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Designing of robust frequency stabilization using optimized MPC-(1+PIDN) controller for high order interconnected renewable energy based power systems

Abstract

The challenge of controlling frequency becomes greater as the complexity of a power network increases. The stability of a power system is highly dependent upon the robustness of the controller. This paper presents automatic generation control (AGC) of a four-area interconnected power system along with integrated renewable energy sources of PV and wind energy. The designed model is a challenge given the increased penetration levels of PV and wind along with a thermal-hydropower system. The addition of a hydropower system as a fourth type results in the pole of the open loop system of the hydropower system being located at the right half side of the s-plan. This demands a robust control. A novel MPC-(1 + PIDN) is designed for high-order interconnected areas (HOIA) to stabilize the frequency in a robust way. The salp swarm algorithm is adopted to optimize the parameters of the PIDN controller. The performance of the proposed controller under HOIA is tested in a unbalanced load environment with uncertainty in the power system. The proposed controller can effectively handle the frequency disruption by stabilizing it in \(0.86 s\) for Area-1, \(1.08 s\) for Area-2, \(0.81 s\) for Area-3, and \(0.84 s\) for Area-4 with an average time of \(0.89 s\) for all the areas, whereas the average time for GWO: PI-PD, MPC/PI and GA-PI is \(3.48 s\), \(10.36 s\) and \(18.47 s\), respectively. The results demonstrate the effectiveness of the controller when compared to other controllers.

1 Introduction

With soaring energy demand, renewable energy sources (RES) are becoming increasingly important. However, the integration of RES into the power system potentially increases frequency degradation. This needs to be countered by optimal controllers. Load frequency control (LFC) has been an important research area for stable operation of power systems [1]. The fluctuation in frequency is mainly due to imbalances in load and generation, and nonlinear factors like generation rate constraints (GRC) [2]. The foremost focus of the research is to ensure the stable operation of the generating power system by stabilizing the frequency. The role of LFC or automatic generation control (AGC) is to control the frequency and interchange of power over a tie-line among the interconnected areas to balance the load and generation [3, 4].

1.1 Literature review

LFC has played an important role in confining the system frequency to a pre-specified limit, and has become a hot research topic because of the increasing challenges of the power system and the need for a robust controller. Studies examine the performance of the proposed controller in a highly complex network. There are many controllers for the LFC problem in the literature. PI/PID controllers are widely adopted for frequency stability. They are optimized by several algorithms, e.g., grey wolf optimization, but for complex power networks they exhibit sluggish response [5]. Different meta-heuristic techniques have been applied to different controllers to extract optimal tunable parameters for the controllers. The commonly applied algorithms in the field of AGC controller design are the bacterial foraging optimization algorithm (BFOA) [6], firefly (FA) [7], genetic (GA) [8], and others algorithms. In addition, other control formations have bee to improve effectiveness. In [9], an integral double derivative (IDD) controller is used for LFC with the help of a lighting searching algorithm (LSA). However, the addition of an extra derivative adds oscillations in the frequency response. The integral order (IO) [10], intelligent [11], and degree of freedom-based controllers which include 2DOF and the new 2DOF [12, 13], have been implemented in the field of LFC to enhance the performance of AGC.

For multi-area power systems, cascaded controllers are applied for LFC [14]. A GWO-based PI-PD controller is implemented, which has shown effective controllability [15], while the bat algorithm (BA)-tuned PD-PID is used for an electric vehicle to subdue frequency oscillations [16]. The Harris hawks optimization algorithm for PD-PID is applied in [17]. It shows promising results but the complexity of the algorithm is high. The hybrid stochastic fractal search and pattern search technique (hSFS-PS) for PI-PD controller is designed to enhance the controlling ability of the controller in [18]. The grasshopper algorithm (GOA) is used to optimize a PI-(1 + PD) controller for a multi-area power system in [19], while a multi-verse optimizing technique for PID + DD is implemented to monitor the usefulness of the controller in [20], but the complexity of the system is compromised. The aforementioned techniques have shown efficient results, but the testing system was fairly simplified and was mostly limited to a two-area system.

