Adaptive Dynamic Surface Control for Generator Excitation Control System

Abstract

For the generator excitation control system which is equipped with static var compensator (SVC) and unknown parameters, a novel adaptive dynamic surface control scheme is proposed based on neural network and tracking error transformed function with the following features: (1) the transformation of the excitation generator model to the linear systems is omitted; (2) the prespecified performance of the tracking error can be guaranteed by combining with the tracking error transformed function; (3) the computational burden is greatly reduced by estimating the norm of the weighted vector of neural network instead of the weighted vector itself; therefore, it is more suitable for the real time control; and (4) the explosion of complicity problem inherent in the backstepping control can be eliminated. It is proved that the new scheme can make the system semiglobally uniformly ultimately bounded. Simulation results show the effectiveness of this control scheme.

1. Introduction

With the development of power system, the requirements of the load are increasing. However, the ability of the transmission network is limited and the electric power system has gradually reached its operating limit; the world power systems tend to extend power grid scale and develop larger power system. However, the power systems are more likely to encounter some problems such as oscillations and complex nonlinear phenomena due to the extension of power grid scale and some emergencies. Therefore, ways of maintaining the reliability and stability of power system are attracting more and more research interest [1–4]. Moreover, the promotion of the control ability of the generator excitation system can help us to monitor the states of system and predict system performance; it is the most effective way to improve the performance of the whole excitation control system and it has been widely of concern in recent years [5–7].

The most common method for designing generator excitation system controller is the direct-feedback-linearization, which transforms the nonlinear model into a linear one by some mathematical ways. However, it is only effective to the system when special conditions are satisfied [8, 9]. Now, more and more nonlinear control techniques are being applied to the generator excitation systems. In [10], an output feedback controller combined with high-order sliding-mode techniques is proposed to guarantee the transient stability of the power systems. In [11], for the nonlinear characteristics of the transient process in power system whose parameters are known, Colbia-Vega and his collaborators designed a controller for steam-valving and excitation system, which can stabilize the voltage and improve power quality. In [12], an adaptive fuzzy zero dynamic and sliding model control is proposed to cope with the unknown parameters, and the fuzzy logic rules are used to approximate the unknown nonlinear functions. For the time-varying and uncertainties of the reactance and voltage in one-machine infinite-bus power system, Zhang and Zhu in [13] designed an adaptive excitation controller by using strong tracking filter in the design procedure of excitation controller of generators, and the stability of the control system is improved when it is accompanied with some disturbances; however, it has the explosion of complicity problem inherent in the backstepping control. In [14], an adaptive filter control for generator excitation system was proposed based on IEEE 14-bus.

Motivated by the previous work ([12, 13], etc.) this paper has successfully fused adaptive dynamic surface control with neural network and tracking error transformed function to design a robust adaptive controller for the nonlinear power system. The features of our control scheme are as follows:(i)the tracking error of the power angle and voltage at the SVC can be transformed to an arbitrarily prespecified performance by an error transformed function;(ii)the neural network based adaptive dynamic surface controller is designed without doing any linearization to the excitation generator model. Therefore, the design procedures and the final control law are greatly simplified and the control accuracy is improved;(iii)by using the neural network, the structure and the parameters of the control system can be totally unknown;(iv)by estimating the norm of the weighted vector of neural network instead of the weighted vector itself, the computational burden is greatly reduced and it is more suitable for the real time control.

This paper is organized as follows. In Section 2, the mathematical model of excitation system of power generator is introduced and the control purpose is formulated. In Section 3, the design procedure of the neural network and the tracking error transformed function based adaptive DSC scheme is presented. Section 4 gives the stability analysis for the proposed scheme. Finally, simulation examples are given to demonstrate the effectiveness of the proposed design method.

2. The Excitation System of Power Generator

2.1. Mathematical Model

Assume that the SVC device is installed in a single-machine infinite-bus system; the structure and the equivalent circuit diagram are shown in Figure 1 [15].

Figure 1 

Single machine infinite bus system with SVC.

To proceed, the following assumptions are made.

Assumption 1. Ignoring the dynamic process of rapid excitation equipment means the control voltage 𝑢_𝑓 is equal to the generator excitation voltage 𝐸_𝑓, 𝐸_𝑓=𝑢_𝑓.

Assumption 2. The mechanical power of generators remains unchanged in transient stability process; that is, Δ𝑃_𝑚=0.

