Load Frequency Controller Design Based on the Direct Synthesis Approach Using a 2DoF-IMC Scheme for a Multi-Area Power System

Abstract

To maintain reliable and sustainable power supply, the frequency should be kept constant under varying load conditions. The two degrees of freedom internal model control (2DoF-IMC) scheme is a robust control technique and is efficient for load-disturbance rejection problems in industrial process control. The asymmetry of IMC compared to other methods regarding controller design is that it does not guarantee the stability of the system by itself but is based on the stability of the controlled system. For the control of less-stable and unstable systems, it is therefore usually supplemented with an additional controller, establishing two degrees of freedom in the overall design. In this manuscript, the load-frequency-regulation problem was investigated using a 2DoF-IMC scheme for a single-area as well as a multi-area power system. In the 2DoF-IMC scheme, two controllers are used to control the set-point response and load-disturbance response separately. The set-point controller is designed through the internal model control (IMC) principle, whereas the load-disturbance rejection controller is designed via the direct-synthesis (DS) approach. In the DS approach, the closed-loop transfer function of the system model is matched with the desired closed-loop transfer function of the system and the disturbance-rejection controller is approximated at a very low-frequency point to obtain the proportional–derivative (PD) controller parameter. The simulation results of the proposed method provide satisfactory performance for load-frequency control (LFC) in the single-area power system and extended to two-area and four-area power systems. The effect of non-linearity, such as generation rate constraint (GRC), was investigated in the single-area power system to establish the efficacy of the proposed method. A random step loading pattern was also considered to confirm the robustness of the proposed method. The overall performance of the proposed control scheme is comparatively better than the recently reported work.

1. Introduction

A large-scale power system network may depreciate its performance because of the presence of fast-burden unsettling influences, vulnerabilities of parameters, operational varieties, and so forth. In this manner, the stability and robustness are the most important parameters for enormous-scale power system networks when designing load-frequency controllers. The power system networks have complexity, non-linearity, and higher-order dynamics. Therefore, it is a difficult task to design a robust as well as a simple controller for such a complex system. In the present scenario of the power system network, the balance between power generation and load demand must be maintained so that the frequency deviation as well as tie-line power exchange are within the pre-specified limit [1]. For a couple of decades, various methodologies utilizing different control techniques have been proposed for the LFC problem in single area power systems (SAPSs) as well as multi-area power systems (MAPSs) [2,3].

The IMC is one of the famous control techniques for the LFC problem with SAPSs and MAPSs [4,5,6]. Tan [4,5] has proposed a proportional–integral–derivative (PID) controller based on the two degrees of freedom internal model control (2DoF-IMC) approach for LFC problem in SAPSs and MAPSs and developed an anti-GRC approach to restrict the problem associated with GRC. The performance for LFC involving the three-area power system as well as the four-area power system (FAPS) has been further improved using the IMC-PID controller by Tan [6].

Different modified structures of the 2DoF-IMC scheme used to eradicate the load-disturbance problem and its application have been discussed in [7,8,9]. Saxena and Hote [7] have designed a 2DoF-IMC controller for the LFC problem via model reduction of the power system using Routh and Padé approximations. Further, Saxena and Hote [8] presented a PID controller design via the IMC scheme utilizing an approximated lower order model of the power system. Recently, a PID controller has been designed by Singh et al. [10] based on the 2DoF-IMC scheme for LFC in single-area and extended to two-area hydro-thermal power systems, but in this case the higher-order power system model is reduced to a lower-order model using logarithmic approximations.

Saxena [9] has proposed a fractional order (FO) controller using an IMC scheme to address the frequency regulation problem in MAPSs. A PID double derivative (PID + DD) controller has been developed by Raju et al. [11] for a three-area thermal power system using the ant–lion optimization technique. Further, the performance has been improved by Guha et al. [12] using a PID + DD controller and the parameters are optimized using the multi-verse optimization technique. A modified control structure with a larger number of controllers has been proposed by Padhan and Majhi [13] for SAPSs and extended to MAPSs. They have used the Laurent series expansion to reduce the plant model for controller design. The direct synthesis (DS) approach is used to design a PID controller in a single loop by Anwar and Pan [14] for SAPSs and MAPSs without reduction of the plant model in the frequency domain.