To overcome the limitations in the above controllers, several studies have proposed fuzzy/adaptive fuzzy-based PI/PID controller tuning but the attachment of a fuzzy controller makes it cumbersome [21, 22]. In [23], an internal model controller (IMC)-based PI controller is developed to cope with the LFC issue, while in [24], a dropout deep neural network with transfer learning technique is applied to ensure the optimal dispatch of AGC. Another dispatch technique, incorporating mileage, total generation and cost based on an MPC framework is proposed in [25], to maximize the Genco total profit by dynamic variation in the AGC. Likewise, a model predictive controller (MPC), which predicts the future behavior each time by minimizing the cost function, is the most extensively used controller in industry [26]. The importance of MPC is due to its fast response and robustness against the load disruption and uncertainty. The use of an MPC can be found in [27], where the MPC is deployed for a two-area thermal-thermal system and for PV-connected thermal systems. However, its performance is not analyzed in a complex network The MPC presents low overshoot with small steady-state error. The performance of MPC is tested for a plug-in hybrid electric vehicle along with a wind system in [28], while an adaptive MPC controller is also used for a PV-thermal two-area system to handle frequency fluctuations in a promising way [29]. In recent research, MPC-PI gain scheduling has been proposed for a renewable-based thermal interconnected two-area system for flexible operation of a power generating system [30]. Adaptive AGC mileage dispatch is proposed with renewable energy integration for better frequency regulation [31]. In [32], a four-area hydro-thermal design is proposed with the intelligent control design of PID, while a Fuzzy-PI controller with the integration of RES is proposed in [33] but the complexity of the studied network is not high. Active disturbance rejection control (LADRC) is designed for a four-area system to solve the problem of LFC, with a hydro-thermal area contained in the model [34].

1.2 Limitations of previous work and aims of the present study

The detailed literature review has indicated the following areas that need to be further investigated and thus has led to the proposed scheme in this paper:

  1. (1)

    Conventional control techniques, e.g., PI/PID controllers, exhibit difficulties in reducing the uncertainty in the system. To improve the performance of the controller, a cascaded formation design has been introduced to overcome the challenge of system stability. However, the cascaded formations of the controller lack the ability to handle uncertainty robustly.

  2. (2)

    Little attention has been devoted to analyzing the high complexity of the power system network, i.e., higher-order areas, and significant gaps exist in the analysis and performance of the controller designs for high-order areas.

  3. (3)

    The newly formulated design MPC-(1 + PIDN) is devoted to enhancing the control ability of the previous (1 + PID) controller. The aim is to alleviate the frequency fluctuations robustly by applying the combination of prediction and conventional control techniques mainly for the higher-order area-based power system.

1.3 Innovation and contribution

The limitations in previous studies have paved the way for the following innovations in this paper:

  1. (1)

    Formulation of high-order interconnected areas (HOIA), e.g., a combination of four areas, where the complexity of the area is enlarged by penetrating RES in the power system.

  2. (2)

    For the HOIA system, the MPC-(1 + PIDN) controller is designed whereby the tunable parameters of the controller are obtained from SSA. The cascaded design applies the property of prediction based on the system constraints. These switch the operating state of the 1 + PIDN controller.

  3. (3)

    The additional advantages of this type of controller as compared to PID are its robustness and ability to handle constraints while countering the noise.

This rest of the paper is organized as follows: Sect. 2 describes the system modeling, in which detailed modelling of wind, PV, thermal power system and hydropower system is established. Section 3 details the design of the proposed MPC-(1 + PIDN) controller, while in Sect. 4, the simulation results of the proposed technique under a complex system model are shown. Section 5 concludes the work.

2 System under investigation

Generally, frequency instability is due to the difference in demand and active power generation. The LFC mechanism is used to maintain the balance between load and generation, while the model of LFC is characterized by uncertainty and nonlinearity. The four-area power system model is depicted in Fig. 1, where the power-sharing among the areas takes place through tie-lines. The model contains three thermal areas along with PV and wind, while the fourth area contains a hydropower system and is also fed with PV and wind.