Then, the mathematical model of the generator excitation control system equipped with SVC can be described as follows [16–18]: 𝛿̇=𝜔−𝜔₀,̇𝜔𝜔=⁰𝐻𝑃_𝑚0−𝐷𝐻𝜔−𝜔₀−𝜔₀𝐻𝐸^′_𝑞𝑉_𝑠𝑋^′_𝑑Σsin𝛿+𝑑₁,̇𝐸^′_𝑞𝑋=−^𝑑Σ𝑇_𝑑0𝑋^′_𝑑Σ𝐸^′_𝑞+1𝑇_𝑑0𝑋_𝑑−𝑋^′_𝑑𝑋^′_𝑑Σ𝑉_𝑠1cos𝛿+𝑇_𝑑0𝑢_𝑓+𝑑₂,𝐵̇_𝐿=1𝑇_𝐶−𝐵_𝐿+𝐵_𝐶+𝑢_𝐵, (1) where 𝛿 is the power angle of the generator running rotor; 𝜔 is the rotor speed of the generator (rad/s); 𝐷 is the damping constant; 𝐻 is the inertia constant; 𝜔₀=2𝜋𝑓₀ is the synchronous machine speed (rad/s); 𝑓₀ is the rated frequency (Hz); 𝑃_𝑚 is the mechanical power of motor output; 𝑇_𝑑0 is the direct axis transient short circuit time constant (s); 𝑇_𝐶 is the time constant of adjusting system and SVC (s); 𝐸^′_𝑞 is the transient electromotive force (EMF) of the generator; 𝑉_𝑠=1 is the voltage of the system; 𝑢_𝑓 is the control voltage of excitation equipment; 𝑢_𝐵 is the input of SVC; 𝐵_𝐿 is the adjustable equivalent susceptance in SVC; 𝐵_𝐶 is the initial value of adjustable susceptance; 𝑑₁ is position bounded torque interference of the rotor; and 𝑑₂ is electromagnetic transient interference of the stator: 𝑋₁=𝑋^′_𝑑+𝑋_𝑇,𝑋₃=𝑋_𝑑+𝑋_𝑇,𝑋_𝑑Σ=𝑋₃+𝑋₂+𝑋₃𝑋₂𝐵_𝐿−𝐵_𝐶,𝑋^′_𝑑Σ=𝑋₁+𝑋₂+𝑋₁𝑋₂𝐵_𝐿−𝐵_𝐶, (2) where 𝑋₂ is the transmission line reactance; 𝑋_𝑑 is the generator direct axis reactance; 𝑋^′_𝑑 is the direct axis transient reactance of the generator; and 𝑋_𝑇 is the transformer reactance.

When the system encounters the external disturbances, the stability of the power angle and SVC access point voltage can be effectively improved by combining the regulation of SVC with generator-excitation adjustment. In this paper, we take increment of the power angle and accessing point voltage as the output value of the system to design our adaptive dynamic surface controller; that is, 𝑦₁=𝛿−𝛿₀,𝑦₂=𝑉_𝑚−𝑉_ref, (3) where 𝑉_𝑚 is the accessing point voltage and 𝑉_ref is the reference voltage. According to the circuit, we have 𝑉_𝑚=𝑋₂𝐸^′_𝑞²+𝑋₁𝑉_𝑠²+2𝑋₁𝑋₂𝐸^′_𝑞𝑉_𝑠cos𝛿𝑋^′_𝑑Σ. (4)

Based on the above descriptions, let 𝑥₁=𝛿−𝛿₀, 𝑥₂=𝜔−𝜔₀, 𝑥₃=𝐸^′_𝑞, and 𝑥₄=𝑉_𝑚−𝑉_ref; then, (1) can be expressed as the following form: 𝑥̇₁=𝑥₂,𝑥̇₂𝜔=−⁰𝐻𝑉_𝑠sin𝑥₁+𝛿₀𝑋^′_𝑑Σ𝑥₃−𝐷𝐻𝑥₂+𝜔₀𝐻𝑃_𝑚0+𝑑₁,𝑥̇₃=1𝑇_𝑑0𝑢_𝑓−𝑋_𝑑Σ𝑇_𝑑0𝑋^′_𝑑Σ𝑥₃+1𝑇_𝑑0𝑋_𝑑−𝑋^′_𝑑𝑋^′_𝑑Σ𝑉_𝑠cos𝛿+𝑑₂,𝑥̇₄𝑉=−^𝑠sin𝛿𝑋₁𝑋₂𝑉_𝑚𝑋^′_𝑑Σ²𝑥₂𝑥₃−𝑋₁𝑋₂𝑉_𝑚𝑋^′_𝑑Σ⋅1𝑇_𝑐−𝐵_𝐿+𝐵_𝐶+𝑋²₂𝑥₃+𝑋₁𝑋₂𝑉_𝑠cos𝛿𝑉_𝑚𝑋^′_𝑑Σ²⋅−𝑋_𝑑Σ𝑇_𝑑0𝑋^′_𝑑Σ𝑥₃+1𝑇_𝑑0𝑋_𝑑−𝑋^′_𝑑𝑋^′_𝑑Σ⋅𝑉_𝑠+𝑢cos𝛿_𝑓𝑇_𝑑0𝑉_𝑚𝑋^′_𝑑Σ²⋅𝑋²₂𝑥₃+𝑋₁𝑋₂𝑉_𝑠+𝑋cos𝛿₁𝑋₂𝑇_𝐶𝑋^′_𝑑Σ−𝑥₄𝑢_𝐵. (5) Let 𝑔₂𝜔=−⁰𝐻𝑉_𝑠sin𝑥₁+𝛿₀𝑋^′_𝑑Σ,𝑔₃=1𝑇_𝑑0,𝑔₄=𝑋₁𝑋₂𝑇_𝐶𝑋^′_𝑑Σ,𝑓₂𝑥₂𝐷=−𝐻𝑥₂+𝜔₀𝐻𝑃_𝑚0,𝑢^′_𝐵=−𝑥₄𝑢_𝐵,𝑓₄𝑥₄𝑉=−^𝑠sin𝛿𝑋₁𝑋₂𝑉_𝑚𝑋^′_𝑑Σ²𝑥₂𝑥₃−𝑋₁𝑋₂𝑉_𝑚𝑋^′_𝑑Σ⋅1𝑇_𝑐−𝐵_𝐿+𝐵_𝐶+𝑋²₂𝑥₃+𝑋₁𝑋₂𝑉_𝑠cos𝛿𝑉_𝑚𝑋^′_𝑑Σ²⋅−𝑋_𝑑Σ𝑇_𝑑0𝑋^′_𝑑Σ𝑥₃+1𝑇_𝑑0⋅𝑋_𝑑−𝑋^′_𝑑𝑋^′_𝑑Σ𝑉_𝑠+𝑋cos𝛿₂₂𝑥₃+𝑋₁𝑋₂𝑉_𝑠cos𝛿𝑇_𝑑0𝑉_𝑚𝑋^′_𝑑Σ²𝑢_𝑓,𝑓₃𝑥₃𝑋=−^𝑑Σ𝑇_𝑑0𝑋^′_𝑑Σ𝑥₃+1𝑇_𝑑0𝑋_𝑑−𝑋^′_𝑑𝑋^′_𝑑Σ𝑉_𝑠cos𝛿, (6) where 𝑥_𝑖∶=[𝑥₁,…,𝑥_𝑖]^𝑇∈𝑅^𝑖, 𝑖=1,…,4, are state vectors. Then, the system (5) can be converted into the following two independent subsystems (7) and (8), whose relative degree is 3 and 1, respectively: 𝑥̇₁=𝑥₂,𝑥̇₂=𝑔₂𝑥₃+𝑓₂𝑥₂+𝑑₁,𝑥̇₃=𝑔₃𝑢_𝑓+𝑓₃𝑥₃+𝑑₂,𝑦₁=𝑥₁,𝑥̇₄=𝑔₄𝑢^′_𝐵+𝑓₄𝑥₄,𝑦₂=𝑥₄. (7)(8)