In most of the above-mentioned literature, the design methods are based on the low order model of the system. Additionally, the complexity of controller configuration as well as control structure has been observed. Afterward, controller parameters are obtained using a suitable approximation technique or some meta-heuristic optimization technique. The higher order systems are usually approximated to a lower order one.

In recent years, an improved control structure has been used for complicated industrial process and in LFC problems [15,16,17,18,19,20,21]. Siddique et al. [15] proposed a cascade control scheme for unstable as well as integrating processes. Gundus and chow [16] have used a PID controller for LFC with a communication delay but its frequency deviation and oscillation are greater. K. Lu et al. [17] proposed a robust proportional–integral (PI) controller by using constrained population extremal optimization for the frequency regulation problem of MAPSs. Cai et al. [18] have presented PID control for MAPSs to regulate its frequency and tie-line power. A parallel control structure has been used by Kumar and Anwar [19] for frequency regulation in MAPS-based systems.

In this regard, a 2DoF-IMC control structure has been used for the LFC problem of the power system. The enhanced load-disturbance rejection is achieved in the 2DoF-IMC control structure where the set-point and load-disturbance response are decoupled through two controllers. The set-point controller is designed through the IMC control scheme. The load-disturbance controller is designed through the direct synthesis approach. The proposed control scheme has been validated in single-area as well as multi-area multi-source power systems. The performance of the proposed control scheme is comparatively better in comparison with the recently reported work. The effect of nonlinearity has also been investigated. The salient features of the proposed control scheme are as follows.

• Enhanced load-frequency regulation has been obtained by using the 2DoF-IMC control scheme.
• The controllers have been designed for the higher-order system without reduction of the system model.
• Proposed method has been tested for various cases of single-area power systems, two-area power systems and extended to the four-area power system.
• The effect of GRC is incorporated into SAPSs to investigate the robustness of the proposed method.
• To analyze the robustness of the proposed controller, ±50% uncertainty in the system parameter and different step loading pattern is considered in the LFC system.

The rest of the research work is arranged in the following section: In Section 2, Modeling of LFC and the problem formulation of the paper are described. The controller design scheme is well elucidated in Section 3. Simulation results and the performance response for different cases are provided in Section 4. Finally, the conclusion of the paper is briefly summarized in Section 5.

2. Description of Control Structure and Problem Formulation of LFC

The control system structure as well as modeling of the power system play vital roles in the control of frequency deviation in the power system. A brief description of the control structure used and the modeling of the power system is discussed in this section.

2.1. Brief Description of IMC and 2DoF-IMC Control Structure

The conventional internal model control structure is shown in Figure 1, where H_p(s) is the actual system and H_m(s) is the nominal model. H_imc(s) is the controller and U_r is the command signal, U_c is the controller effort, U_d is the undesired disturbance, and Y is the output of the system.

Figure 1. IMC control structure.

The output in the IMC scheme may be represented as

$$y=\left[\begin{array}{l}\frac{H_p(s)H_{imc}(s)}{1+H_{imc}(s)[H_p(s)-H_m(s)]}r\\ +\frac{1-H_m(s)H_{imc}(s)}{1+H_{imc}(s)[H_p(s)-H_m(s)]}d\end{array}\right]$$

(1)

For the case of perfect modeling, i.e., $H_p(s)=H_m(s)$, the output becomes:

$$y=H_p(s)H_{imc}(s)r+[1-H_m(s)H_{imc}(s)]d.$$

(2)

It can be seen from Figure 1 and Figure 2 that the IMC control scheme behaves like an open-loop control system in the case of perfect modeling. In the case of modeling errors and disturbances in the system, the feedback action takes place. In the open loop control system stability is guaranteed if stability is controlled by the stable controller. This feature is well utilized in the IMC control scheme and makes the control structure robust against modeling error.

Figure 2. The 2DoF-IMC control scheme.

The controller in the IMC scheme is designed by factoring the system model into two parts as given by

$$H_m(s)=H^{+}(s)H^{-}(s)$$

(3)

where $H^{-}(s)$ includes all poles and zeros lie in the left-hand sides whereas $H^{+}(s)$ consists of the non-minimum phase part in the system.

