1. Introduction
Maintaining the frequency of a power system within acceptable limits is essential for a power system to be stable and to operate with reliability. Due to the fluctuation in consumer demand, the power varies and any change in real power mostly influences the system frequency. A synchronous generator (SG) is used in conventional power generating stations, particularly in steam, hydro, gas, or combined-cycle power plants, to produce electricity with the aid of a turbine and speed governor [
1]. Under both static and dynamic conditions, the generators in the control zone periodically change their combined speeds (acceleration or deceleration) to maintain the frequency and the corresponding power angle at predetermined levels [
2,
3]. Load frequency control (LFC) refers to the issue of regulating the actual electrical output of producing units in response to alterations in system frequency and the exchange of patch line power within predetermined constraints. The LFC stabilizes the frequency and power profile of the connection line after controlling power generation in the demand-load line to reduce the temporal average of the zone control error [
4,
5].
Automatic generation control (AGC), pole placement design, and optimal control design are currently the most advanced techniques for preserving the power system’s frequency against variations in active power load. In optimal control design, the control system employs the state-variable feedback technique, which is implemented by diminishing the system variables’ performance indices [
6]. Proportional–integral–derivative (PID) controller, linear quadratic regulator (LQR), model predictive control (MPC), and sliding mode control (SMC) are some optimal controllers which can be used for load frequency control [
7,
8,
9,
10]. Additionally, a variety of artificial intelligence methods have been developed for implementation in LFC, with fuzzy logic and artificial neural networks (ANN) being the most frequently used.
To enhance the overall system dynamics, a PID controller was suggested in [
11]. The suggested controller had a negligible overshoot but the settling time required was approximately 6 s. However, there is a zero steady-state error in a PID-controlled system. It was shown that the frequency response would stabilize significantly more quickly than a PID-controlled system by using an advanced optimization technique. In [
12], a novel stability criterion was developed with a proportional–integral (PI)-based LFC to assess the influence of the time-varying delay on the stability of power systems. Active disturbance rejection control (ADRC) for a wind-integrated power system LFC was presented in paper [
13] considering delays in communication. In order to enhance the stability of frequency control, equivalent input disturbance (EID) compensation is utilized to eliminate the impact of load variation. Based on a frequency-domain approach, paper [
14] presented a fractional-order PID controller cascaded with a first-order filter for the delayed LFC system. In order to stabilize the system, a PID-based secondary controller was suggested using a modified variant of particle swarm optimization in [
15]. In [
16], authors analyzed the methods for dealing with the uncertainty of renewable energy and controller for a multi-area power system. In [
17], a tuning-free PID control method was presented for a second-order plus time delay (SOPTD) system. In paper [
18,
19,
20], an LFC power system was implemented with a PI-type SMC. Because of the use of PI, the required settling time is comparatively high.
Thus, by replacing PID controllers with optimal control techniques, the system response could be improved more in terms of robustness. To avoid the issues faced by a classical I or PI or PID controller, an optimal control may be used to control the system. In [
10], an optimal control method called a linear quadratic regulator (LQR) was suggested where the transient responses of the control system were significant as the required settling time was approximately 0.7 s. Despite offering better settling time performance, the LQR controller has a notable steady-state error.
Furthermore, the application of the aforementioned controllers is not limited to the LFC of conventional power systems. A new optimization technique named the African Vulture Optimization Algorithm was presented to ensure the performance of a cascaded controller in [
21]. To mitigate the problem of the renewable energy source (RES) for a two-area interconnected network, this controller was introduced. Including battery and pump hydro as new applications, ref. [
22] described the frequency and power control of a microgrid power system. For the control technique, the optimal tuning of the PI controller using the quasi-Newton method was considered. In [
23], the quasi-Newton algorithm for hybrid power plant operation was presented and contrasted to a standard PID controller. In [
24], the fractional-order proportional–integral–derivative (FOPID) controllers were proposed in relation to regulating the temperature in ambulances, regulating bioreactors, induction heating systems, and the advancement of temperature control systems. This optimal PID performs better than traditional PID in terms of responses according to the timing information. The LQR controller has additional applications beyond LFC. In [
25], an energy management system for battery supercapacitor hybrid systems containing a battery (lithium-ion) and a supercapacitor along with a bidirectional DC-DC converter was presented. For three-area power systems, ref. [
26] analyzed the performance of model-based LQR and data-driven fuzzy controllers. Based on the comparison, it can be seen that the LQR strategy outperforms the fuzzy control strategy.
For the effective control of system frequency with respect to frequency deviation, steady-state error, and settling time, a new optimal control technique is proposed in this paper for the mitigation of frequency fluctuations in a single-area power system implementing an LQR-based proportional damping compensator.
