Abstract
To improve power quality in power systems vulnerable to current disturbances and unbalanced loads, a hybrid control scheme is proposed in the present paper. A hybrid adaptive robust control strategy is devised for an SMIB power system equipped with a static VAR compensator to ensure robust transient stability and voltage regulation (SVC). High-order sliding mode control is combined with a dynamic adaptive backstepping algorithm to form the basis of this technique. To create controllers amenable to practical implementation, this method uses a high-order SMIB-SVC model and introduces dynamic constraints, in contrast to prior approaches. Improved transient and steady-state performances of the turbine steam-valve system are the goals of the dynamic backstepping controller. A Lyapunov-based adaptation law is developed to address the ubiquitous occurrence of parametric and nonparametric uncertainty in electrical power transmission systems due to the damping coefficient, unmodeled dynamics, and external disturbance. High-order sliding mode (HOSM) control is used for generator excitation and SVC devices to construct finite-time controllers. The necessary derivatives for HOSM control are calculated using high-order numerical differentiators to prevent simulation instability and convergence issues. Simulations demonstrate that the suggested method outperforms conventionally coordinated and hybrid adaptive control schemes regarding actuation efficiency and stability.
1. Introduction
Today’s power systems are far more loaded than in the past and are frequently operated at capacity, resulting in technical constraints, equipment deterioration, and economic losses. Load changes, transmission line outages, and short circuits continue to be the typical operating conditions that cause generators to experience poorly damped oscillations, loss of synchronism, and instability. Designing adequate control methods is one of the most effective strategies to address these difficulties. Stability and voltage management of power systems are critical, particularly for long transmission lines and large power plants [1, 2].
Single-machine infinite bus (SMIB) is a crucial technology for real-time voltage management and stability. It helps improve dependability, stability, and transmission efficiency. The coordination between SMIB and shunt-type flexible alternating current transmission systems (FACTSs), such as static VAR compensator (SVC) provides suitable control solutions for their interdependent parameters. Since 1970, SVC technology has been used as a realistic solution to the problem of continually generating or absorbing reactive power, and it is still utilized in AC power systems. SVC devices can exchange capacitive or inductive currents. The integrated SMIB-SVC system provides effective frequency oscillation damping, enhances transient stability, and controls the voltage and reactive power [3]. However, the complexity of modern power systems complicates and challenges the design of coordinated controllers for SMIB-SVC.
Modern power systems improve high-order multivariable processes that involve highly nonlinear electrical elements and parametric uncertainty. Changes in their inherent nonlinear properties are frequently time-varying, rendering fixed-parameter control algorithms incapable of delivering sufficient performance. Therefore, high-order dynamic models and adaptive parameter update laws are required for the control design. Adaptive robust control is crucial to appropriately compensate for nonlinear dynamics, fast unmodeled dynamics, and disturbances. However, several signal processing-related technical challenges arise when designing adaptive controllers, and designers must consider all settings and variables when working in an industrial environment. Much research has been devoted to developing adaptive generator excitation and steam-valve controllers in recent years. For example, Wan et al. [4] designed a nonlinear SMIB-SVC controller using the immersion and invariance (I&I) approach. The authors implemented class-K functions to improve transient and steady-state performance. Milla and Duarte-Mermoud [5] created a predictive optimization adaptive (POA) method for computing PSS device parameters. The oscillations of the SMIB were enhanced by the POA-PSS method. Liu et al. [6] suggested a continuing fraction-based method for online simulation and control of power systems using SMIB. Bux et al. [7] examined the damping effect of the VSC stabilizer and its influence on electromechanical oscillation modes. To compensate for parameter sensitivity in SMIB, Roy et al. [8] integrated feedback linearization with adaptive control. The control strategy based on Lyapunov was superior to previous exciting partial feedback linearizing schemes. Kamari et al. [9] presented an optimal PSS-PID controller for suppressing low-frequency oscillations in SMIB. Using a PID controller, the authors tuned the controller’s parameters using chaotic practical optimization (CPSO). CPSO-PID effectively minimized overshooting and decreased the transient response’s settling time. Mijbas et al. [10] presented the multiobjective particle swarm optimization (PSO) technique to enhance the generator’s power angle stability. They demonstrated that changing the controller parameters using PSO improves the stability of the SMIB-SVC system. Kumar et al. [11] utilized ACO-based SSSC to attenuate low-frequency oscillations and voltage variations in multimachine power networks. SSC effectively eliminated interarea subsynchronous oscillations and voltage variations by adopting ACO-based control with the voltage source. Muhammad et al. [12] optimized PSS parameters using the PSO algorithm. The perfect performance of PSS ensured the stability of the SMIB rotor’s frequency response and power angle. However, the probabilistic character of population-based approaches may result in a loss of precision due to overestimating control benefits. Recently, linearized model-based techniques, such as optimum control and direct adaptive control, have been employed to provide high bandwidth and resilient performance in the presence of uncertainties (see [13–17]). Meanwhile, adaptive robust control design sometimes requires more than a linearized system model can provide.
