1\mathrm{p}\mathrm{u}$, while the RESs are integrated in both area 2 and area 3. Figure 16 shows the occurrence of ACEs. As explained earlier, in case 1 and case 2, this changes the LFSP, which alters the prime mover output to a new setpoint. To ensure effective control and maintain system stability, APID effectively minimizes the ACEs, as seen in Figure 16 and

Table 6. ITAE Case 3.

Performance Criterion	ZN_PI	ZN_PID	PSO_PI	PSO_PID	APID
ITAE	19.51	8.485	9.917	3.437	1.672

compared to other controllers optimized with the PSO and ZN algorithms. This restores the system frequency Figure 14 and tie-line power Figure 15 to their nominal values. This demonstrates the successful control and stability of the system in terms of rise time, settling time, and overshoot. The other meta-heuristically optimized controller under-performs when compared with APID. This is because of the inherent nature of the APID controller to improvize the system changes and quickly mitigate those changes.

8. Conclusions

This study emphasizes the importance of developing effective strategies for minimizing changes in power systems caused by variations in load demand. The use of secondary control, tie-line power exchange, and optimization techniques are proposed as effective means to achieve this goal. By reducing frequency deviations and stress on the controllers, these strategies help maintain system stability while ensuring that the power demand is met efficiently. The results suggest that the adaptive-PID controller can be used to modify the values of various parameters in power systems to adapt to changes in load demand while preserving system stability, and this controller outperforms various benchmark techniques such as the Ziegler–Nichols algorithm and particle swarm optimization algorithm. This method offers a more efficient and effective alternative to the traditional hit and trial methodology, which can be time-consuming and tedious. In addition to the gradient descent method, other meta-heuristics can be applied to enhance the performance of controllers used in AGC for future advancements. Overall, this study provides valuable insights into the strategies that can be employed to minimize changes in power systems and maintain system stability. By implementing these strategies, power system operators can improve the efficiency and reliability of the power supply, thus contributing to the sustainable development of the energy sector.