Adaptive Fuzzy Dynamic Surface Control for Multi-Machine Power System Based on Composite Learning Method and Disturbance Observer

A composite learning dynamic surface control is proposed for a class of multi-machine power systems with uncertainties and external disturbances by using fuzzy logic systems (FLSs) and disturbance observer (DOB). The main characteristics of the proposed strategy are as follows: (1) The approximation ability of FLSs for nonlinear model of multi-machine power systems is enhanced considerably by using the composite learning method and providing additional correction information for the FLSs. These findings differ considerably from previous designs that focus directly on the system’s tracking performance. (2) The filtering errors caused by the utilizations of the first-order low-pass filters in dynamic surface control (DSC) are compensated effectively by designing the compensating signals in the control law design process. (3) The compound disturbances including the FLSs’ approximation error and external disturbances are estimated and mitigated by constructing DOB. Finally, the proposed control algorithm is verified on the StarSim Hardware-in-loop experimental platform, and the experimental results validate the effectiveness of the proposed control strategy in suppressing disturbances and enhancing the robustness of the controller.


I. INTRODUCTION
With the expansion of the power grid's scale, modern power systems have gradually formed a strong coupling dynamic nonlinear system. Power systems are more likely to encounter oscillations because of its complex nonlinear characteristics and some accidents, such as three phase voltages short circuit fault; maintaining a safe and stable operation is also more difficult [1]- [4]. Therefore, improving the control ability of multi-machine excitation system has aroused considerable concern among researchers. An increasing number of The associate editor coordinating the review of this manuscript and approving it for publication was Mohammad Alshabi . nonlinear control methods have been applied to multi-machine power systems [5]- [9].
The traditional excitation control strategy usually adopts the PID control method [10]. However, this strategy is no longer suitable for a highly nonlinear multi-machine power system. Backstepping control method provides a systemic framework for tracking and regulating problem of nonlinear systems, which are utilized widely in the excitation control field. In [11], stability sensitive parameters are brought into the multi-machine power system model and a robust adaptive backstepping excitation controller that can which overcome the over-parameterization problem of stability sensitive parameters was design. In [12], by introducing the sliding mode surface into the dynamic surface controller, VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ the robustness and anti-interference ability of the system are improved. Fuzzy logic systems (FLSs) and neural networks (NNs) are usually used to approximate the system's uncertainties because of their good approximation capabilities [13]- [21]. In [22], FLSs and NNs are introduced into the design of backstepping controllers to provide feedback unknown information. However, there is an inherent ''explosion of complexity'' problem exists in backstepping method.
With the increases of system order, the derivative times of virtual control law increases, causing a sharp increase in the complexity of the control law. Dynamic surface control introduces the first-order low-pass filter into backstepping control, and filters the virtual control law to avoid the expansion of its terms [23]- [25]. Therefore, DSC method have been utilized in control multi-machine power system [26]- [30]. In [26], NNs and tracking error transformed functions are used in the design of an adaptive DSC scheme for the SIMS. In [28], an approximation-based adaptive controller for uncertain stochastic nonlinear system with dead-zone and output constraint was proposed, where NNs are utilized to approximate the uncertainties. Although great progress has been made in DSC strategy based on FLSs/NNs, it is designs directly to address the system's tracking performance but ignores how the FLSs/NNs work as an approximator and the approximated FLSs/NNs models are not accurate and interpretable.
The multi-machine power system is affected easily by external disturbance, which can influence the tracking performance. For the parameters uncertainty and disturbance problem, various solutions, such as robust control [31]- [33], online policy iteration [34], [35], adaptive control [36]- [39], parametric based dynamic compensator [40] and active disturbance rejection control [41]. In [31], a discrete-time sliding mode controller is designed for a class of conic-type systems, and the disturbance attenuation level achieved the prescribed performance. In [34] and [35],by using the neural network technique, a novel online policy iteration scheme has been developed, compared with the previous offline method, this method obtained an excellent calculation accuracy while retaining the nonlinear characteristics of the system. Reference [41] designed an active disturbance rejection controller for hydraulic servo systems, where the problem of difficult combination of parametric adaptive control and disturbance observer was solved. Also in [38], a multilayer NNs based RISE controller was designed for hydraulic system with various disturbances, and obtained a dynamic tracking accuracy of 0.2% through experiments. However, controller design based on disturbance observer (DOB) has caused some concerns in recent years [42]- [47]. In [43], a backstepping control in combination with a DOB to estimate the complex disturbance for a class of uncertain strict-feedback nonlinear system with unknown external disturbance was proposed. In [48], DOB was combined with fuzzy logic systems to realize the composed estimation for a class of uncertain nonlinear system in the presence of actuator saturation and external disturbances. In [49], an extended sliding mode disturbance observer was proposed to compensate the strong disturbances of permanent magnet synchronous motor servo system and achieve high tracking performance. In [50], an adaptive DOB control scheme was proposed to overcome the frequent incompatibility and harmonic interference in the distributed static compensator systems.
Previous works have focused on system uncertainties and external disturbances. However, the works usually focused on the system's progressive tracking stability, while the accuracy of the approximated models was neglected. In this article, the main contributions are listed as follows.
1) The approximation ability of FLSs for nonlinear model of multi-machine power systems is enhanced by introducing the composite learning method and providing additional correction information. This is achieved by introducing a serial−parallel estimation model (also called a state predictor in some literature) to provide prediction errors, which is designed in the composite learning of FLSs as an additional adjustment information to update the weight vector.
2) The compensating signals are introduced to overcome the filtering error caused by using a first-order low-pass filter in DSC, thereby enabling the compensated tracking error to be obtained. The compensated tracking error signal is introduced together with the prediction error to update the weight vectors of the FLSs.
3) The DOB is employed to estimate compound disturbances that include the approximation error of the FLSs and the external disturbances in the multi-machine power systems. Thus, in addition to the system's external disturbances, the approximation error of FLSs is also considered, which further improves the accuracy of the multi-machine power systems.
The rests of this article are as follows. Section 2 expounds on the mathematical model of power generator excitation system and fuzzy logical systems. In Section 3, the composite learning dynamic surface control is designed and the stability analysis of the entire control system discussed. The experimental results of the proposed control strategy are given in Section 4. Section 5 concludes this article.