Fig. 1
figure 1

Four-area power system model under investigation

2.1 Thermal system modeling

The thermal generator consists of a governor, generator, steam turbine and re-heater. It also comprises of nonlinearity factors like generation rate constraint (GRC) and droop [34]. The governor controls the mechanical motion of the turbine while maintaining valve operation and recovering the re-heater lost energy. Droop performs the function of controlling the speed of the prime-mover that is attached to the generator which converts mechanical energy into electrical energy [35]. The installed system is rated at \(2000\mathrm{ MW}\) where the nominal operational load is \(1000\mathrm{ MW}\) in each area. The assumed base power is \(1000\mathrm{ MVA}\).

The governor output \(\Delta {P}_{gi}\left(s\right)\) is the difference of the reference power \(\Delta {P}_{ref}\) and frequency change \({\Delta f}_{i}(s)\) passed through droop control \(\frac{1}{R}\), as:

$${\Delta P}_{gi}\left(s\right)={\Delta P}_{ref}\left(s\right)-\frac{1}{R}{\Delta f}_{i}(s)$$
(1)

Power transfer between the areas (i and j) can be summarized as:

$${\Delta P}_{tieij}=\sum_{j=1}^{n}\frac{{T}_{ij}}{S}\left({\Delta f}_{i}-{\Delta f}_{j}\right) i\ne j$$
(2)

Area central error (ACE) is stated as:

$${ACE}_{i}={B}_{i}{\Delta f}_{i}+{\Delta P}_{tie,ij} i \ne j$$
(3)

where \({B}_{i}\) is the frequency bias factor parameter in p.u. of MW/Hz, and \({\Delta P}_{tie,ij}\) is the change of tie-line power between the two areas.

Output \({\Delta f}_{i}\) is expressed as:

$$\Delta f_{i} \left( s \right) = G_{p} \left( s \right){ }\left[ {\mathop \sum \limits_{j = 1}^{n} \Delta P_{Rij} - \Delta P_{d,i} - \Delta P_{tie,i} } \right]$$
(4)

where \({G}_{p}\left(s\right)=\frac{1}{{M}_{i}s+{D}_{i}}\)

2.2 Hydro generator modeling

The fourth area comprises of a hydropower plant along with PV and wind energy resources. The hydropower plant contributes to meeting the energy demand of the overall power generating system, while the PV and wind also contribute to meeting the energy demand. The nominal operation load on the hydro power plant is \(1000\mathrm{ MW}\), while the installed capacity of the hydro power plant is \(2000 \mathrm{MW}\). The hydropower plant exchanges power with Area-1 that is further connected with Area-2 and Area-3.

The hydro turbine transfer function is expressed as:

$${G}_{t}\left(s\right)=\frac{{K}_{r}{T}_{r}s+1}{{T}_{r}s+1}.\frac{-{T}_{w}s+1}{{0.5T}_{w}s+1}$$
(5)

For Area-4, the open loop transfer function can be written as:

$$G(s)=\frac{{K}_{p4}({T}_{2}s+1)(-{T}_{w}s+1)}{{({T}_{1}s+1)({T}_{3}s+1)({T}_{4}s+1)(0.5T}_{w}s+1)}$$
(6)

Figure 2 shows the pole-zero mapping of Area-4 as given in (6), with the zero on the right half plane making the system more difficult to control. The design of the robust controller is thus crucial for controlling the overall four-area system.

Fig. 2
figure 2

Pole-zero mapping of the fourth area

The model symbols along with their values are listed in Table 1

Table 1 Model symbols

2.3 Photovoltaic system

The complexity of the power system in the four areas further increases because of PV penetration. PV is a DC nonlinear output source and is highly dependent on sun irradiation.

To connect PV generation to the power system, several conversions units are used. Figure 3 shows the designed PV model. These subsystems ensure that the PV runs at maximum power and transforms DC power into AC [36].