High-order sliding mode (HOSM) control is gaining popularity in building resilient power network controllers due to load disturbances and measurement noises. Wan and Jiang [18] created a super-twisting sliding model-based control strategy for SMIB to provide robust stability and high-performance control. Han and Liu [19] designed a control technique for perturbed triple integrator chains using HOSM control. Trip et al. [20] designed a load frequency controller for power networks using sliding mode control of the second order. Although the controller displayed excellent tracking performance, it could not adequately adjust for frequency measurement disturbances. Adirak and Ekkachi [21] combined backstepping control and first-order sliding mode control to improve the transient stability and voltage regulation of SMIB. Compared to I&I and conventional backstepping control, the suggested control method demonstrated appropriate performance and enhanced closed-loop control stability. Neither external disturbances nor measurement noises were accounted for in this system. Under load disturbances, Dev et al. [22] proposed an adaptive super-twisting sliding mode controller for two sections of interconnected power networks. The authors implemented the dynamic adaptive rule proposed by Gutierrez et al. [23] to deal with the unknown limits of disturbances and prevent overestimating control advantages.
This study presents a novel solution to the problem of transient stability and voltage regulation in power systems with an SVC device. An SMIB-SVC backstepping-HOSM coordinated control technique is intended to simultaneously regulate the synchronous generator and SVC device. The nonlinear coordinated control of SMIB-SVC has received only a few important contributions until recently. Using a simplified SMIB-SVC model, Kanchanaharuthai and Mujjalinvimut [24] suggested a nonlinear coordinated control approach resembling backstepping. The authors demonstrated that their design methodology outperforms I&I control and conventional backstepping techniques. To recover the output voltage and remove power angle variations, Psillakis and Alexandridis (2020) utilized backstepping and feedback linearization control approaches. Keskes et al. [25] created a nonlinear coordinated control for the SMIB-SVC by employing input-output linearization and pole-assignment techniques. The authors demonstrated that the suggested control scheme outperformed traditional PSS and noncoordinated controllers regarding oscillation damping and voltage regulation. As with most known design methods, these control algorithms are unrestricted, fulfill asymptotic stability, and frequently exhibit a chattering effect that is difficult to implement in practice.
This study stands out in comparison to similar ones conducted recently. The following are the principal contributions of the current paper. It starts with an eighth-order mathematical model of the SMIB-SVC system. A high-order SMIB-SVC model is required to improve transient and steady-state performance. The dynamics of mechanical power, exciter dynamics, and SVC dynamics are all considered in the model, as are the impacts of subflux and transitory flux couplings along the d and q axes. This model is a significant improvement over its predecessors in the literature when applied to power systems that are inextricably linked to one another (e.g., [4, 9, 18, 25, 26]). Second, a dynamically constrained backstepping (ADBS) controller is developed to enhance steady-state performance and attain transient stability. Third, parameter uncertainties and insufficiency in load disturbance compensation are addressed by adding a Lyapunov-based adaptation law to the load controller. Fourth, the synchronous generator and SVC device are controlled in tandem using controllers based on the finite-time HOSM model. The suggested ADBS-HOSM control strategy ensures more enhanced transient stability, decreased susceptibility to parameter uncertainties, and resilience against disturbances and measurement noises compared to recent contributions (e.g., [4, 18, 25]). Furthermore, the ADBS-HOSM control is chattering-free, shielding the actuators from the erratic high-frequency oscillations induced by feedback linearization and traditional sliding mode controllers. According to the authors, this is the first citation for finite-time backstepping-HOSM coordinated control design based on a comprehensive SMIB-SVC model.