A. MODEL DYNAMICS AND PROBLEM FORMULATION
A large-scale power system consists of n generators connected by transmission lines and equipped with SVC. The mathematical model is expressed as follows [26], [51]: Therein, u i is the control signals of the generator with P ei = E qi I qi , Q ei = −E qi I di , E qi , I qi , I di can be calculated and the actual control signal E fi can obtain from (2). h i (δ, ω) represents the interconnection term with According to [51], the interconnection term h i (δ, ω) satisfies: where and p 1ij , p 2ij are constants with values either 1 or 0. The notation for the power systems is given as follow. δ i is the power angle of the ith generator(rad), ω i is the rotor speed of the ith generator(rad/s), ω i0 = 2πf 0 is the synchronous machine speed(rad/s), P mi is the mechanical power, P ei is the electromagnetic power, D i is the damping constant, H i is the inertia constant(s), E qi is the transient electromotive force of the ith generator, T d0i represent the direct axis transient short-circuit time constant(s), T Ci is the time constant of adjusting system and SVC(s), Q ei is the reactive power, u i is the control voltage of excitation equipment, u Bi is the input of SVC, B L is the adjustable equivalent susceptance in SVC, B Ci is the initial value of the adjustable susceptance, B ij is the i th row and jth column element of nodal susceptance matrix at the internal nodes after eliminating all physical buses, and d i1 is the position bounded torque interference of the rotor. Let where V mi is the accessing point voltage and V refi is the reference voltage. According to [26], (1) can be expressed as following two subsystems: . . , 4, are state vectors and y i1 and y i2 are the outputs of the large-scale power system and the SVC equipment, where with where X 1i , X 2i , X 3i are transmission line reactances, X di is the generator direct axis reactance, X di is the direct axis transient reactance of the generator, and X Ti is the transformer reactance. Define following assumptions: where κ 0i and κ 1i are positive constants.
Assumption 2: The control gain functions g ij , j = 2, 3, 4, can be written as known part g ijN and unknown part g ij , and Assumption 3: The reference signals y ri are smooth and bounded functions. Its first and second derivatives are exist and satisfy y 2 ri +ẏ 2 ri +ÿ 2 ri χ for positive real number χ . Remark 1: The controlled object system includes two subsystems, which are excitation subsystem (6) and Static var compensator subsystem (7). There is coupling between the two controllers: the SVC subsystem (6) includes the state variables x i1 , x i2 , x i3 and control signal u i of the excitation system. Therefore, considering the coupling effect, the target system in this article is defined as a multi-machine power system with SVC.