Fig. 3
figure 3

PV system modeling

In the design, it is considered that PV is operating at its maximum power point [37]. The transfer function of the boost converter is given as:

$${\mathrm{g}}_{1}(\mathrm{s})=\frac{1}{{\mathrm{m}}_{1}}$$
(7)

where \({\mathrm{m}}_{1}\) is:

$${\mathrm{m}}_{1}=\frac{{\mathrm{v}}_{2}}{{\mathrm{v}}_{1}}=\frac{{\mathrm{i}}_{1}}{{\mathrm{i}}_{2}}$$
(8)

Similarly, the inverter transfer function is denoted as:

$${\mathrm{g}}_{2}\left(\mathrm{s}\right)=\frac{{\mathrm{I}}_{\mathrm{ac}}\left(\mathrm{s}\right)}{{\mathrm{I}}_{2}(\mathrm{s})}=\frac{{\mathrm{s}}^{2}}{{\mathrm{s}}^{2}+{\upomega }^{2}}$$
(9)

where \(\omega =2\pi f=2\pi \left(50\right)=314.12rad/sec.\)

The transfer function of the instantaneous power is given as:

$$\mathrm{P}\left(\mathrm{s}\right)=\frac{{\mathrm{v}}_{\mathrm{m}}{\mathrm{i}}_{\mathrm{m}} }{2\mathrm{s}}+ \frac{{\mathrm{v}}_{\mathrm{m}}{\mathrm{i}}_{\mathrm{m}}}{2} \frac{\mathrm{s}}{{\mathrm{s}}^{2}+({2\upomega )}^{2}}$$
(10)

and the gain of the instantaneous power is provided as:

$${\mathrm{g}}_{3}\left(\mathrm{s}\right)=\frac{\mathrm{p}(\mathrm{s})}{{\mathrm{I}}_{\mathrm{ac}}(\mathrm{s})}={\mathrm{v}}_{\mathrm{m}}\left(\frac{\left({\mathrm{s}}^{2}+{\upomega }^{2}\right)\left({\mathrm{s}}^{2}+({2\upomega )}^{2}\right)}{{\mathrm{s}}^{2}({\mathrm{s}}^{2}+({4\upomega )}^{2})}\right)$$
(11)

The average power is expressed as:

$${p}_{avg}\left(s\right)= \frac{{v}_{m}{i}_{m}}{2s}$$
(12)

and the average power gain is described as:

$${g}_{4}\left(s\right)=\frac{{p}_{avg}(s)}{p(s)}$$
(13)

2.4 Wind system

Wind energy meets the energy requirement via sharing the power through a tie line in an interconnected area. However, variation in the wind speed causes power imbalance and imposes disruption in frequency from the nominal value. In general, the interconnection or penetration of wind energy into the power system causes the LFC issue [38]. The modeling of the wind system is carried out and its schematic structure is depicted in Fig. 4. The mechanical output power of the wind turbine (WT) including the wake effect is expressed as [39]:

$${P}_{w}=\frac{1}{2}\rho {A}_{T}{C}_{p}(\lambda ,\beta ){V}_{wake}^{3}$$
(14)

where

Fig. 4
figure 4

Wind energy conversion model

$$\left\{\begin{array}{c}{V}_{wake}={V}_{m}\left[1-\sum_{i=1}^{n}\left(1-\sqrt{\left(1-\frac{{F}_{T}}{\rho \pi {r}^{2}{V}_{m}^{2}}\right)}\right){\left(\frac{r}{r+k\Delta x}\right)}^{2}\right]\\ {C}_{p}\left(\lambda ,\beta \right)={C}_{1}\left(\frac{{C}_{2}}{{C}_{I}}-{C}_{3}\beta -{C}_{4}\right){e}^{\frac{-{C}_{5}}{{\lambda }_{I}}}+{C}_{6}{\lambda }_{T}\\ \frac{1}{{\lambda }_{I}}=\frac{1}{\lambda +0.08\beta }-\frac{0.035}{{\beta }^{3}+1}; {\lambda }_{T}=\frac{{w}_{r}r}{{V}_{w}}; {A}_{T}=\frac{\pi {d}^{2}}{4}\end{array}\right.$$

\({C}_{p}\) is the power coefficient, \(\beta\) is the blade pitch angle, and \(\rho\) denotes the air density (\(\frac{Kg}{{m}^{3}}\)). The rotor diameter is represented by d (m), and the tip speed ratio (TSR) is denoted by \({\lambda }_{T}\), and \({\lambda }_{I}\) is the intermittent TSR. \({F}_{T}\) is the trust force, k denotes the decreasing coefficient (for onshore WT, k = 0.075, and for offshore WT, k = 0.05), \(\Delta x\) is the distance between two consecutive WTs, and \({V}_{w}\) is the effective wind velocity.