Now, we can design our adaptive dynamic surface control law of the power system based on the following assumptions.

Assumption 3. The value of 𝑔₂ is unknown but the sign of 𝑔₂ is known; it is assumed that there exist two constants 𝑔_𝑚 and 𝑔_2max satisfying 0<𝑔_𝑚≤𝑔₂≤𝑔_2max. (9)

Assumption 4. The reference signal 𝑦_𝑟 is smooth and bounded; [𝑦_𝑟,𝑦̇_𝑟,𝑦̈_𝑟]^𝑇 belong to a compact set for all 𝑡≥0.

Remark 5. Assumptions 1 and 2 which are the same as those in [15] are used to construct the model of the excitation system of power generator. Assumption 3 is used in Section 4, such as in the proof of (59), (65), and (67). We also emphasize that 𝑔_𝑚 and 𝑔_2max are not required in the implementation of our control law but are required for analysis only. Assumption 4 is common in all dynamic surface control schemes.

2.2. RBF Neural Network

In general, the neural network is a multi-input single-output system [19]; the mathematical expression is 𝑌=𝜗^𝑇𝜓(𝜉), (10) where 𝜉∈ℝ^𝑛 is the input of RBF neural network; 𝑌∈ℝ is the output of RBFNN, 𝜗∈ℝ^𝑁 is an 𝑁-dimensional parameter vector, and here 𝑁 is the number of neurons; 𝜓(𝜉)∶ℝ^𝑛→ℝ^𝑁 is nonlinear vector function and 𝜓(𝜉)=[𝜓₁(𝜉),…,𝜓_𝑁(𝜉)]^𝑇 and 𝜓_𝑘(𝜉) is the Gauss function.

Lemma 6. RBFNN is a universal approximator in the sense that, given any real continuous function ℱ(⋅)∶Ω_𝜉→ℝ with Ω_𝜉⊆ℝ^𝑛 a compact set and any 𝛿_𝑚>0, by appropriately choosing 𝜎_𝑘 and 𝜁_𝑘, 𝑘=1,…,𝑁, for some sufficiently large integer 𝑁, there exists an RBF network [20], ∀𝜉∈Ω_𝜉⊆ℝ^𝑛, |𝛿|≤𝛿_𝑚, ℱ(𝜉)=𝜗^∗𝑇𝜓(𝜉)+𝛿(𝜉), (11) where 𝜗^∗ is the optimal weight vector of 𝜗 and 𝛿(𝜉) is approximation error and can be defined as 𝜗^∗=argmin_{𝜗∈ℝ𝑛}sup_{𝜉∈Ω𝜉}𝑌(𝜉)(𝜉,𝛿(𝜉)(𝜉)−ℱ)=ℱ−𝜗^∗𝑇𝜓(𝜉). (12)

2.3. Error Transformed Function

We define the tracking error: 𝑒∶=𝑦−𝑦_𝑟, (13) where 𝑦_𝑟 is the desired trajectory. According to [21, 22], the performance function and error transformed function are defined as follows: a performance function 𝜛(𝑡)∶ℝ₊→ℝ₋−{0} is defined as a positive smooth decreasing function, such that, for all 𝑡≥0, (𝑡)(𝑡)(𝑡)(0)(𝑡)(𝑡)(𝑡)(0)−𝜎𝜛<𝑒<𝜛,if𝑒>0,−𝜛<𝑒<𝜎𝜛,if𝑒<0, (14) where 0<𝜎<1 and lim_𝑡→∞=𝜛_∞>0 and 𝜛_∞ is the maximum allowable tracking error of steady-state. To convert (14) into an equivalent unconstrained one, define the error transformed function 𝑒(𝑡)∶=𝜛(𝑡)Φ𝑆₁, (15) where 𝑆₁ is the transformed error and Φ(𝑆₁) is a smooth and strictly monotone increasing function; the inverse function has the following properties: −𝜎<Φ𝑆₁(0)𝑆<1,if𝑒>0,−1<Φ₁<𝜎,if𝑒(0)<0, (16) where lim_{𝑆1→−∞}Φ𝑆₁=−𝜎,lim_𝑆1→∞Φ𝑆₁=1,if𝑒(0)>0,lim_{𝑆1→−∞}Φ𝑆₁=−1,lim_𝑆1→∞Φ𝑆₁=𝜎,if𝑒(0)<0. (17)

