In this paper, a commonly used non-reheat thermal power plant connected by tie-line is considered for system under study (Ali and Abd-Elazim, 2011; Panda et al., 2013; Padhan et al., 2014b; Sahu et al., 2016). A speed-governing system, a turbine, and a generator are present in each section of a power plant having two outputs and three inputs. The inputs are the tie-line power error ( Δ P T i e ), load disturbance ( Δ P D ), and controller input (written as u). Area Control Error and Generator Frequency ( Δ F ) are the outputs (ACE). Each area has a rating of 2000 MW and a nominal load of 1,000 MW. A C E 1 and A C E 2 are area control errors; B 1 and B 2 are the frequency bias parameters; u 1 and u 2 are the control outputs from the controller; R 1 and R 2 are the governor speed regulation parameters in p.u. Hz; T G 1 and T G 2 are the speed governor time constants in sec; T T 1 and T T 2 are the turbine time constant in sec; Δ P D 1 and Δ P D 2 are the load demand changes; Δ P T i e is the incremental change in tie line power in p.u; K P 1 and K P 2 are the power system gains; T P 1 and T P 2 are the power system time constant in sec; T 12 is the synchronizing coefficient and Δ F 1 and Δ F 2 are the system frequency deviations in Hz. The nominal parameters of the system are given in Appendix.

The two-area power system has IPIDF controllers available in each region to regulate the frequency. Standard PID controllers with static values do not offer satisfactory accuracy across a wide variety of operating situations (Sahu et al., 2016). IPIDF controller design therefore enters the picture for the purpose of enhancing system performance. The IPIDF controller combines the PIDF controller and the traditional I controller. Where KII is the I controller’s integral gain.

In contrast to PID controller, IPIDF controller is displayed in Figure 1 _. It has a transient response to instruction intake ratio that is good over a larger range of plant component fluctuation, is simple to calibrate, and has more reliable regulation. In addition to being simpler than PID, IPIDF designing and tuning is also quicker. The transfer function of the IPIDF controller is shown in Eq. 1. T F I − P I D F = K I I s + K P + K I s + K D N C s s + N C (1)

In order to build controllers that use optimization techniques and to fine-tune controller parameters in accordance with performance indices, objective functions are used. Control designs often employ one of four basic types of objective functions (Shabani et al., 2013)_. The target function ITAE is used in this paper because it shortens peak overshoot and settling rate. It cannot be done using IAE or ISE-based correction, though. The mathematical formulas for the ITAE are shown in Eq. 2 (Sahu et al., 2016). J = I T A E = ∫ 0 t s i m Δ f 1 + Δ f 2 + Δ P T i e ⋅ t ⋅ d t (2)

The MATLAB/SIMULINK environment was used to create the model of the system under study depicted in Figure 3 and a DE program was constructed (. mfile). The present work chooses −2.0 and 2.0 as the lowest and maximum values for controller parameters. The objective function, which is computed in the. m file, is used by the optimization process. In the current study, the population size NP = 100, generation number G = 100, step size F = .8, and crossover probability CR = .8 have all been used. With a situation of a 10% shift in the burden in Area-1 only at t = 0 s, the output/gain of the controller are optimized here by the DE method, and their control parameters are displayed in Table 1. The final value that is determined by repeating the optimization procedure 50 times will be determined by the optimal value for each parameter.

Three examples can be taken into account in order to examine the dynamic performance of the system under examination. In the first scenario, Area-1 is the only one to receive the 10% step load; in the second, Area -2 is the only one to receive the 10% load disturbance; and, in the third scenario, Area -1 and Area -2, respectively, are the recipients of 10% and 20% step load. The effectiveness indicator values are given in Table 2 for the first case, where a 10% shift in the burden is applied to Area -1 at time t = 0 s. Figure 4 displays the associated frequency change in the area and Power variations along the tie line. In contrast to GA PID (Sahu et al., 2016), PSO PID (Sahu et al., 2016), FA PID (Sahu et al., 2016), DE: PID (.3391) (Sahu et al., 2016), and DE: PIDF (.1764), the IPIDF controller yields a reduced ITAE (.1238). When compared to other controllers, the proposed IPIDF controller reduces the ITAE value by 12.38%. The change in error is then displayed in Figure 5 when a similar operation is carried out with a 10% shift in the burden in Area -2 only at t = 0 s. Finally, Area -1 and Area -2, respectively, receive 10% and 20% step disruption. In Figure 6, the dynamic reaction is displayed. The figure clearly shows that the IPIDF controller functions more effectively analyzing peak overshoot and undershoot, as well as settling time. The yielded results of the IPIDF controller are therefore evidently better than that of the PIDF controller.