The controller in the IMC scheme is selected as given by

$$H_{imc}(s)=\frac{1}{H^{-}(s)}f(s)=\frac{1}{H^{-}(s){(L_{1}s+1)}^{n_1}}$$

(4)

where $f(s)=\frac{1}{{(L_{1}s+1)}^{n_1}}$ is the low-pass filter with order n₁. The order of the filter is chosen to make the IMC controller $H_{imc}(s)$ suitable. The filter time constant is the only tuning parameter and its value is related to the desired closed-loop system speed. The lower value of L₁ results in a faster response with a loss of robustness whereas the higher value is responsible for a slower response with better robustness.

The achievable set-point performance through the IMC scheme is better but the load-disturbance response is poor especially for a sluggish system. To obtain a better load-disturbance response the 2DoF-IMC control scheme is used which decouples the set-point and load-disturbance rejection response. The schematic diagram of the 2DoF-IMC scheme is shown in Figure 2. An additional load-disturbance controller H_d(s) was used to attenuate the disturbance entering the system.

The output in the 2DoF-IMC scheme is obtained as:

$$Y=\left[\begin{array}{l}\frac{H_p(s)H_{imc}(s)[1+H_m(s)H_d(s)]}{1+H_{imc}(s)[H_p(s)-H_m(s)]+H_p(s)H_d(s)}{U}_r\\ +\frac{H_p(s)[1-H_m(s)H_{imc}(s)]}{1+[H_p(s)-H_m(s)]H_{imc}(s)+H_p(s)H_d(s)}d\end{array}\right]$$

(5)

With the perfect model of the system, i.e., $H_m(s)=H_p(s)$ the Equation (5) becomes

$$Y=H_p(s)H_{imc}(s)U_r+\frac{H_p(s)[1-H_m(s)H_{imc}(s)]}{1+H_p(s)H_d(s)}d$$

(6)

From (2) and (6) it can be seen that the use of an addition controller in the 2DoF-IMC attenuates the effect of disturbance through feedback. An appropriate selection of this controller will enhance the load-disturbance rejection characteristic of the overall system. Here, a proportional–derivative (PD) controller such as $H_d \left(s\right)$ is considered to for the LFC control problem and its transfer function is written as:

$$H_d\left(s\right)=K_c+K_{d^s}$$

(7)

where $K_c$ is the proportional gain and $K_d$ is the derivative gain.

2.2. Modeling of Single- as Well as Multi-Area Power Systems

The third-order model of a single-area thermal power system consisting of a governor, turbine, generator, and load is shown in Figure 3. The simplified transfer function of the nominal model from Figure 3 is written as

$$H_m=\frac{\Delta f}{u}=\frac{T_g{T}_t{T}_p}{1+T_g{T}_t{T}_p/R}$$

(8)

where T_g, T_t, and T_p are the time constant of governor, turbine and generator, and load-transfer function, respectively. R represents the speed drop of the governor.

Figure 3. Single-area thermal-power system model.

The block diagram representation of a multi-area power system (MAPS) is presented in Figure 4. The area control error (ACE) is a measure of performance of LFC in the MAPS. It incorporates the information of frequency deviation and tie-line power exchange. The ACE of the ith area is defined as

ACE_i = ∆p_tie,i + β_i ∆f_i

(9)

where ∆p_tie,i represents the change in the tie-line power exchange and β_i is a constant bias factor. The tie-line power exchange between area ith and other areas is given by

$$\Delta p_{tie,i}={\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^N\Delta p_{tiei,j}}=\frac{1}{s}\left[{\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^N{t}_{ij}\Delta f_i-{\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^N{t}_{ij}\Delta f_j}}\right]$$

(10)

where t_ij represents the synchronizing power coefficient of the MAPS. The closed-loop system model for the MAPS is obtained as

$$M_{ni}={\beta }_i\frac{T_{gi}{T}_{ti}{T}_{pi}}{1+T_{gi}{T}_{ti}{T}_{pi}/R_i}$$

(11)

Figure 4. Block diagram of the ith control area.

3. Controller Design Scheme

In this paper, a 2DoF-IMC control scheme was implemented in the LFC problem. The set-point controller is considered the IMC controller and its design procedure is discussed in Section 2. The load-disturbance controller is taken as the PD controller and its parameters are designed using a direct synthesis (DS) approach. In this approach, the desired response transfer function is matched with the closed-loop system model to be designed including an unknown controller. The expression of the controller in terms of the desired system and a closed-loop transfer function of the system is obtained analytically. In general, this controller is further approximated to a suitable form using some approximation technique.