As (17) shows, if 𝑆₁∈ℒ_∞, (16) holds. Take 𝜛(𝑡)>0 and (15) into consideration, and we can have −𝜎𝜛(𝑡)<𝜛(𝑡)Φ(𝑆₁)<1 (if 𝑒(0)>0) or −𝜛(𝑡)<𝜛(𝑡)Φ(𝑆₁)<𝜎𝜛(𝑡) (if 𝑒(0)<0); that is, (14) holds. Hence, from the above analysis, to achieve the prespecified tracking performance, we just only need to show that 𝑆₁∈ℒ_∞, where the strictly increasing property of Φ(𝑆₁) guarantees that we can obtain 𝑆₁=Φ⁻¹𝑒(𝑡)𝜛(𝑡). (18)

Note that the case 𝑒(0)=0 can be incorporated into 𝑒(0)>0 or 𝑒(0)<0. At the same time, 𝜎 cannot be chosen to be zero due to 𝑆₁(0) being infinite.

3. Adaptive Dynamic Surface Control of Generator Excitation System

According to the mathematical model of the generator excitation system that is shown in (7), we design adaptive dynamic surface controller about power angle subsystem as follows.

Step 1. Let the first surface error and the tracking error as (18) be defined: 𝑆̇₁−𝜛̇=Ψ𝜛𝑒+𝑥₂−𝑦̇_𝑟,1Ψ∶=𝜛𝜕Φ⁻¹𝜕(),𝑒/𝜛 (19) where 𝑦_𝑟 is an ideal output of the power angle; we suppose that 𝑦_𝑟 is a smooth and continuous function, 𝑒∶=𝑦₁−𝑦_𝑟=𝑥₁−𝑦_𝑟.
Consider the following quadratic function: 𝑉₁=12𝑆²₁. (20) We can obtain that 𝑉̇₁=𝑆₁Ψ−𝜛̇𝜛𝑒−𝑦̇_𝑟+Ψ𝑥₂−𝑥_2𝑑+Ψ𝑥_2𝑑, (21) where 𝑥_2𝑑 is the virtual control signal that will be designed; choose the virtual control law as follows to stabilize (21): 𝑥_2𝑑=−𝑘₁𝑆₁−Ψ−𝜛̇/𝜛𝑒+Ψ𝑆₁−𝑦̇_𝑟Ψ. (22)
In order to avoid “differential explosion,” the new variable 𝑧₂ is obtained by first-order low pass filter: 𝜏₂𝑧̇₂+𝑧₂=𝑥_2𝑑, (23) where 𝜏₂ is the filter time constant.

Step 2. Define the second surface error: 𝑆₂=𝑥₂−𝑧₂. (24)
Consider the following quadratic function: 𝑉₂=12𝑆²₂+𝑔_𝑚𝛾_𝑣2𝑣̃²₂. (25) The time derivative of 𝑉₂ is 𝑉̇₂=𝑆₂𝑔₂𝑥₃+𝑓₂𝑥₂+𝑑₁−𝑧̇₂+𝑔_𝑚𝛾_𝑣2𝑣̃₂𝑣̇̂₂. (26) We need to use neural network to approximate the unknown terms on a compact set Ω_𝜉2: 1𝑔₂𝑓₂𝑥₂−𝑧̇₂+𝑑₁+32𝑔²₂𝑆₂=𝜃_𝛿^∗𝑇₂𝜓₂𝜉₂+𝛿^∗₂𝜉₂, (27) where 𝜉₂∶=(𝑥₁,𝑥₂,𝑥_2𝑑,𝑧₂)∈Ω_𝜉2⊂ℝ⁴; define 𝜐₂=𝜐₂−𝜐^∗₂, and 𝛾_𝜐2 is a positive design parameter. Let the estimated value of 𝜐^∗₂=(1/𝑔_𝑚)‖𝜃_𝛿^∗𝑇₂‖² be 𝜐₂, according to inequality 𝑔₂𝑆₂𝜃_𝛿^∗𝑇₂𝜓₂≤𝑔_𝑚𝛼²₂𝑆²₂𝜐^∗₂𝜓^𝑇₂𝜓₂2+𝑔²_2max2𝛼²₂,𝑔₂𝑆₂𝛿₂≤12𝑔²₂𝑆²₂+12𝛿₂^∗2𝜉₂. (28)(29) From (27) and (28), we have 𝑉̇₂≤𝑆₂𝑔₂𝑥₃−𝑥_3𝑑+𝑔₂𝑥_3𝑑+𝑔_𝑚𝛼²₂𝑆₂𝜐^∗₂𝜓^𝑇₂𝜓₂2−𝑔²₂𝑆²₂+12𝛿₂^∗2+𝑔²_2max2𝛼²₂+𝑔_𝑚𝛾_𝑣2𝑣̃₂𝑣̇̂₂, (30) where 𝑥_3𝑑 is the virtual control signal to be designed; let the power angle actual control law and parameter adjustment law, respectively, be as follows: 𝑥_3𝑑=sgn𝑔₂−𝑘₂𝑆₂−𝛼²₂𝑆₂𝜐₂𝜓^𝑇₂𝜓₂2,𝑣̇̂₂=𝛾_𝑣2𝛼²₂𝑆²₂𝜓^𝑇₂𝜓₂2−𝜎_𝑣2𝑣̂₂. (31)(32) The new variable 𝑧₃ is obtained by first-order low pass filter: 𝜏₃𝑧̇₃+𝑧₃=𝑥_3𝑑, (33) where 𝜏₃ is the filter time constant. In order to avoid “differential explosion” problem brought by backstepping control, we use 𝑧₃ instead of 𝑥_3𝑑 in the following.