The paper is formatted as follows: Section 2 explains an enhanced dynamic SMIB-SVC model. Section 3 presents the design of the nonlinear backstepping controller and adaptive parameter law. Section 4 develops HOSM controllers for generator excitation and SVC devices. In Section 5, numerical simulations are performed to validate the theoretical conclusions and assess the performance of the suggested control approach. The conclusion is presented in Section 6.
2. Dynamic Modeling and Problem Statement
The control laws designed with the traditional third-order power system model in mind have been shown in the literature to be a common cause of power oscillations and instability. As a result, only control laws based on a high-order model can guarantee optimal transient and steady-state performance. Here, we provide a nonlinear eighth-order SMIB-SVC dynamic model that accounts for both parametric and nonparametric forms of uncertainty.
2.1. Mechanical Load Dynamics
where , , , denote the generator rotor angle, angular speed, synchronous speed, mechanical power input, active power, and governor value position. denote the inertia constant, damping coefficient, time constant, and throttle pressure.
2.2. Exciter Dynamics
where denote the q-axis and d-axis voltage and transient voltages, , and denote the infinite bus, field, and terminal voltages of the generator, respectively, and are the time constant and exciter parameter, , , and denote the d-axis generator, transient reactance, and equivalent transient reactance, respectively, and are the time constants of the excitation winding, and are positive constants.
2.3. SVC Dynamics
where , , , and denote the initial susceptance, time constant, gain, and the equivalence output of the SVC controller, and are the reactance of the transmission lines, denotes the transformer reactance, and and are the susceptance of the inductor of TCR (i.e., thyristor-controlled reactor) and the susceptance of the capacitor in SVC, respectively (see [26, 27]). Model (1)–(3) presents a complete and more accurate SMIB-SVC model that can be used in designing advanced controllers. It is worth noting that since the power system has an SVC electronic compensator, its d-axis equivalent reactance is time-varying even in the absence of disturbances.
3. Dynamic Feedback Backstepping for Load Control
3.1. Load Control Model Parametrization
For designing a dynamic adaptive backstepping load controller, dynamics (1) are described in a parametric form, with , as follows:withwhere and denote some fictive controls and represents the external disturbance. The state is selected to be the tracked output. Model (4) is parameterized using the following vector :
The control vector is designed to asymptotically drive the states to their zero level despite parameter uncertainties and disturbances. The output is subjected to the following asymptotically stable constraints:where . The control input must guarantee a satisfactory performance level and maintain a sufficient robustness margin under the following assumptions:
Assumption 1. The functions and are uncertain functions due to the uncertainties of the parameters and .
Assumption 2. The uncertainties and have bounded Euclidian norms: and with .
Assumption 3. The nonparametric disturbance satisfies that .
Assumption 4. The control input is a Lebesgue measurable bounded signal.
Assumptions 1 and 2 are made according to the requirements of control-affine nonlinear systems (i.e., nonlinear dynamics systems with a linear control input), which is consistent with the proposed nonlinear backstepping control type. In adaptive nonlinear-affine control theory, the functions f(x) and g(x) must be smooth and bounded, with a known local relative degree to the control input u. Numerous control designs, including backstepping, feedback linearization, and sliding mode control techniques, require such assumptions [28].
For Assumption 3, external disturbances and parameter uncertainties are inevitably present in power systems, including SMIB systems. They are typically unknown and unmeasurable, drastically degrading the desired control’s performance. Observers must be designed with disturbance rejection or compensation. In such observers’ design, the upper limit of the disturbance must be specified. The practical applicability of the designed control law is contingent on the fourth assumption. A Lebesgue measurable function must represent an admissible signal control. The Lebesgue condition ensures that the control law has a physical meaning and is applicable [29].
3.2. Backstepping Controller Design
We consider the parametric model (4) as a cascade connection of three subsystems (8) and define, for each subsystem, a candidate Lyapunov function (CLF), as shown in (9),where denotes the state error vector with and being the expected states of the model (4).
Theorem 1. Let Assumptions 1–4 hold. The output of model (4) asymptotically converges to its reference value under the dynamic constraints (7) if the effective control input is chosen asand the control gains are appropriately selected such that with
Proof. The control law (10) is designed using recursive dynamic backstepping. To do so, we consider the following three-step stabilizing scheme.