The two-mass model of gear coupling is represented as [40]:

$$\left\{\begin{array}{c}{\dot{\omega }}_{WT}=-\left(\frac{{D}_{WT}+{D}_{shaft}}{{J}_{WT}}\right){\omega }_{r,WT}+\left(\frac{{D}_{shaft}}{{J}_{WT}}\right){\omega }_{gen}-\left(\frac{1}{{J}_{WT}}\right){T}_{int}\\ +\left(\frac{1}{{J}_{WT}}\right)d(t)\\ {\dot{\omega }}_{gen}=\left(\frac{{D}_{shaft}}{{2J}_{WT}}\right){\omega }_{r,WT}+\left(\frac{{D}_{gen}+{D}_{shaft}}{{J}_{gen}}\right){\omega }_{gen}-\left(\frac{1}{{{n}_{gear}J}_{gen}}\right){T}_{int}\\ +\left(\frac{1}{{J}_{gen}}\right)u(t)\\ {\dot{T}}_{int}={K}_{shaft}\left({\omega }_{WT}-\frac{{\omega }_{gen}}{{n}_{gear}}\right)\end{array}\right.$$
(15)

where \({J}_{WT}\) and \({J}_{gen}\) represent WT’s moment of inertia of the rotor and generator, respectively. Shaft internal torque is represented by \({T}_{int}\), WT rotor and generator speed is denoted by \({\omega }_{r,WT}\), and gear turns ratio is represented by \({n}_{gear}.\) \({D}_{WT}\), \({D}_{shaft}\) and \({D}_{gen}\) are the damping constant of the WT, coupling shaft and generator, respectively.

Assuming the output power \({P}_{w}\) is ideal and the generator is lossless, thus there are [40]:

$${P}_{w}={T}_{gen}{\omega }_{g}$$
(16)
$$\frac{d{T}_{gen}}{dt}=\left(-\frac{1}{{\tau }_{gen}}\right){T}_{gen}+\left(\frac{1}{{\tau }_{gen}}\right){T}_{contol}$$
(17)

where \({\tau }_{gen}\) denotes the time constant of the generator, and \({T}_{contol}\) represents the control torque that manipulates the output power of the WT against uncertain wind speed.

3 Proposed control strategy

The control operation of the four-area hybrid interconnected power system is designed. A robust control strategy is proposed where a cascaded design is formulated as MPC-(1 + PIDN) to deal with the LFC problem, originating from a mismatch in load and generation. The proposed controller contains the MPC that handles the constraints and minimizes the cost function robustly, while the (1 + PIDN) controller has the property of further expediting the controlling process by minimizing the settling time and handling disruption in frequency effectively. The MPC-(1 + PIDN) control structure is depicted in Fig. 5.

Fig. 5
figure 5

MPC-(1 + PIDN) controller design

3.1 Design of MPC

The MPC controller is widely used in industry because of its robustness and was first introduced in 1978 [41]. In an MPC model, there are mainly three blocks, i.e.: predictive block, optimization solver, and cost function block. The function of these blocks is to shift the system response towards the reference trajectory based on the past and future predictions.