Step 3. Define the third surface error: 𝑆₃=𝑥₃−𝑧₃. (34) Consider the following quadratic function: 𝑉₃=121𝑔₃𝑆²₃+1𝛾_𝑣3𝑣̃²₃. (35) The time derivative of 𝑉₃ is 𝑉̇₃=𝑆₃𝑢_𝑓+𝑓₃𝑥₃𝑔₃+𝑑₂𝑔₃−𝑧̇₃𝑔₃+1𝛾_𝑣3𝑣̃₃𝑣̇̂₃. (36) Similar to second step, we use neural network to approximate the unknown term in a compact set: 1𝑔₃𝑓₃𝑥₃−𝑧̇₃+𝑑₂=𝜃_𝛿^∗𝑇₃𝜓₃𝜉₃+𝛿^∗₂𝜉₃, (37) where 𝜉₃∶=(𝑥₁,𝑥₂,𝑥₃,𝑥_3𝑑,𝑧₃)∈Ω_𝜉3⊂ℝ⁵; let 𝜐₃=𝜐₃−𝜐^∗₃, 𝛾_𝜐3 is positive design parameter, and 𝜐₃ is the estimate value of 𝜐^∗₃=‖𝜃_𝛿^∗𝑇₃‖². Consider the hypothesis; we have 𝑉̇₃≤𝑆₃𝑢_𝑓+𝛼²₃𝑆₃𝜐^∗₃𝜓^𝑇₃𝜓₃2+𝑆₃2+12𝛿₃^∗2𝜉₃+12𝛼²₃+1𝛾_𝑣3𝑣̃₃𝑣̇̂₃. (38) Let the power angle actual control law and parameter adjustment law be, respectively, as follows: 𝑢_𝑓=−𝑘₃𝑆₃−𝛼²₃𝑆₃𝜐₃𝜓^𝑇₃𝜓₃2−𝑆₃2,𝑣̇̂₃=𝛾_𝑣3𝛼²₃𝑆²₃𝜓^𝑇₃𝜓₃2−𝜎_𝑣3𝑣̂₃. (39)(40)
Similarly, we design controller for the voltage subsystem (8); surface error is defined in (18): 𝑆̇₄=Ψ₂−𝜛̇₂𝜛₂𝑒₂+𝑔₄𝑢^′_𝐵+𝑓₄𝑥₄−𝑦̇_𝑟2,Ψ₂1∶=𝜛₂𝜕Φ⁻¹𝜕𝑒₂/𝜛₂, (41) where 𝑒₂∶=𝑦₂−𝑦_𝑟2=𝑥₄−𝑦_𝑟2, 𝑦_𝑟2 is the ideal voltage reference signal; consider the following quadratic function: 𝑉₄=121𝑔₄𝑆²₄+1𝛾_𝑣4𝑣̃²₄. (42) The time derivative of 𝑉₄ is 𝑉̇₄=𝑆₄Ψ₂𝑔₄−𝜛̇₂𝜛₂𝑒₂−𝑦̇_𝑟2+Ψ₂𝑓₄𝑥₄𝑔₄+Ψ₂𝑢^′_𝐵+1𝛾_𝑣4𝑣̃₄𝑣̇̂₄. (43) We use neural network to approximate 𝑓₄(𝑥₄): 1𝑔₄Ψ₂𝑓₄𝑥₄−𝜛̇₂𝜛₂𝑒₂−𝑦̇_𝑟2=𝜃_𝛿^∗𝑇₄𝜓₄𝜉₄+𝛿^∗₄𝜉₄, (44) where 𝜉₄∶=(𝑥₄,𝜛₂,𝜛̇₂,𝑒₂,𝑦̇_𝑟2)∈Ω_𝜉4⊂ℝ⁵; let 𝜐₄=𝜐₄−𝜐^∗₄, 𝛾_𝜐3 is a positive design parameter, and 𝜐₄ is an estimate value of 𝜐^∗₄=‖𝜃_𝛿^∗𝑇₄‖²: 𝑉̇₄≤𝑆₄𝛼²₄𝑆₄𝜐^∗₄𝜓^𝑇₄𝜓₄2+Ψ₂𝑢^′_𝐵+12𝑆₄+12𝛿₄^∗2𝜉₄+12𝛼²₄+1𝛾_𝑣4𝑣̃₄𝑣̇̂₄. (45) Design voltage control law and parameter adjustment law are, respectively, as follows: 𝑢^′_𝐵=1Ψ₂−𝑘₄𝑆₄−12𝑆₄−𝛼²₄𝑆₄𝜐₄𝜓^𝑇₄𝜓₄2,𝑣̇̂₄=𝛾_𝑣4𝛼²₄𝑆²₄𝜓^𝑇₄𝜓₄2−𝜎_𝑣4𝑣̂₄. (46)

Remark 7. We emphasize our dynamic surface control scheme which, combined with tracking error transformed function (15), is also suitable for the half-car active suspension system that is described in [23, 24].