Step 1. As a starting step, a virtual state feedback law is sought to stabilize the first subsystem asymptotically. This is done if there exists a specific gain that fulfills the following inequality:Substituting in expression (12) givesUsing the first equation of model (4), one can find that the condition (13) is guaranteed by the following virtual control :It is clear that the intermediate control law (13) asymptotically steers towards for a selected gain .
Step 2. Similarly, it can be easily proven that there exists a gain for whichand with , the second subsystem is asymptotically stable under the following virtual control:The control law (16) guarantees the global asymptotical stability of the first and second subsystems. However, is only a virtual control law, which implies the following step.
Step 3. The final step in the backstepping procedure is to find the accurate control input under which the system (4) reaches its coordinate origin or operating point with desired performances and under prescribed dynamic constraints. Taking the time derivative of CLF ,and then, with , if there exists a constant such thatIt results that the real control is deduced from (4), (17), and (18) as defined in form (10).
This is the end of the proof.
Remark 1. The feedback control law (10) is nonsingular since , , where denotes the operating space of model (4).
Remark 2. The case corresponds to the unconstrained model.
3.3. Adaptive Dynamic Backstepping Controller
In Subsection 2.3, the backstepping controller (10) was derived under nominal conditions with known parameters and . In practice, the values of these parameters change with the system’s parameters. To introduce the uncertainties , a new parameter error vector is defined with and being the estimators of and , respectively. The uncertain form of the vector-valued function in equation (4) is defined as follows:with
Theorem 2. Suppose that the unknown estimators and are bounded, the closed-loop control (19) is robustly stable under the law (21) conjointly with the parameter update law (22) if the vector c fulfills the output condition (23).
The control law is The adaptive law is The output condition is with and denoting adaptive gain coefficients.
Proof. Considering the following CLF ,and the time derivative is given asAs, from (15), , it remains to find the condition for which . Using the adaptive law (22), we obtainUsing condition (23), it follows that , , and , where and denote the system’s operating and parameters spaces, respectively.
This is the end of the proof.
4. High-Order Sliding Mode for Exciter Control
For dynamic models (2) and (3) for the exciter and SVC devices, we define the following sliding variables:where denotes a field voltage reference, while denotes the initial susceptance of the SVC controller. For designing robust HOSM-based feedback controllers and that provide finite-time convergence of and , the following assumptions are made.
Assumption 5. Both systems (2) and (3) have a well-known defined relative degree to their tracking output .
Assumption 6. In both cases, the following r-sliding set is a nonempty integral set:
Assumption 7. A homogeneous r-sliding mode controller can be obtained if there exists a function , which is continuous everywhere except for the set
Lemma 1 (see [30]). Let Assumptions 5–7 hold. Local finite-time convergence of the r-sliding mode can be achieved and maintained by the following homogenous HOSM controller:
The constant p is the least common multiple of the set , the parameters are the controller parameters, and and are intermediate functions (for the proof of Lemma 1, see [30]).
With applied as in model (2) and as in model (3), the field voltage and susceptance converge to their references or desired final values. The successive time-derivatives in the recursive algorithm (29) are computed using the following robust high-order sliding mode differentiator (see [31]):where is a Lipschitz constant and denotes the differentiator parameters.
5. Simulation Results
In this section, different scenarios are simulated for desired operating points to demonstrate the effectiveness and robustness of the proposed ADBS-HOSM control scheme. The values of the parameters are as follows: , , , , , , , and The control technique employs two distinct sets of parameters. Adaptive backstepping parameters (i.e., ) are chosen to ensure a robust transient response with correction for parameter uncertainties. The operating point of the mechanical load is regulated using the three gains , while parameter uncertainties are adjusted by . The second group of parameters pertains to the h exciter’s high-order sliding mode HOSM controller. The gain G is a tuned gain, whereas are chosen in accordance with the HOSM control paradigm [32].
5.1. Performance and Effectiveness
The proposed ADBS-HOSM control scheme is applied with , , and as initial conditions and as the operating conditions. The values of the control parameters are chosen as follows: , , and for controllers (23); and for adaptive laws (24) and (25); and for HOSM controllers (29) with ; and and for HOSM observers (30). Figures 1–4 show the time history of the transient response provided by both the ADBS-HOSM controller and the controller proposed by Wan and Jiang [18].