The desired closed-loop transfer for load-disturbance rejection is selected as

$$H_{LD}(s)=\frac{Ks}{{(L_{2}s+1)}^{n_2}}$$

(12)

Here, L₂ is the time constant, which is the selected to have the desired closed-loop load-disturbance rejection speed. n₂ is the order of the desired model and $K=\frac{n_1{L}_1{k}_{p}R}{R+k_p+k_{p}R{K}_c}$ is a constant gain. One zero at the origin in (12) ensures the load-disturbance rejection.

The load-disturbance transfer function from d to $\Delta f$ may be obtained from (6) by considering the set-point input U_r = 0 as

$$H_{d,f}(s)=\frac{H_p(s)[1-H_m(s)H_{imc}(s)}{1+H_p(s)H_d(s)}$$

(13)

The controller expression may be obtained as

$$H_d(s)\cong \frac{[1-f(s)]}{H_{LD}(s)}-\frac{R+T_g{T}_t{T}_p}{R{T}_g{T}_t{T}_p}$$

(14)

The expression of $H_d(s)$ is approximated to the PD controller using a frequency-response-matching method presented in [14,15].

$$K_c+K_{d}s=\left[\begin{array}{l}\frac{{(L_{2}s+1)}^{n_2}[{(L_{1}s+1)}^{n_1}-1]K_c}{n_1{L}_{1}s{(1+L_{1}s)}^{n_1}}-\frac{R+T_g{T}_t{T}_p}{R{T}_g{T}_t{T}_p}\\ +\frac{(R+k_p){(1+L_{2}s)}^{n_2}[{(1+L_{1}s)}^{n_1}-1]}{n_1{L}_1{k}_{p}Rs{(1+L_{1}s)}^{n_1}}\end{array}\right]$$

(15)

Assuming $X_1=\frac{{(L_{2}s+1)}^{n_2}[{(L_{1}s+1)}^{n_1}-1]K_c}{n_1{L}_{1}s{(1+L_{1}s)}^{n_1}}$ and $X_2=\frac{(R+k_p){(1+L_{2}s)}^{n_2}[{(1+L_{1}s)}^{n_1}-1]}{n_1{L}_1{k}_{p}Rs{(1+L_{1}s)}^{n_1}}-\frac{R+T_g{T}_t{T}_p}{R{T}_g{T}_t{T}_p}$.

By considering $s=j\omega$ in Equation (15), the following relation may be obtained

$$K_c+j(K_d\omega )=(K_c\mathrm{Re}[X_1]+\mathrm{Re}[X_2])+j(K_c\mathrm{Im}[X_1]+\mathrm{Im}[X_2])$$

(16)

By equating real and imaginary parts in (16) the following equation in matrix form is obtained

$$\left[\begin{array}{cc}1-\mathrm{Re}[X_1]& 0\\ -\mathrm{Im}[X_1]& \omega \end{array}\right]\left[\begin{array}{c}{K}_c\\ K_d\end{array}\right]=\left[\begin{array}{c}\mathrm{Re}[X_2]\\ \mathrm{Im}[X_2]\end{array}\right]$$

(17)

The solution of the linear equation in (17) will provide the parameters of the PD controller.

4. Results and Discussion

In order to investigate the performance of the proposed controller via the 2DoF-IMC technique, six different cases were studied and compared against the newly reported design methods. In this section, the first four cases considered are single-area power systems (SAPSs), and the fifth case is a two-area power system (TAPS). The sixth case is a four-area power system (FAPS). The proposed controller design method via the 2DoF-IMC technique shows superior performance compared to the other design methods.

4.1. Case 1

In order to validate the effectiveness of the proposed method, a non-reheated thermal turbine (NRTT)-based SAPS is considered with the system parameters provided in A1 in Appendix A [14]. The performance indices considered for comparison are the integral of square error (ISE), the peak value in the frequency deviation, and the settling time.