Changes to the system’s operating circumstances and system parameters are made as part of the sensitivity study, which examines the proposed system’s resilience (Ali and Abd-Elazim, 2011; Panda et al., 2013; Padhan et al., 2014b; Sahu et al., 2016). Here, the reliability of the system is evaluated by altering the value of T_G, T_T, T₁₂, and B in the range of ±25% for the same controller value. Here, under changing load conditions, a sensitivity study is conducted for 10% and 20% shift in burden in Area 1 and Area 2 respectively. Table 3’s listing of the system characteristics demonstrates how the IPIDF controller is resistant to various parameter variations.

The SMES’s capacity to store electrical energy in the form of magnetic energy and its ability to deliver enormous amounts of power instantly are two of its key capabilities. Because all of a SMES unit’s components are static, it is more stable than other power storage devices. Pradhan et al. (2016); model the SMES and connect in the power system to investigate the system. The model for SMES is shown in Figure 7. Two SMES units are set up in areas 1 and 2 in the current study to stabilize frequency oscillations, as depicted in Figure 9. The input signal of the SMES controller is p.u. frequency deviation (ΔF) and the output is change in control vector (ΔP_SMES). The values of the time constant T_SMES and the controller gains K_SMES are .03 s and .12, respectively.

A HVDC link is taken into consideration in parallel with the HVAC system in order to enhance the dynamic performance of the power system. Figure 8 depicts the single line diagram of a two-area power system with parallel HVAC/HVDC linkages. The HVDC link’s control system responds promptly to a step load disruption by suppressing the peak value of the transient frequency deviation. The governors then eliminate the steady state inaccuracies of the frequency deviation. The dynamics of the governors in both areas can be ignored for the purpose of simplicity in the control design of the HVDC link. The change in output in area-1 of an HVDC link can be expressed as follows for a sudden step load perturbation: ∆ P D C = K D C 1 + S T D C ∆ F 1 Where K_DC is gain of a HVDC link and T_DC is time constant of HVDC link in seconds.

To demonstrate the efficacy of the suggested IPIDF controller, the study is further extended with SMES and HVDC link (Pradhan et al., 2016; Dekaraja et al., 2022; Ramoji, 2022; Sivadanam et al., 2022) as shown in Figure 9. The MATLAB/SIMULINK environment was used to create the model of the system under study depicted in Figure 9, and a DE program was constructed (. mfile). The present work chooses −2.0 and 2.0 as the lowest and maximum values for controller parameter for K_II, K_P, K_I, and K_D. The lowest and maximum value for the filer (N_C) is taken 1 and 300 respectively. With a situation of a 10% shift in the burden in Area -1 only at t = 0 s, the output/gain of the controller are optimized here by the DE method. The same DE method, where the 50 best final solutions from the 50 optimization runs were utilized to establish the controller’s settings. The top 50 concluding responses from the 50 runs are shown in Table 4.

Table 5 shows the outcome of the system DE-optimized IPIDF controller for the fast scenario, where Area -1 is subjected to a burden of 10% load at time t = 0 s. Performance of the proposed IPIDF controller is compared to that of PIDF. Table 5 makes it very evident that the IPIDF controller produces a lower ITAE value than the PIDF (ITAE = .3279 vs. .1330). Area -2 is the only location in the second scenario to have a burden of 10%. Finally, to investigate the dynamic performance of system 10% and 20% shift in burden is given to Area -1 and Area -2 accordingly. The dynamic reaction is seen in Figure 10–12. It is obvious that, in terms of output performance, the IPIDF controller surpasses the PIDF controller (minimal Peak undershoot, frequency settling durations, and power variations in tie-line).

Individual system performance of the system with just SMES and only HVDC connection is compared to the system with both SMES and HVDC link in order to demonstrate the effectiveness of adding both SMES and HVDC link to the systems. The change in errors for the 10% and 20% loads to Areas 1 and 2 is presented in Figure 13. From Figure 13, it is evident that the system clearly performs better when SMES and HVDC links are included in the system.

The reliability of the system (Figure 9) is evaluated by altering the value of T_G, T_T, T₁₂, and B in the range of ±25% for the same controller value. Here, under changing load conditions, a sensitivity study is conducted for 10% shift in burden in Area 1 only. Table 6’s listing of the system characteristics demonstrates how the IPIDF controller is resistant to various parameter variations. Table 6 makes it evident that the performance index values fluctuate within allowable limits and are typically identical to the standard value. Dynamic nature of the system for different parameter variation is shown in Figure 14.

A performance analysis of the suggested system is also done for several load perturbations, such as step load and pulse load to Area 1 (Sahu et al., 2016). Figure 15A depicts the application of a step load to region 1 that has a period of 140 s and a breadth of 10 s (Sahu et al., 2016). Figure 15B depicts the corresponding tie-line power exchange. Area 1 is then put under a pulse (Sahu et al., 2016) change with an initial magnitude of 10% P.U and a frequency of .025 Hz shown in Figure 16A. Figure 16B depicts the change in tie-line power. We can infer from all of the findings that the IPIDF controller reduces oscillation.