5.2. Robustness
Measurement noises are a common problem in power systems, reducing their performance and reliability ([4]; Ghahremani and Kamwa, 2011). We consider realistic scenarios where corrupting noises are present in the available outputs for measurement. In this example, white Gaussian noise with a variance of 0.01 is used to mimic the conditions in the situation described in [4]. The robustness of the proposed ADBS-HOSM controller and the IARK controller is compared in Figures 5 and 6. It is clear from this comparison that the ADBS-HOSM controller is superior to the IARK controller in terms of disturbance compensation.
5.3. SMIB-SVC Coordinated Control
In this scenario, the ADBS-HOSM coordinated control scheme is compared to two other recently published coordinated control schemes, the input-output linearization controller (IOL) and the adaptive variable-structure (AVR)/point-synchronous (PSS)/point-integral (PI) controller (SVC) [25, 33]. Figures 7–10 display the results of running the simulation with the same data as in [25]. The simulation results show that the proposed coordinate control scheme outperforms those created with traditional control methods like input-output linearization and proportional-integral-derivative (PID). More specifically, ADBS-HOSM improves actuation efficiency in terms of settling time and stability performance.
5.4. Critical Clearing Time and Angle
It is recognized that the size and length of disturbances impact the stability of power systems and can lead to synchronization loss. The critical clearing angle and critical clearing time (CCT) are essential to the power system’s stability. Therefore, the defect must be rectified prior to CCT; otherwise, the system will be unstable. The second scenario was resimulated to calculate CCT. As illustrated in Figure 11(a), the system was stable for 0.24 seconds before becoming unstable at 0.25 seconds, as depicted in Figure 11(b). Therefore, 0.24 second is regarded as the system’s CCT. In addition, the phase-plane frequency angle is presented to check the system’s stability.
(a)
(b)
The four simulated scenarios demonstrate the effectiveness and performance of the presented coordinated SMIB-SVC control strategy. Practical cases, including control in the presence of parameter uncertainties (scenario 5.1), the effect of external disturbances and measurement noises (scenario 5.2), coordination of the SMIB and SVC systems (scenario 5.3), and the critical clearing time and angle (scenario 5.4) were considered. The four scenarios address the essential practical concerns related to the real-time control of modern power systems.
6. Conclusion
A novel robust adaptive control framework was created to simultaneously design an SMIB generator excitation controller with an SVC controller. To provide a robust and implementable controller, uncertainties in physical parameters were considered. Nonlinear controllers were developed for adaptive dynamic backstepping steam-valve load control and high-order sliding mode-based (HOSM) generator excitation control. Using a high-order power model and parameter estimator law, transient stability and voltage regulation were improved. The system’s resilience is ensured by the fact that the adaptive control law is designed to adjust for parametric uncertainties, while HOSM is applied to compensate for nonparametric uncertainty, unmodeled dynamics, and external disruptions. Our findings contribute to the existing literature by considering a more realistic damping model of the synchronous generator related to the system’s state variables. As a result of our work, the current literature is expanded to include a more realistic damping model of the synchronous generator. The proposed control scheme maintains the nonlinear features of the underlying SMIB-SVC system model by employing high-order modeling. The research findings demonstrated, via simulation, the effectiveness, viability, and superiority of the proposed control scheme in each simulated case. The modular character of the proposed method makes it adaptable to various FACTS devices that can be described in a manner comparable to the one investigated in this study. This can comprise SMIB systems with a static synchronous series compensator (SSSC), a unified power flow controller (UPFC), and a thyristor-controlled reactor (TCR). However, future works can consider some limitations, such as frequency stability for large-scale power systems.
The Kundur and large-scale systems feature a topology with several buses and regions. Both systems can be considered multiload systems, making them appropriate for the multiagent control paradigm. Future works include analyzing the transient response of the entire network and applying the proposed control mechanism to multimachine systems employing FACTS devices. Renewable energy and power grids rely largely on the synchronization and stability of multimachine power systems; hence, employing a multiagent control paradigm, the method will also extend to the distributed coordinated control of networked power systems.
Data Availability
The simulation data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. G-1436-135-504. The authors, therefore, acknowledge with thanks DSR technical and financial support.