The output predictive vector is represented by \({Y}_{p}(k)\) as:

$${Y}_{p}\left(k\right)=\phi Z\left(k\right)+\psi \Delta U\left(k\right)+{\psi }_{I}\Delta {U}_{I}(k)$$
(18)

The control law is obtained by the gradient descent method, i.e., \(\frac{\partial J(k)}{\Delta U\left(k\right)}=0\). To obtain the control law \(u\left(k\right)\) the following equations are applied:

$$\Delta U\left(k\right)={\left({\psi }^{T}Q\psi +R\right)}^{-1}{\psi }^{T}Q\left({Y}_{r}\left(k\right)-\phi Z\left(k\right){-\psi }_{I}\Delta {U}_{I}\left(k\right)\right)$$
(19)
$$\Delta u\left(k\right)=\left({E}_{{N}_{u}} {O}_{{N}_{u}\left(P-1\right)}\right)\Delta U\left(k\right)$$
(20)
$$u\left(k\right)=\Delta u\left(k\right)+u\left(k-1\right)$$
(21)

In the design of the cost function for the LFC problem, the constraint for the MPC is expressed as:

$$minJ\left(k\right)=min\left\{{\left({Y}_{p}\left(k\right)-{Y}_{r}\left(k\right)\right)}^{T}Q\left({Y}_{p}\left(k\right)-{Y}_{r}\left(k\right)\right)+{\left(\Delta u\left(k\right)\right)}^{T}R(\Delta u\left(k\right))\right\}$$
(22)

The parameters Q and R are the weighting vectors to balance the square future control and performance predictive error.

3.2 Design of (1 + PIDN) and optimized by the Salp swarm algorithm

The modified structure of the PID controller with a modified (1 + PIDN) is implemented. The gains of the (1 + PIDN) controller, i.e., \({K}_{P}\),\({K}_{I}\),\({K}_{D}\), are optimized by the salp swarm algorithm (SSA). The control signal \(u\left(k\right)\) that is the output of the MPC is fed to (1 + PIDN), and the (1 + PIDN) controller processes the optimal signal of the MPC to further refine the signal to achieve the optimal control output for the LFC problem, as:

$$output={(1+K}_{P}+\frac{{K}_{I}}{s}+\frac{{K}_{D}Ns}{N+s})\times u\left(k\right)$$
(23)

where \(u\left(k\right)\) is the output signal of the MPC.

3.2.1 Salp swarm algorithm (SSA)

The salp swarm optimization algorithm is designed from the transparent bodied slap which transforms itself into a chain-like structure during the search for food as proposed in [42]. SSA is mainly comprised of two sections: (i) leader and (ii) followers. The algorithm is translated in the form of a block diagram shown in Fig. 6.

Fig. 6
figure 6

Block diagram of SSA

The leader movement of the SSA is decided by:

$${K}_{j}^{i}=\left\{\begin{array}{c}{M}_{i}+{C}_{1}\left(\left({ub}_{j}-{lb}_{j}\right){C}_{2}+{lb}_{j}\right), {C}_{3}\ge 0\\ {M}_{i}-{C}_{1}\left(\left({ub}_{j}-{lb}_{j}\right){C}_{2}+{lb}_{j}\right), {C}_{3}<0\end{array}\right.$$
(24)

where \(\mathrm{M}\) and \(\mathrm{K}\) signify the target food and two-dimensional salp position, respectively. \({ub}_{j}\) and \({lb}_{j}\) denote the respective upper and lower bounds of the \({j}^{th}\) dimension. Similarly, \({C}_{2}\) and \({C}_{3}\) are uniform coefficient numbers and \({C}_{1}\) is further expressed as:

$${C}_{1}={2e}^{-{\left(\frac{4t}{{t}_{max}}\right)}^{2}}$$
(25)

where t denotes the current iteration and \({t}_{max}\) signifies the maximum number of iterations. The position is updated by:

$${K}_{j}^{i}=\frac{1}{2}{at}^{2}+{v}_{0}t$$
(26)

4 Results and discussion

This section presents the effectiveness of the proposed control technique under HOA simulated in MATLAB 2017a, using a CoreTMi7 CPU at 2.2 GHz and 8 GB of RAM.

The initialization of the algorithms of GA, GWO, SSA, and MPC details are shown in Table 2. Moreover, the optimized controller parameters for the four-area based power system is formulated in Table 3.