B. FUZZY LOGIC SYSTEMS (FLSs)
An FLS is composed of a fuzzy rule base as following form: where x i (i = 1, 2, · · · , n) are the input variables, A m i and B m are fuzzy sets in R, and µ A m i (x i ) and µ B m (y) are fuzzy membership functions. With singleton fuzzifier, product inference, centroid defuzzifier, the FLS can be constituted in the follow- VOLUME 8, 2020 ing form [52]: where Define the fuzzy basis functions as Then, FLS can be expressed as where ω ∈ R M is the adjustable weight vector and ( is the fuzzy basis function vector. Lemma 1 [52], [53]: Let continuous function f (x) : R n → R defined on compact set x ⊂ R n , and any constants ε > 0, an FLS exists such that the following formula hold, Then, f (x) can be approximated by FLS in the compact set x , described as where ε * is the approximate error and satisfying |ε * | ε M .

III. DESIGN OF ADAPTIVE FUZZY DYNAMIC CONTROLLER BASED ON COMPOSITE LEARNING AND DISTURBANCE OBSERVER
According to (6) and (7), the difficulty of controller design lies in the unknown nonlinear functions     26). In addition, k ij , (j = 1, . . . , 4) and η il , L fil , c il , γ υ il , γ izl , (l = 2, 3, 4) are positive design parameters.ω ij , (j = 2, 3, 4) is the estimation of ω * ij . ω * T ij and ij (x ij ) are the ideal weight vector and fuzzy basis function vector of FLSs.D ij , (j = 2, 3, 4) are the observed value of the DOBs. In this article, DOB is designed as following: where p i (x ij ) is nonlinear function to be observed, K ij > 0 are positive design gain of DOBs, and it should meet K ij x ij = dp i (x ij )/dt. The observation error of DOB is defined asD ij = D ij − D ij . Remark 2: According to Assumption 2, D ij ≤ ϕ ij0 , Ḋ ij ≤ ϕ ij , j = 2, 3, 4, where ϕ ij0 and ϕ ij are positive parameters.

IV. STABILITY ANALYSIS
The proof of the control system stability will be discussed in this section. Select the Lyapunov function as with whereω ik =ω ik − ω * ik , (k = 2, 3, 4). Theorem 1: Consider a closed-loop control system consisting of the object power systems (6), (7) and the actual controllers (T1.15), (T1.22). If assumptions 1-4 are satisfied and the initial conditions satisfy which can make all signals of the closed-loop system semi-global become bounded uniformly and the system tracking error can converge to arbitrarily small residual set.
Proof: The proof is given in the Appendix. Remark 3: To make it easy for readers to understand, the parameters of the controller proposed in this article can be determined by the following steps: Step 1, according to the actual application, the time constants τ ij (i = 1, 2; j = 2, 3) in (T1.3) and (T1.8) can be selected in the range of 0.001 to 0.1.
Remark 4: The improved FLSs are used as universal approximators to deal with systems' uncertainties. As shown in (T1.13), (T1.20) and (T1.27), the additional adjustment term prediction error e ip is added when designing the weight update law of FLS. The compensated tracking error E i guarantees the output tracking performance of the system, and the prediction error e ip guarantees the authenticity of FLSs approximation. Furthermore, DOB is introduced to handle the external disturbances. In addition to observing the external disturbances, DOB also observes the approximation error of FLS. DOB and FLS work together to solve all unknown nonlinear functions of the system. Through the Lyapunov stability analysis, the stability of the control system is ensured.
Remark 5: The fuzzy logic system is used as a general approximator to approximate unknown nonlinear functions in the system (1), i.e. [16], [19], [21]- [23], [47] and [48]. It is worth noting that the fuzzy system does not need to accurately approximate the unknown functions, but to make the system output accurately track the desired signal. In other words, the estimated values of the norm of fuzzy weight vector will be updated online as the output error of the system changes.