The percent improvement of the proposed method compared to the other design method is defined as:

$$\%\mathrm{Improvement}=\frac{\mathrm{reported}\mathrm{peak}\mathrm{undershoot}-\mathrm{proposed}\mathrm{peak}\mathrm{undershoot}}{\mathrm{reported}\mathrm{peak}\mathrm{undershoot}}\times 100$$

The proposed controllers are designed considering L₁ = 0.1, r₁ = 3, L₂ = 0.1, and n₂ = 3. The PD controller for the load-disturbance response is derived as K_c(s) = 44.72 + 4.1668 s. The obtained controllers are applied in the 2DoF-IMC scheme and a load change as ΔP_d = 0.01 p.u. is considered at t = 1 s in the SAPS. Figure 5 shows the frequency regulation of the nominal system in case 1 and confirms that the proposed method has a smaller undershoot and faster transient response compared to Saxena and Hote [8], Padhan and Majhi [13], and Anwar and Pan [14]. Figure 6 shows the frequency response of the perturbed system (50% increment in K_p and T_p gain) and it is observed that the proposed 2DoF-IMC controller provides a better frequency response compared to Saxena and Hote [8], Padhan and Majhi [13], and Anwar and Pan [14]. A comparative analysis of the nominal model and perturbation in the nominal model is provided in Table 1. From Table 1, it is clearly shown that the proposed method has a better settling time (t_s) and a smaller percent undershoot (US) for nominal as well as perturbed systems.

Figure 5. Frequency deviation response of the nominal system in case 1.

Figure 6. Frequency regulation of the perturbed system in case 1.

Table 1. Comparative analysis of the LFC power system in case 1.

Method	kc	ki	Kd	Nominal Plant				Perturbed Plant
				Under-Shoot (US) (×10−3)	% Improvement in Terms of US Value w.r.t Padhan and Majhi	ts	ISE (×10−5)	Peak Value (×10−3)	ts	ISE (×10−5)
Proposed PD	44.724	-	4.17	−0.23	97.6	1.92	0.19	−0.22	1.98	0.19
Saxena and Hote [8]	66.96	171.6	15.91	−3.4	64.6	6.1	0.14	−3.4	5.1	0.14
Anwar and Pan [14]	1.52	2.50	0.27	−9.02	6.04	2.87	3.5	−9.02	2.87	3.6
Padhan and Majhi [13]	1.49	1.30	0.235	−9.6		4.07	2.6	−9.6	6.90	2.6

4.2. Case 2

The dynamic performance of the proposed 2DoF-IMC method was analyzed by considering a SAPS with a re-heated turbine (RTD) with the system parameter provided in A2 in Appendix A [14]. The frequency regulation response for nominal along with the perturbed system is obtained to verify the efficacy of the proposed controller. The transfer function of RTD is provided with $T_t(s)=\frac{q{t}_{r}s+1}{(t_{r}s+1)(t_{t}s+1)}.$ Where $t_r$ is the time delay constant for charging the reheat section of the thermal boiler and q is the torque developed in the high-pressure section of the thermal turbine.

The desired IMC filter and desired load-disturbance transfer function (LDTF) model is considered with L₁ = 0.3, r₁ = 3, and L₂ = 0.3, n₂ = 3, respectively. The proposed controller via the 2DoF-IMC technique is derived as K_c(s) = 16.0328 + 4.7448 s. The obtained controller performance is observed with a unit step load (i.e., ΔP_d = 0.01 p.u.) at t = 1 s in a SAPS. Figure 7 shows the frequency regulation response of the nominal system in case 2 and confirms that the proposed method has a smaller percent undershoot as well as faster transient frequency response compared against Padhan and Majhi [13] and Anwar and Pan [14]. Figure 8 shows that the frequency regulation of the perturbed system (50% increment in K_p and T_p gain) is approximately the same as the nominal system and shows that the proposed controller via the 2DoF-IMC technique provides a better frequency response against Padhan and Majhi [13] and Anwar and Pan [14]. The comparative analysis of the nominal along with the perturbed system is provided in Table 2. The proposed method has a superior settling time (t_s), integral of square error (ISE), and smaller percent undershoot (%US) for the nominal model and perturbation in the nominal model is provided in Table 2. Frequency regulation in Figure 8 illustrates that the proposed controller utilizing 2DoF-IMC is more robust against the latest reported method.

Figure 7. Frequency regulation of the nominal system in case 2.

Figure 8. Frequency regulation of the perturbed system in case 2.

Table 2. Comparative analysis of the LFC power system in case 2.

