# Coordinated Switching Control of the Excitation and Steam Valve System Based on State and Input Constraints

Authors: Nan Jiang; Ting Liu; Haiyan He; Xin Li
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

A single unit infinite system of the excitation and steam valve control was proposed based on Barrier Lyapunov theory of restrictive log type. The input amplitude constraint of the steam valve control was considered, and the coordinated nonlinear backstepping controller was designed by switching mechanism. At the same time, the generator rotor effect was considered to be an external unknown large disturbance on the output system, and the conservativeness of the simple estimates for the upper and lower bounds and scaling disturbance was reduced by Minimax. The Minimax method also ensured that the output of the controller and the power angle were within the prescribed range and inhibited the system output effect of disturbance as much as possible. Finally, simulation results of the generator disturbance of mechanical power in the single unit infinite system show that the control scheme effectively improves the transient stability of the dynamic processes of power systems.


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## Body

## 1. Introduction

Turbine generator excitation control and steam valve control are two key methods for improving the stability of power systems. The rising speed of the excitation current is limited to the constant of the field winding, and excitation control is limited to the maximum value of the excitation current (the maximum value of the excitation current of a generator is generally 1.8 up to 2.0 times its rated current). Requiring excessive maximum value excitation current will increase the manufacturing costs of generators [1], and relying just on excitation control to enhance system stability results in limited improvement.A prime mover using nonlinear control for opening the steam valve is often effective for further improving the transient stability of power systems. In view of the large power reheat turbine generator sets used in power systems, mechanical hydraulic governors are increasingly being replaced by power-frequency electrohydraulic governors. The capacity and load adaptability of the primary frequency are improved by the opening steam valve control in the reheat turbine generator set. Thus, an increasing number of engineering researchers are focusing on the stability of power systems.Many advanced control methods have been recently applied to the excitation and steam valve controller design of generators. Reference [2] described a nonlinear model of power systems that conducts precise linearization of state variable feedback. In this model, the nonlinear controller of the opening steam valve was designed by differential geometry method, but for the state variables feedback exact linearization method was applied based on the exact state space model; the method was not robust to uncertainty of the parameters and models.The nonlinear robust controller designed by adaptive backstepping was discussed in [3]. Although the internal and external disturbances of the system were considered, disturbance was just binding scaling, and the bound properties of the control variable were not considered. The nonlinear dispersion steam valve controller was designed by limiting the control variable for multimachine systems, which was difficult to achieve in [1] because of excessive computations. The adaptive backstepping method was used to design a nonlinear adaptive robust controller that simultaneously considered the input constraints of the main valve and the fast valve [4]. The steam valve controller was designed without considering the coordinated excitation and valve control. A design was proposed for a nonlinear adaptive robust controller based on adaptive backstepping in which coordinated excitation and valve control were simultaneously considered under input constraints [5]. However, disturbance was simply limited to the scaling of estimate bounds without consideration of state constraints. Although the state constraints of the main steam valve were considered, the impact of disruption perturbation was not considered in the design processes of [6–9].Modern power systems are very complex, and they contain a wide variety of devices. These devices differ in physical properties and operating regions. Thus, the different constraints of these various devices should be considered in the automation of power systems [10]. Existing research results were obtained by common Lyapunov function, and the switching mechanism was introduced to obtain a better solution by establishing the constraints of control variable input. However, these studies did not indicate any limit to the state constraints for the system. The control problem for a class of constrained systems was solved by constructing a log type of symmetric Barrier Lyapunov Function [11]. A class of control problems with strictly limited output feedback nonlinear systems was studied in [12]. In the present paper, we propose a new type of log BLF (Barrier Lyapunov Function) to expand the range of the initial selection. This method is convenient and easy for the controller design process.In addition, this paper introduces the system constraints of the status rotor running angleδ by constructing Barrier Lyapunov Function to establish the state bounds of the system. First, a nonlinear controller is designed by backstepping. Then, external disturbance is combined with Minimax to reduce the conservativeness of simple scaling disturbances. By using parameter-mapping mechanism, the adaptive law is designed by the upper and lower bounds of the parameters. Then, we estimate the uncertain parameters of the system and establish the coordinated excitation and valve control model. The switching mechanism is introduced to deal with the input amplitude constraint problems of the main steam and fast steam valve control in the system. To satisfy the system’s state constraints, the controller is designed with backstepping which is nonlinear and simultaneously combined with relevant constrained signal and constraint values. The construction of a restrictive log type of Barrier Lyapunov Function guarantees that the state power angle set will be within the prescribed range. The Simulation results reveal that amplitude of the state response curve for the system is small, inhibitory effect of disturbance is good, and the limited control inputs are within the set range.