The proposed control technique comprises a proportional controller, a damping compensator, and an LQR. To enhance the system’s tracking performance, an optimal state feedback controller is designed using the LQR, and damping compensator, and it is then paired with a proportional controller. The damping compensator’s purpose is to minimize low-frequency oscillations that occur due to load variations. Based on the relevant literature survey and the proposition of the paper, it is evident that this article brings a new look to the existing literature in the following areas: (I) introduction of robust hybrid control technique (II) experimental analysis of complicated power system model, and (III) wide range of responses of a power system model. The key contributions of the paper are summarized as follows:
A new LQR-based proportional damping compensator is proposed to mitigate the frequency fluctuation of a single-area power system;
A hybrid multiprocessor-based processor-in-loop (PIL) technique is introduced in the paper to validate the performance of the proposed control strategy; and
The performance of the proposed controller is evaluated both in simulation and experimental environments in terms of frequency deviation, settling time, and steady-state error for various load conditions.
The rest of the paper is organized as follows: the modeling of the single-area power system is presented in
Section 2, while
Section 3 describes the design of the proposed controller; simulation result analysis is presented in
Section 4 and experimental analysis is included in
Section 5; finally,
Section 6 concludes the paper.
2. Modeling of the Single-Area Power System
LFC is a crucial control problem because it controls the system frequency and inter-area tie-line power to determine the quality of power generation [
27]. The single-area power system stands for a power system network consisting of a governor, turbine, and rotating mass with load. This is the basic unit of a power system network. If more than one single-area network is connected in a cascaded manner, the resulting power system network is said to be a multi-area power system. In this work, the mitigation of frequency fluctuation is targeted on a single area network using an LQR-based proportional damping compensator-based control technique.
The key characteristic of a single-area power system is that it is not directly connected to other power systems or a large grid. It can operate independently and does not share electrical power with neighboring areas or the grid. The fundamental components of a single-area power system are the governor, turbine, alternator, rotating mass, and load.
Figure 1 depicts the plant model of a single-area power system consisting of a governor, speed regulator of the governor (R), turbine, mass, and load connected in a closed-loop network. This system is called a single-area power system which is the basic unit of an interconnected power system network.
Among the three categories of turbines used in power systems: non-reheat, reheat, and hydraulic turbines, non-reheat turbines are represented here. By using turbines, renewable energy sources such as steam or water can be converted into mechanical power, which can then be directly supplied to the generator. Based on each type, the following transfer function models are provided:
The turbine’s mechanical energy is transferred to generators, where it is transformed into electrical energy. By adjusting the turbine inputs, such as the characteristic for speed regulation (
R) and the governor time constant (
τg), governors are utilized in power systems to detect frequency biases caused by fluctuations in load and eradicate them.
The open-loop transfer function of the block diagram for the single-area power system and the closed-loop transfer function relating the load change, Δ
PL, to the frequency fluctuation, Δ
fd, are provided below.
The nomenclature section presents the details of LFC model parameters utilized for a single-area power system. The values of the turbine and governor time constant are τT = 0.5 s and τg = 0.2 s, respectively. The governor inertia constant (H) is set to 5 s and the frequency sensitive load coefficient (D) is set to 0.8. Load change varies from 0.2 p.u. to 1 p.u. throughout the simulation study.
The development of the state-space model of a single-area power system is described here. For isolated turbo-generator power systems, the state-space equations are developed as follows.
The state-space matrix is constructed as shown below from the above-mentioned three equations.
The power system frequency fluctuates as a result of variations in load. The impact of changing the load on an open-loop isolated power system’s frequency fluctuation is illustrated in
Figure 2. It is notable from
Figure 2 that, in an open-loop scenario, the frequency fluctuation varies according to the load applied. So, it is expected to mitigate the frequency deviation of single-area power system using a closed-loop control strategy. This is the main motivation of the work.
3. Design of the Proposed Controller
The design of the proposed controller has three steps. Firstly, the LQR has to be designed. After that, the proportional damping compensator will be designed. Finally, the LQR has to be integrated with the proportional damping compensator to obtain the proposed controller.
Figure 3 depicts the mathematical development of the proposed control technique for a single-area power system. In the proposed technique, the LQR controller is connected in a feedback loop. Both the proportional controller and damping compensator are connected in a series with the plant. Integrating the LQR with the proportional controller and damping compensator forms the proposed controller which significantly improves the system response in terms of frequency deviation, settling time, and steady-state error.
3.1. Design of the LQR
The LQR controller is an idealized control regulator that considers an appropriate control law to minimize a quadratic objective function while enhancing system dynamics by using a feedback gain. It uses a set of linear differential equations to model system dynamics, and the performance indicator is provided by a quadratic function, hence the name LQR. The single-area plant can be described as follows:
where,
is the state vector’s derivation and
U(t), and
Y(t) are input and output vectors, respectively, and the ranges are
,
, and
.
For LQR controller, the quadric objective function is mentioned below.