Method	kc	ki	Kd	Nominal Plant				Perturbed Plant
				Under-Shoot (US) (×10−3)	% Improvement in Terms of Peak Value w.r.t Tan Padhan and Majhi	ts	ISE (×10−6)	Peak Value (×10−3)	ts	IAE (×10−5)
Proposed PD	16.03	-	4.74	−0.45	95.3	2.45	0.11	−0.48	2.35	1.42
Anwar and Pan [14]	10.6	2.50	2.57	−4.75	50.5	6.80	9.7	−4.76	6.8	0.70
Padhan and Majhi [13]	6.16	1.93	1.16	−9.6		11.8	1.39	−7.4	11.8	1.95

4.3. Case 3

In case 3, the SAPS with a non-reheated thermal turbine (NRTT) is considered to validate the dynamic performance of the proposed controller via the 2DoF-IMC technique and the system parameters are mentioned in A3 in Appendix A [14].

The desired IMC filter and desired LDTF model are considered with L₁ = 0.2, r₁ = 3, and L₂ = 0.2, n₂ = 3, respectively. The proposed controller via the 2DoF-IMC technique is derived as K_c(s) = 274.2 + 54.19 s. The performance of the proposed controller utilizing the 2DoF-IMC technique is observed with a unit step load (i.e., ΔP_d = 0.01 p.u.) at t = 1 s in a SAPS, which is shown in Figure 3. In case 3, the frequency regulation of the nominal system is shown in Figure 9 and shows the smooth and fast response as compared to Anwar and Pan [14] and Gundes and Chow [16]. Figure 10 shows the frequency regulation of +50% perturbation in K_p and T_p gain and confirm that the proposed PD controller provides a much superior frequency response as compared to Anwar and Pan [14] and Gundes and Chow [16]. The detailed analyses of the nominal model and perturbation in the nominal model are mentioned in Table 3. From Table 3, it is clearly shown that the proposed PD controller via 2DoF-IMC has a significantly improved settling time and low undershoot for the nominal and perturbed systems. Frequency regulation in Figure 10 illustrates that the proposed controller utilizing the 2DoF-IMC technique has more robustness against the other design method.

Figure 9. Frequency regulation of the nominal system in case 3.

Figure 10. Frequency regulation of the perturbed system in case 3.

Table 3. Comparative analysis of the LFC power system in case 3.

Method	kc	ki	Kd	Nominal Plant				Perturbed Plant
				Under-Shoot (US)	% Improvement in Terms of Peak Value w.r.t Tan Gundes and Chow	ts	ISE (×10−6)	Peak Value (×10−3)	ts	IAE (×10−5)
Proposed PD	274.2	-	54.19	−0.04	97.68	7.07	0.11	−0.04	7.15	0.01
Anwar and Pan [14]	0.267	2.50	9.70	−1.51	12.7	41.5	9.7	−1.57	41.8	1.17
Gundes and Chow [16]	1.50	1.05	0.44	−1.73		62.6	1.39	−1.81	63	1.8

4.4. Case 4: Single Area Non-Reheat Thermal Power System with GRC

In case 4, the effect of generation rate constraint (GRC) in a SAPS is considered to demonstrate the effectiveness of the proposed controller via the 2DoF-IMC technique. The system parameters are reported in A1 [14]. The stability and reliability are a major concern for the power system. The control of LFC becomes difficult in the presence of non-linearity like GRC. The GRC is the limit on the generation rate within a specified time. It produces oscillation and may lead to system instability. In this case, Figure 11 shows the SAPS with a GRC block, which cascades to the transfer function of the turbine model.

Figure 11. Single-area thermal power system with a GRC block.

In this case, the parameter of GRC is taken as 0.1 p.u./min = 0.0017 p.u./s [14]. The desired IMC filter and load-disturbance transfer function model are considered with L₁ = 0.1, r₁ = 3, and L₂ = 0.1, n₂ = 3, respectively. The proposed controller via the 2DoF-IMC technique is derived as K_c(s) = 44.72 + 4.1668 s. The frequency regulation of the LFC system with GRC is performed satisfactorily, which is shown in Figure 12. The proposed controller via the 2DoF-IMC technique outperforms with consideration of the GRC effect.

Figure 12. Frequency regulation of case 4 with GRC.