4. Stability Analysis

This section will conduct stability analysis for dynamic surface control scheme on the generator excitation system. Although the control law is simple, the stability analysis is relatively complex due to the introduction of first-order low-pass filters. For the first subsystem in (7), to begin with, we define 𝑦_𝑒2=𝑧₂−𝑥_2𝑑=𝑧₂+𝑘₁𝑆₁+Ψ−𝜛̇/𝜛𝑒+Ψ𝑆₁−𝑦̇_𝑟Ψ,𝑦_𝑒3=𝑧₃−𝑥_3𝑑=𝑧₃±𝑘₂𝑆₂+𝛼²₂𝑆₂𝜐₂𝜓^𝑇₂𝜓₂2, (47)(48) where 𝑥_2𝑑 and 𝑥_3𝑑 are designed in (22) and (31), respectively. From (23) and (33), we have 𝑧̇₂=𝑥_2𝑑−𝑧₂𝜏₂𝑦=−^𝑒2𝜏₂,𝑧̇₃=𝑥_3𝑑−𝑧₃𝜏₃𝑦=−^𝑒3𝜏₃. (49) Then, the time derivative of (47) and (48) is 𝑦̇_𝑒2𝑦=−^𝑒2𝜏₂+𝑘₁𝑆̇₁Ψ+𝑘₁𝑆₁Ψ̇Ψ²+𝜛̈𝜛𝜛̇𝑒−²𝜛²𝜛̇𝑒+𝜛𝑒̇+Ψ̇𝑆₁+Ψ𝑆̇₁−𝑦̈_𝑟𝑦=−^𝑒2𝜏₂+𝐵₂𝑆₁,𝑆₂,𝑦₂,𝜛,𝜛̇,𝜛̈,𝑦_𝑟,𝑦̇_𝑟,𝑦̈_𝑟,𝑦̇_𝑒3𝑦=−^𝑒3𝜏₃±𝑘₂𝑆̇₂+𝛼²₂𝑆̇₂𝜐₂𝜓^𝑇₂𝜓₂2+𝛼²₂𝑆₂𝜐̇₂𝜓^𝑇₂𝜓₂2+𝛼²₂𝑆₂𝜐̇₂𝜓₂ײ_𝑗=1𝜕𝜓₂𝜕𝑥_𝑗𝑥̇_𝑗+𝜕𝜓₂𝜕𝑆₂𝑦=−^𝑒3𝜏₃+𝐵₃𝑆₁,𝑆₂,𝑆₃,𝑦₂,𝑦₃,𝜐₂,𝜛,𝜛̇,𝜛̈,𝑦_𝑟,𝑦̇_𝑟,𝑦̈_𝑟, (50) where 𝐵₂ and 𝐵₃ are continuous functions.

Theorem 8. Consider the closed-loop control system composed by object (7), first-order low-pass filter (23), (33), the actual control law (39), and parameter adjustment law (40): 𝑉=𝑉₁+𝑉₂+𝑉₃+12𝑦²_𝑒2+12𝑦³_𝑒3, (51) where 𝑉₁, 𝑉₂, and 𝑉₃, respectively, are defined by (20), (25), and (35). We suppose that for the given positive constant 𝛿^∗_𝑖, (27), (37), and (44) meet |𝛿_𝑖(𝜉_𝑖)|≤𝛿^∗_𝑖, 𝑖=2,3,4, in a compact set. For any given positive parameter 𝑝, if 𝑉(0)≤𝑝, (52) then through the proper selection of the design parameters 𝑘₁, 𝑘₂, 𝑘₃,𝛾_𝜐2,𝛾_𝜐3, 𝜎_𝜐2, and 𝜎_𝜐3 all the variables in the closed-loop system are uniformly ultimately bounded. The tracking performance of the system can be specified in advance.