The value of
P and
K is established through the appropriate selection of the
Q and
R matrices where
Q is state-weighting matrix and
R is the control-weighting matrix. The selection of the
Q and
R matrices during the LQR design process is crucial and is determined by the control objectives and system characteristics. Tuning the values of
Q and
R matrices typically involves trial and error, engineering discretion, and system knowledge. Matrix
A is the state matrix of order
, matrix
B is the input matrix of order
,
C and
D are the output matrix of order
, and feedforward matrix of order
respectively. The values of the above-mentioned matrix for this plant are
Matrix D is considered a zero-matrix due to the absence of a direct feedforward mechanism. The value of K(t) significantly affects the output of the load frequency control. The roots of the system will be positioned in the most precise location with the assistance of LQR. Thus, by minimizing a cost function, the LQR controller provides optimal performance, ensuring a rapid response to load changes and stable frequency regulation. It provides disturbance resistance and maintains stable frequency regulation even in the presence of uncertainty. The LQR controller is adaptable and flexible, allowing for simple tuning to satisfy specific control objectives and system requirements in load frequency control.
3.2. Design of the Proportional Damping Compensator
A damping compensator is a control mechanism used in power systems to improve the stability of the system by providing additional damping to oscillations.
Figure 4 shows the proportional damping compensator where the op-amp structure of a basic damping compensator is connected in a series with the proportional controller.
In the majority of instances, the compensator is designed as a feedback control loop that modifies the power system’s parameters in response to shifting system conditions. The transfer function of a damping compensator is provided as follows:
The LFC loop can be improved by selecting proper values for the resistor (
Ri), capacitor (
Ci), and inductor (
Li). To analyze the effects of the resistor, capacitor, and inductor, three compensators are considered with different values for
Ri,
Ci, and
Li [
28].
3.3. Integrating the LQR with Proportional Damping Compensator
The proportional (P) controller is a form of feedback controller that modifies the output based on the discrepancy between the actual and target system frequency and the damping compensator’s goal is to increase the system’s damping ratio, which is a gauge of its capacity to withstand oscillations and return to its stable equilibrium condition following a disturbance. To improve the stability and performance of the system and the further development of the performance of the control technique, a proportional controller is incorporated with a proportional damping compensator along with the LQR.
In the proposed control technique, the proportional compensator and the damping compensator are connected in a series but the LQR is incorporated as a feedback loop as depicted in
Figure 5. The transfer function of the open-loop system is given as
By combining the benefits of the LQR, P-controller, and the damping compensator, the proposed controller is formed.
It has been demonstrated that using the LQR-based proportional damping compensator in LFC boosts transient performance, dampens frequency oscillations better, and increases robustness to changes in load and system parameters.
Figure 6 shows the root locus diagram of a closed-loop system with a proposed control technique. The figure shows that the poles and zeros of the closed-loop system lie on the left side of the real axis which indicates the system’s stability. The transfer function of the closed-loop single-area power system with proposed control technique is calculated.
5. Experimental Analysis
To validate the performance of the proposed controller, a hybrid multiprocessor-based processor-in-loop (PIL) technique is introduced. The presented technique is different from traditional PIL techniques as the presented PIL technique employs multiple single microcontroller units (MCUs) for conducting individual tasks during real-time simulation. The complete setup of the hybrid multiprocessor-based processor-in-loop (PIL) technique is depicted in
Figure 10. The step-by-step implementation procedure is described here. At first, the plant model is deployed in TMS320F28335 DSP MCU. The state-space model of the plant is built in MATLAB/Simulink software environment. After that, the corresponding mathematical model is transferred to a code composer studio-integrated development environment (CCS-IDE) and finally deployed to the TMS320F28335 DSP MCU. Secondly, the proposed controller as well as other existing controllers are implemented utilizing slave MCU as depicted in
Figure 13. STM32F401CCU6 MCU is chosen as the slave which is connected to the host PC and real-time plant system through the I2C communication protocol. Finally, the digital responses of the plant and controllers are converted into analog signals through a digital to analog converter (DAC) interface. The performances of the plant and controllers are captured through a GDS1104B 4-channel digital oscilloscope from the DAC output terminals. The frequency fluctuation response against different load conditions for a single-area power system is shown in
Figure 14. From
Figure 14, it is clear that different loads have been applied on the single-area power system which leads to a frequency fluctuation at open-loop conditions. The experimental result of
Figure 13 complies well with the simulated result of
Figure 2. The frequency deviation response of three different existing LQR-based compensators is depicted in
Figure 15. The experimental result of
Figure 15 agrees well with the simulation result of
Figure 8. The frequency deviation response of
Figure 16 shows the frequency variation for the open loop, PID, LQR, and the proposed controller, while a step change of load is applied on the system. From
Figure 16, it is clear that the proposed controller takes the minimum time to recover from the transient period as compared to the open loop, PID, and LQR. The simulation result of
Figure 10 is similar to the experimental result of
Figure 16. The dynamic load variation response of the proposed controller is shown in
Figure 17. Three different loads have been applied at different time instants as cleared from
Figure 17 which is similar to the simulation result of
Figure 12. The proposed controller tries to recover the load transient quickly. From the experimental findings, it is evident that the responses of the different existing and proposed controllers coincide with the simulation results.