4.5. Case 5

In order to validate the performance of the proposed method in case 5, a two-area power system (TAPS) with a non-reheated thermal turbine (NRTT) was considered and the system parameter is reported in A4 in Appendix A [17]. Five different cases of TAPSs with NRTT are discussed in this section. In sub-case 5, frequency regulation and tie-line power interchange between two areas were observed to verify the efficacy of the proposed PD controller.

The desired IMC filter and desired LDTF model were considered with L₁ = 0.2, r₁ = 3, and L₂ = 0.2, n₂ = 3, respectively. The proposed controller via the 2DoF-IMC technique was derived for areas 1 and 2 as K_c1(s) = K_c2(s) = 24.508 + 4.4225 s, which was applied for six sub-cases in a TAPS.

4.6. Case 5a: 10% Step Load in Area 1 and Zero Percent in Area 2

In the case of 5a, the 10% step load in area 1 and zero percent in area 2 were considered to demonstrate the performance of the simulation results in the TAPS. The system parameter for the TAPS is reported in A4 in Appendix A. Figure 13 shows that the proposed controller via the 2DoF-IMC technique has much better performance than the other reported literature such as Anwar and Pan [14] and K. Lu et al. [17]. The proposed controller utilizing the 2DoF-IMC approach has small frequency regulation as well as smoother transient speed. A detailed analysis of the comparison performance in case 5a is mentioned in Table 4. Table 4 concludes that the settling time and percent undershoot in area 1, area 2, and tie-line power are superior to Anwar and Pan [14] and K. Lu et al. [17]. Similarly, the change in frequency deviation is small in area 2 and the settling time is also better than K. Lu et al. [17] and Anwar and Pan [14], which is shown in Figure 14. The tie-line power exchange between two areas is smoother and faster, which is shown in Figure 15. To analyze the quality of the proposed method, the performance indices’ integral of square error (ISE) is small, which indicates a better control scheme in terms of performance response.

Figure 13. Frequency regulation of area 1 in case 5a.

Table 4. Comparative analysis of the LFC power system in case 5a.

Method	Kp	Ki	Kd	Area 1 $\mathbf{\Delta }{\mathit{f}}_1$			Area 2 $\mathbf{\Delta }{\mathit{f}}_2$		Tie-Line Power $\mathbf{\Delta }{\mathit{p}}_{\mathit{t}\mathit{i}\mathit{e}}$
				%US (×10−2)	ts	ISE (×10−3)	%US (×10−2)	ts	% US (×10−3)	ts
Proposed PD	24.5	-	4.425	−0.22	0	0.45	−0.05	0	0.70	0
Anwar and Pan [14]	3.55	5.95	1.22	−6.4	0.44	0.33	−2.5	0.69	−8.42	0.4
Tan [4]	1.569	2.397	0.525	−25.4	3.14	33	−0.22	4.26	−0.078	3.64

Figure 14. Frequency regulation of area 2 in case 5a.

Figure 15. Tie-line power interchange in case 5a.

4.7. Case 5b: 10% Step Load in Area 2 and Zero Percent in Area 1

In case 5b, 10% step load in area 2 and zero percent in area 1 were considered to describe the dynamic performance of the simulation results in the TAPS. The system parameter for the TAPS is provided in A4 in Appendix A. Figure 16 shows that the proposed controller via the 2DoF-IMC technique has much better performance than the other reported literature such as Anwar and Pan [14] and K. Lu et al. [17]. The proposed controller utilizing the 2DoF-IMC technique has a small frequency deviation as well as smoother transient speed. Similarly, the change in frequency difference is small in area 2 and settling time is also better than Anwar and Pan [14] and K. Lu et al. [17], which is shown in Figure 17. The tie-line power exchange between two areas is smoother and faster, which is shown in Figure 18.

Figure 16. Deviation in frequency of area 1 in case 5b.

Figure 17. Deviation in frequency of area 2 in case 5b.

Figure 18. Tie-line power interchange in case 5b.

4.8. Case 5c: 10% Step Load in Both Areas

In case 5c, the 10% step load was considered in area 1 and area 2 to illustrate the dynamic performance of the TAPS. The system parameters are provided in A4 in Appendix A. In case 5c, the deviations in frequency of area 1 and area 2 are the same. Figure 19 shows that the proposed controller via the 2DoF-IMC technique has a smoother and better response than Anwar and Pan [14] and K. Lu et al. [17].