Proof. The derivative of 𝑉 by considering (51) is 𝑉̇=𝑉̇₁+𝑉̇₂+𝑉̇₃+𝑦_𝑒2𝑦̇_𝑒2+𝑦_𝑒3𝑦̇_𝑒3. (53) From (24) and (47), we have 𝑥₂=𝑆₂+𝑦_𝑒2+𝑥_2𝑑. (54) And consider the inequality Ψ𝑆₁𝑆₂≤12Ψ²𝑆²₁+12𝑆²₂,Ψ𝑆₁𝑦_𝑒2≤12Ψ²𝑆²₁+12𝑦²_𝑒2. (55) Substituting (54) and (55) and virtual control law (22) into (21), we have 𝑉̇₁≤−𝑘₁𝑆²₁+12𝑆²₂+12𝑦²_𝑒2. (56) Using (34) and (48), 𝑥₃=𝑆₃+𝑦_𝑒3+𝑥_3𝑑. (57) At the same time, 𝑔₂𝑆₂𝑆₃≤12𝑔²₂𝑆²₂+12𝑆²₃,𝑔₂𝑆₂𝑦_𝑒3≤12𝑔²₂𝑆²₂+12𝑦²_𝑒3. (58) Substitute (57) and (58), virtual control law (31), and parameter adjustment law (32) into (30): 𝑉̇₂≤−𝑘₂𝑔₂𝑆²₂+12𝑆²₃+12𝑦²_𝑒3+12𝛿₂^∗2+𝑔²_2max2𝛼²₂−𝑔_𝑚𝜎_𝑣2𝑣̂₂𝑣̃₂. (59) From (39) and (40), (38) can be transformed as 𝑉̇₃≤−𝑘₃𝑆²₃+12𝛿₃^∗2+12𝛼²₃−𝜎_𝑣3𝑣̂₃𝑣̃₃. (60) According to Assumption 4, Π∶=𝑦_𝑟,𝑦̇_𝑟,𝑦̈_𝑟∶𝑦²_𝑟+𝑦̇²_𝑟+𝑦̈_𝑟≤𝐵₀ (61) is a compact set in ℝ³, and 𝐵₀>0. 𝑀₂ is the maximum of |𝐵₂| in Π; 𝑀₃ is the maximum of |𝐵₃| at Π. For any 𝑝>0, using inequality, we can obtain that 𝑦_𝑒2𝐵₂≤𝑦²_𝑒2𝐵²₂+𝜇2𝜇2≤𝑦²_𝑒2𝑀²₂+𝜇2𝜇2,𝑦_𝑒3𝐵₃≤𝑦²_𝑒3𝐵²₃+𝜇2𝜇2≤𝑦²_𝑒3𝑀²₃+𝜇2𝜇2, (62)(63) where 𝜇 is a positive constant parameter 1𝜏₂=12+𝑀²₂2𝜇+𝛼₀,1𝜏₃=12+𝑀²₃2𝜇+𝛼₀. (64) Furthermore, −𝑔_𝑚𝜎_𝑣2𝑣̂₂𝑣̃₂𝜎≤−^𝑣2𝑔_𝑚2𝑣̃²₂+𝜎_𝑣2𝑔_𝑚2𝑣₂^∗2,−𝜎_𝑣3𝑣̂₃𝑣̃₃𝜎≤−^𝑣32𝑣̃²₃+𝜎_𝑣32𝑣₃^∗2. (65)(66) Taking (56), (59), (60), and (62)–(66) into consideration, (53) can be transformed into the following equation: 𝑉̇≤−𝑘₁𝑆²₁−𝑘₂𝑔_𝑚−12𝑆²₂−𝑘₃−12𝑆²₃−𝛼₀𝑦²_𝑒2−𝛼₀𝑦²_𝑒3−𝜎_𝑣2𝑔_𝑚2𝑣̃²₂−𝜎_𝑣32𝑣̃²₃+𝐶, (67) where 𝜎𝐶=^𝑣2𝑔_𝑚2𝑣₂^∗2+𝜎_𝑣32𝑣₃^∗21+𝜇+2𝛿₂^∗2+12𝛿₃^∗2+𝑔²_2max2𝛼²₂+12𝛼²₃. (68) And 𝛼₀ is a positive design parameter: 𝛼₀≤min𝑘₁,𝑘₂𝑔_𝑚−12,𝑘₃−12,𝛾_𝑣2𝜎_𝑣22,𝛾_𝑣3𝜎_𝑣32. (69) From (67), we have 𝑉̇≤−2𝛼₀𝑉+𝐶. (70) Let 𝛼₀>𝐶.2𝑝 (71) We can see that when 𝑉=𝑝, 𝑉̇≤0. So, 𝑉≤𝑝 is an invariant set. If 𝑉(0)≤𝑝, then, for all 𝑡≥0, 𝑉(𝑡)≤𝑝. From (70), ∀𝑡≥0, (𝑡)≤𝐶0≤𝑉2𝛼₀+𝑉(0)−𝐶2𝛼₀𝑒^{−2𝛼0𝑡}. (72) Then, lim_𝑡→∞𝑉(𝑡)=𝐶2𝛼₀. (73)
Similar to the second subsystems, we substitute (46) into (45): 𝑉̇₄≤−𝑘₄𝑆²₄−𝜎_𝜐42𝜐²₄+12𝛿₄^∗2+12𝛼²₄+𝜎_𝜐42𝜐₄^∗2. (74) Let 𝑘₄≥𝑎𝑔₄,𝜎_𝑣4≥2𝑎𝛾_𝑣4,𝐶₂=𝜎_𝑣42𝑣₄^∗2+12𝛿₄^∗2+12𝛼²₄. (75)(76)(77) We have 𝑉̇₄≤−2𝑎𝑉+𝐶₂,lim_𝑡→∞𝑉₄(𝑡)=𝐶.2𝑎 (78)(79)
As a result, all the signals such as 𝑆₁, 𝑆₂, 𝑆₃, 𝑆₄, 𝑘₁, 𝑘₂, 𝑘₃, 𝑘₄, 𝜎_𝑣2, 𝜎_𝑣3, 𝜎_𝑣4, 𝛼₀, and 𝑎 in the closed-loop system are uniformly bounded. In addition, (69), (75), and (76) show that, by selecting appropriate design parameters, 𝛼₀, 𝑎 can be infinite; (73) and (79) show that the tracking error can converge to an arbitrary small value.