V. EXPERIMENTAL RESULTS
The StarSim Hardware-in-loop testing platform is used to demonstrate the performance of the proposed control scheme. The experiment environment consists of four parts: Host Computer, MT RTS(Real-Time Simulator), Adapter plate and MT RCP (Rapid Control Prototype), and the structure of the test platform is shown in FIGURE 2. The hardware configuration is listed in TABLE 2.
The RTS is utilized to run large-scale power electronic system controlled object. The RTS is built by Simulink or StarSim software on FPGA chips. It can simulate the characteristics of the controlled object and send the response signals to the controller box in real-time. The RCP is used to run   the control algorithm download from StarSim RCP software in real-time and send control signals to RTS to complete the closed-loop experimental system. The adapter plate is used to realize the signal connection between the RCP and RTS. Host computer is used to realize the online parameters adjustment and real-time response signals observation.
In order to verify the effectiveness of the proposed composite learning DSC scheme, a model of two-machine excitation system with SVC equipment (see FIGURE 3). The experiment will be carried out under two different situations: 1) a three-phase short-circuit fault occurs on one transmission line between the two generators and 2) the operation point of power system is changed. The numerical values of the generators and the transmission line are presented in [26]. The traditional tracking error-based NNs method was used as a comparison.
The design parameters of the controllers are chosen as follows: k 11 = 11, k 12 = 43, k 13 The interference signal is chosen as smooth function, d i1 = 0.01cos(t).
The experimental results are shown in FIGUREs 4-12. FIGURE. 4 shows the tracking error of the power angle of the proposed method (denoted as ''DOB-CL'') and the general tracking error based NNs adaptive DSC control method (denoted as ''Error-NNs''). FIGURE 5 shows the response curves of power angle, both methods can track the reference power angle. However, the proposed control method can suppress the disturbance better and obtain a better tracking performance, while the response under Error-NNs is with oscillation. Define the generalized disturbances as FIGUREs.6 and 7 show that the compound estimationˆ i with DOB and FLSs can better ''comprehend'' the true unknown information i . The FLSs     is designed with composite learning that adds the prediction error to the new-type update laws. FIGUREs 8 and 9 show the response curves of rotating speed and the electrical power angles. FIGURE 10 shows the access point voltage response curves of SVC equipment. FIGUREs 11 and 12 show the control input signals for the two generators and the two SVC equipment, respectively.
Case 2: Control response to change of operation point In this case, the system is running steadily and the equilibrium point (EP) is changed at t = 8s. The equilibrium     Remark 6: It is worth noting that the occurrence of power system failures is difficult to predict, and most of them occur when the power system is in a stable working state. In order to make the experimental results in this article closer to the actual working conditions of the power system, the two cases selected in the experimental section are the random faults that occur when the power system is stable. Therefore, the tracking error of power angle in FIGURE 4 is zero at the beginning of the simulation.

Remark 7:
In the experiment, the measurement noise is introduced into the system when the signals are transmitted between MT RTS and MT RCP. The presence of the noise in the actual control signal makes it difficult to select the design parameters of the controller, and it also has an adverse effect on the entire control system. In order to solve this problem, the control algorithm to reduce the measurement noise will be considered in the next research work, such as the control   scheme combining the desired compensation technique [36] and the adaptive dynamic surface.
Remark 8: It is worth noting that the verification of the algorithm in this article is carried out on the semi-physical experimental platform of our laboratory. Due to the particularity of the experimental conditions, only the improved FLSs DSC method and the general NNs DSC method are compared. It is verified that the proposed method in this article has better tracking performance. Furthermore, comparing experiments with other researchers' advanced control algorithms is one of our future research works.

VI. CONCLUSION
In this article, an adaptive fuzzy dynamic surface control method based composite learning and DOB is proposed for large-scale power systems with uncertainties and external disturbances. The approximation capability of the FLSs is influenced considerably when employing the composite learning method. In addition, compensating signals are introduced in the design of the control law and the filtering error caused using the first-order low-pass filter in the DSC was eliminated. Furthermore, DOB is applied to estimate the generalized disturbance, which includes not only external disturbances but also the FLSs' approximation error. Consequently, the accuracy of the control system is improved further. The effectiveness of the proposed method is verified on a Hardware-in-the-loop Testing Platform. Furthermore, in spirited by the previous works, the designing a discrete-time dynamic surface control [31], online policy iteration control [34], [35] and multilayer NNs based active disturbance rejection control [38], [41] for multi-machine power system will be considered in the future works.