Table 2 Algorithms parameters setting
Table 3 Optimized controller parameters

After applying the setting in the simulation environment the performance of the proposed controller is simulated by applying the load perturbation as shown in Fig. 7. The load fluctuates between \(1\) and \(5\mathrm{ \%}\), where \(5\mathrm{\%}\) load is considered to be the worse load changing condition.

Fig. 7
figure 7

Abrupt load deviation presentation

The frequency responses for Area-1, Area-2, Area-3 and Area-4 are depicted in Figs. 8, 9, 10 and 11, respectively. The proposed controller MPC-(1 + PIDN) shows robust results compared to other controllers, as it can effectively handle the frequency disruption by stabilizing under unbalanced loading. The highlighted parts of Figs. 8 and 9 are further explained to provide a better view of the responses obtained under the applied load condition. The effectiveness of the proposed controller on the fourth area which has a zero on the right half plane requiring a robust controller for the handling of such issue, is also evident from Fig. 11. Overall, the proposed controller provides optimal results and handles the complexity of the four-area system effectively.

Fig. 8
figure 8

Area-1 frequency response of the controller

Fig. 9
figure 9

Area-2 frequency response of the controller

Fig. 10
figure 10

Area-3 frequency response of the controller

Fig. 11
figure 11

Area-4 frequency response of the controller

The results are compiled to analyze the effectiveness of the proposed controller and to compare with other state-of-the-art controllers as listed in Table 4. The deviations in frequency are plotted in Fig. 12. As seen, the proposed controller shows a minimum frequency deviation that is close to zero, while the other controllers have significant deviations. The tie-line power-sharing response of the controller is shown in Figs. 13, 14, 15, 16. The results indicate that the proposed controller shares the power effectively among the areas, while controllers such as FA-PI show an ineffective response.

Table 4 Performance analysis of the different controllers
Fig. 12
figure 12

Frequency deviation of the controllers

Fig. 13
figure 13

Tie-line_12 response

Fig. 14
figure 14

Tie-line_23 response

Fig. 15
figure 15

Tie-line_13 response

Fig. 16
figure 16

Tie-line_14 response

The response time (ST) of the proposed MPC-(1 + PIDN) controller is \(0.86 \mathrm{s}\) and has an undershoot response (US) of \(-0.001\) for Area-1. In comparison, the response of GWO: PI-PD controller is \(3.3 \mathrm{s}\) while also exhibiting US and an overshoot (OS) of \(-\hspace{0.17em}0.03\) and \(0.01\) respectively in Area-1. Similarly, the response of MPC/PI presents a settling time (ST) of \(9.57 \mathrm{s}\) and a US response of \(-\hspace{0.17em}0.03\) with no OS. For Area-1, the response manifested by GA-PI is \(17.87\) s to settle the frequency fluctuation, while US and OS are \(-\hspace{0.17em}0.18\) and \(0.01\), respectively.

In the case of Area-2, the proposed controller shows an ST of \(1.08 s\) and OS of \(0.01\) with no US. In comparison, the ST, US and OS of GWO: PI-Pd are \(2.93 s\), \(0\) and \(0.05\), respectively. In the case of the MPC/PI controller, the ST is \(9.57 s\), US is \(-\hspace{0.17em}0.03\) with no OS. The GA-PI controller shows ST of \(20.1 s\) and the responses of US and OS are \(-\hspace{0.17em}0.01\) and \(0.18\), respectively.

The simulated response of the MPC-(1 + PIDN) in Area-3 shows an ST of 0.81 s, US of \(-\hspace{0.17em}0.01\) with no OS. Similarly, in Area-3, the ST responses of GWO: PI-PD, MPC/PI and GA-PI are \(4.45 s\), \(9.76 s\) and \(17.93 s\), respectively. The US of the GWO: PI-PD, MPC/PI and GA-PI are \(-\hspace{0.17em}0.04\), \(-\hspace{0.17em}0.02\) and \(-\hspace{0.17em}0.18\), respectively. Further, the OS response of GWO: PI-PD is \(0.01\), while the MPC/PI presents no OS and GA-PI exhibits \(0.01\) OS response.