Figure 19. Deviation in frequency of area 1 in case 5c.

4.9. Case 5d: 50% Change in t_g

To validate the robustness of the proposed controller via the 2DoF-IMC technique the time constant of the generator T_g was varied. In case 5d, ±50% changes in T_g were considered and from Figure 20 it was observed that frequency deviation is the same as the nominal model in the perturbed system. Therefore, we can easily conclude that the proposed controller utilizing the 2DoF-IMC technique is more robust.

Figure 20. Deviation in frequency of area 1 in case 5d.

4.10. Case 5e: 50% Change in T_t

To validate the robustness of the proposed controller via the 2DoF-IMC technique the time constant of the turbine in the synchronous generator T_t was changed. In case 5e, ±50% changes in T_t were considered. From Figure 21, it was observed that the frequency deviation lies within the specified limit in the perturbed system. Therefore, we can easily conclude that the proposed controller utilizing the 2DoF-IMC technique is more robust.

Figure 21. Deviation in frequency of area 1 in case 5e.

4.11. Case 5f: Random Loading in the TAPS

To verify the robustness of the proposed controller via the 2DoF-IMC technique random step loads at different times in the TAPS were used, which is shown in Figure 22. In case 5f, from Figure 23 and Figure 24 it was observed that the frequency deviation still lies within the specified limit in the perturbed system. Figure 25 shows the tie-line power exchange within a specified limit under the random loading pattern. Therefore, we can easily conclude that the proposed controller utilizing the 2DoF-IMC technique is more robust.

Figure 22. Random loading pattern of case 5f.

Figure 23. Deviation in frequency of area 1 in case 5f.

Figure 24. Deviation in frequency of area 2 in case 5f.

Figure 25. Tie-line power interchange in case 5f.

4.12. Case 6

In case 6, the TAPS LFC problem is extended to a four-area power system (FAPS) in order to investigate the dynamic performance of the proposed design method. Figure 26 shows the structure of the four-area power system (FAPS), Area No. 1, 2, and 3 are from the non-reheated thermal turbine (NRTT), which are interconnected to others, while Area No. 4 is also from the NRTT, which is connected to area No. 1. The typical system parameters are provided in A5 in Appendix A [18]. In case 5, frequency regulation and tie-line power interchange between other areas are investigated to verify the efficacy of the proposed controller via the 2DoF-IMC technique.

Figure 26. Four-area thermal power system.

The desired IMC filter and LDTF model are considered with L₁ = 0.3, r₁ = 3, and L₂ = 0.3, n = 3, respectively, for the FAPS. Regarding the proposed controller via the 2DoF-IMC technique, areas 1, 2, 3, and 4 are derived as K_c1(s) = 11.9489 + 3.285s, K_c2(s) = 6.13 + 1.5755 s, K_c3(s) = 10.8148 + 3.0035 s, and K_c4(s) = 7.06 + 2.204 s, respectively.

The step-change in load is ($\Delta P_{L1}=\Delta P_{L2}=0.01 \mathrm{p}.\mathrm{u}.MW$) for areas 1 and 3 at t = 0 s and t = 20 s, respectively. From a critical analysis of Figure 27, it was observed that the proposed controller via the 2DoF-IMC technique has better frequency regulation in each area compared with Cai et al. [18]. Figure 28 confirms that the proposed controller has a faster response in tie-line power exchange between other areas and its controller performance is much better than Cai at al. [18]. Obviously, the proposed PD controller outperforms Cai et al. [18].

Figure 27. Deviation in frequency of each area in case 6.

Figure 28. Tie-line power exchange of each area in case 6.

5. Conclusions

The two degrees of freedom internal model control (2DoF-IMC) scheme was used to design load-frequency controllers for the SAPS, TAPS, and FAPS. The proposed method improves the disturbance rejection problem associated with LFC and performs well under consideration of GRC and parametric uncertainty. The robustness of the proposed method was demonstrated against uncertainty in system parameters, random loading patterns, and GRC. In this paper, various cases were considered to approve the viability and performance of the proposed method by reducing the percent undershoot, settling time, and ISE in the frequency regulation response. Overall, the proposed controller has a small percent undershoot as well as fast response in frequency regulation and tie-lie power exchange compared to the latest existing method.