5. Simulation Analysis

The single machine infinite bus power system with SVC is shown in Figures 1 and 2, assuming that the amplitude and frequency of receiving end bus voltage are not changed.

Figure 2 

The equivalent circuit diagram.

System parameters are 𝑋^′_𝑑=0.344, 𝑋_𝑑=0.982, 𝑋_𝑇=0.1, 𝐷=2, 𝑋^′_𝑑Σ=0.6967, 𝐵_𝐿=1.55, 𝐵_𝐶=0.5, 𝑇_𝑑0=5, 𝑇_𝐶=2, 𝐻=13, 𝐸^′_𝑞=1.059, 𝛿₀=40, 𝜔₀=100𝜋 rad/s, 𝑃_𝑚0=1, 𝑉_𝑠=1, 𝑉_ref=1, 𝑋₂=0.35, and 𝑋_𝑑Σ=1.4.

In the simulation the initial values of the states are 𝑥₁(0)=40, 𝑥₂(0)=100𝜋, 𝑥₃(0)=1.05, and 𝑥₃(0)=1. For NN Gaussian function 𝜓_𝑗(𝜉_𝑖)=exp[−(𝜉_𝑖−𝜁_𝑗)^𝑇(𝜉_𝑖−𝜁_𝑗)/𝜂²_𝑗], 𝑖=2,3,4, we choose 21 nodes with the centers of the basis functions 𝜁_𝑗 evenly spaced in [−40,+40] × [−314,+3144] × [−2,+2] × [−1,+1] and width 𝜂_𝑗=1,for𝑗=1,…,21. Also, we choose a smooth and strictly monotone increasing function, Φ₂(𝑆₄)=(2/𝜋)arctan(𝑆₄), 𝜛(𝑡)=0.65𝑒^−8𝑡+0.002 and 𝑆₄=tan((𝜋/2)⋅(𝑒₂(𝑡)/𝑤₂(𝑡))), which means that Ψ₂=(𝜋/2)⋅(1/𝜛₂)⋅cos⁻²((𝜋/2)⋅(𝑒₂/𝜛₂)).

We suppose that three-phase short-circuit occurred on the terminal of the line at 𝑡=8 s and eliminate the fault at 𝑡=9 s; the simulation results are shown in Figures 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13. Figure 3 shows that, by introducing the error transformed function, the system can be stabilized promptly when it encountered some emergencies such as three-phase short-circuit which means that our control scheme makes the generator excitation control system have strong robustness. The tracking error of power angle can be prespecified by using the performance function 𝜛(𝑡)=0.65𝑒^−8𝑡+0.002. Figures 4 and 5 are actual control of power angle and actual voltage control law of the access point voltage, respectively. From Figure 6–Figure 8 we can know that all the states in the control system recover to their desired values in a short time by using our dynamic surface control scheme. Figure 9 to Figure 11 show the estimations of weight vector norm of the neural network. Figure 12 is the access point voltage increment 𝑉_𝑚−𝑉_ref, whose tracking process can also be arbitrarily prespecified by using performance function as that in Figure 3.

Figure 3 

Tracking error and the performance function.

Figure 4 

The actual control law 𝑢𝑓 of power angle.

Figure 5 

The actual voltage control law 𝑢′𝐵 of access point voltage.

Figure 6 

The rotor angle 𝛿 of generator.

Figure 7 

The rotor speed 𝜔 of generator.

Figure 8 

The transient EMF 𝐸′𝑞 of the generator.

Figure 9 

The estimated norm of the weighted vector 𝜐2, with 𝜐∗2=‖𝜃𝛿∗𝑇2‖2.

Figure 10 

The estimated norm of the weighted vector 𝜐3, with 𝜐∗3=‖𝜃𝛿∗𝑇3‖2.

Figure 11 

The estimated norm of the weighted vector 𝜐4, with 𝜐∗4=‖𝜃𝛿∗𝑇4‖2.

Figure 12 

The access point voltage increment 𝑉𝑚−𝑉ref.

Figure 13 

The comparison of our NN based dynamic surface control scheme and backstepping.

Remark 9. The controllers that have been proposed in [12, 13] are based on backstepping technique, which can cause the explosion of complicity problem that means the controller will become complex when the relative degree of the control system is high. Also, our novel neural network based adaptive dynamic surface controller is designed without doing any linearization to the excitation generator model. Therefore, the design procedures and the final control law are greatly simplified and the control accuracy is improved. Figure 13 has shown the comparison of the tracking error between our control scheme and backstepping control scheme.

6. Conclusion

This paper proposed a novel neural network and tracking error transformed function based on adaptive dynamic surface control scheme for the generator excitation control system which is equipped with SVC. In this scheme, both the designing procedures and the control law are simplified because it does not need to make any transformation to the excitation generator model. The prespecified tracking performance can be guaranteed by using the tracking error transformed function. The norm of the weighted vector of neural network is estimated which replaces the estimation of the weighted vector itself; therefore, the computational burden is greatly reduced. Also, the explosion of complicity problem inherent in the backstepping control can be eliminated. It is proved that the new scheme can make the system semiglobally uniformly ultimately bounded.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research is supported by Natural Science Foundation (no. 61304015), China Postdoctoral Science Foundation (no. 2013M540839), Outstanding Young Scholar Project of Jilin City (no. 2013625002), “Twelfth Five Year” Scientific Research Plan of Jilin Province (no. [2014]111), and Nature Science Foundation of Jilin Province (no. 20140101059).