Finally, the responses of the MPC- (1 + PIDN), GWO: PI-PD, MPC/PI and GA-PI in Area-4 show STs of \(0.84\) s,\(3.26 s\), \(13.5 s\) and \(18 s\), while the US are \(0\), \(-\hspace{0.17em}0.02\), \(-\hspace{0.17em}0.006\) and \(-\hspace{0.17em}0.12\), respectively. In case of the OS responses, the MPC- (1 + PIDN), GWO: PI-PD, MPC/PI and GA-PI depict \(0.02\), \(0.06\), \(0.05\), and \(0.24\), respectively.

The competence of the MPC–(1 + PIDN) controller is further tested by applying a random changing load between 1 and 6% variations as shown in Fig. 17. This load imbalance causes frequency distortion and leads to issues of system stability. As seen, the proposed controller demonstrates its robustness by suppressing the frequency fluctuation in minimum time compared to other controllers. The frequency responses in Areas1, 2, 3 and 4 are shown in Figs. 18, 19, 20, 21, respectively. The power between the adjoining areas is shared through a tie-line and the responses of the proposed controller show efficient sharing of power as sown in Figs. 22, 23, 24, 25.

Fig. 17
figure 17

Random load variation

Fig. 18
figure 18

Area-1 frequency response of the controller

Fig. 19
figure 19

Area-2 frequency response of the controller

Fig. 20
figure 20

Area-3 frequency response of the controller

Fig. 21
figure 21

Area-4 frequency response of the controller

Fig. 22
figure 22

Tie-line_12 response

Fig. 23
figure 23

Tie-line_23 response

Fig. 24
figure 24

Tie-line_13 response

Fig. 25
figure 25

Tie-line_14 response

The accumulations of the power from different sources such as PV and wind into the grid are shown in Fig. 26A and B, respectively, whereas the collective power responses of PV and wind into thermal and hydropower plants are shown in Fig. 26C and D, respectively.

Fig. 26
figure 26

Collective output power response A PV B wind C with thermal D with hydro

5 Conclusion

The research aimed to design a control strategy for a high-order interconnected area (HOIA). The complexity of the HOIA is enhanced by penetrating the RES into the power system. The addition of the fourth area of a hydro-power plant causes a zero of the open loop system into the right half plane, and this makes control difficult. To handle such a complex network, a robust controller is required to suppress the frequency disruption due to load unbalance. A cascaded design with a combination of an MPC and 1 + PIDN is proposed. The designed controller MPC-(1 + PIDN) helps deal with the requirement of the HOIA system in a complex structure. It also mitigates the frequency fluctuation during load imbalance in less than \(0.86 s\) as compared to other controllers, e.g., \(3.3 s\) in case of GWO: PI-PD controller, \(9.57 s\) for MPC/PI controller and \(17.87 s\) for GA-PI controller, respectively. Finally, the effectiveness of the controller is investigated under different load-changing conditions and the proposed controller shows superiority over other state-of-the-art controllers.

In future studies, complex system designs will be considered, designs where the number of areas integrated may be enhanced to represent the increasing complexity of the power system network. Efficient and robust control designs will then be further discussed.

Availability of data and materials

Not applicable.

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Acknowledgements

This publication is based upon work supported by the King Fahd University of Petroleum and Minerals. The author at KFUPM acknowledge the Interdisciplinary Research Center for Renewable Energy and Power Systems for the support received under grant no. EC221008.

Funding

The author at KFUPM acknowledge the Interdisciplinary Research Center for renewable Energy and Power Systems for the support received under grant no. EC221008.

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Correspondence to Muhammad Majid Gulzar.

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Gulzar, M.M. Designing of robust frequency stabilization using optimized MPC-(1+PIDN) controller for high order interconnected renewable energy based power systems. Prot Control Mod Power Syst 8, 11 (2023). https://doi.org/10.1186/s41601-023-00286-x

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Keywords

  • MPC-(1 + PID) controller
  • Salp swarm algorithm
  • High order interconnected area
  • Automatic generation control