## 2. Establishment of the Coordinated Excitation and Valve Control Model

The dynamic equations of generator excitation winding are as follows:(1)Vf=Td0′E˙q′+Eq-Efds,Eq=xdΣxdΣ′Eq′-xd-xd′Vscos⁡⁡δxdΣ′,where Vf is the excitation voltage, Td0′ is the time constant of the excitation winding, Eq′ is the q axis transient voltage, Eq is the no load electromotive force of generator, Efds is given the excitation voltage of the stable operation, Vs is the infinite bus voltage, xdΣ is the equivalent reactance between Eq and Vs, xdΣ′ is the equivalent reactance between Eq′ and Vs, and xd and xd′ are the d-axis reactance and transient reactance, respectively.Thus, the excitation equation is(2)δ˙=ω-ω0,ω˙=-DHω-ω0+ω0HPm-Eq′VsxdΣ′sin⁡δ+ξ1,E˙q′=1Td0′-xdΣxdΣ′Eq′+xd-xd′xdΣ′Vscos⁡⁡δ+Efds+Vf+ξ2,where δ is the rotor angle of generator, ω is the rotor speed of the generator, ω0 is the generator synchronous speed, H is the inertia of generator rotor, D is the uncertain parameter of the system model, that is, damping coefficient which cannot be accurately measured, ξ1 is external disturbance of the generator rotor, ξ2 is external disturbance of the generator transient potential, and ξ1 and ξ2 are considered external disturbances of the system.The model is based on the excitation control model of power system. The proposed model is relatively close to the actual model because of two factors for consideration. One is the uncertainty derived from the damping coefficient which is difficult to measure in the actual system. The second is the influence of external disturbances both on generator potential transient speed and on transient voltage.Nowadays, middle reheat is widely used by large-capacity turbine generator sets, and its physical structure is shown in Figure1.Figure 1
Physical structure model of reheat steam turbine.The high-pressure cylinder and the middle-pressure cylinder are equivalent to an inertia link. The equivalent time constants areTHL and TML. The equivalent power distribution coefficients are CH and CML, with mechanical power output of PH and PML, respectively. The total mechanical power output of the prime mover is Pm, which is equal to the mechanical power output of the high-pressure cylinder and the middle-pressure cylinder expressed as PH and PML. Figure 2 shows the block diagram of the value control system transfer function.Figure 2
Block diagram of valve control system transfer function.From Figure2, we can determine practical valve control system differential equations as follows:(3)P˙H=-1THΣPH-CHPm0+CHTHΣu1+ξ3,u1min⁡≤u1≤u1max⁡,(4)P˙ML=-1TMHΣPML-CMLPm0+CMLTMLΣu2+ξ4,u2min⁡≤u2≤u2max⁡,(5)THΣ=TH+THg,TMLΣ=TML+TMLg,where TH and THg are the time constants of the high-pressure cylinder and the main valve oil motive, respectively; TML and TMLg are the time constants of the low-pressure cylinder and the regulating valve oil motive, respectively; PH and PML are the mechanical power outputs of the high-pressure cylinder and the middle-pressure cylinder, respectively; THΣ is the sum time constant of the high-pressure cylinder and the main valve controller; TMLΣ is the sum time constant of the low-pressure cylinder and the fast valve controller; CH and CML are the power distribution coefficients of the high-pressure cylinder and the low-pressure cylinder and are 0.7 and 0.3, respectively; u1 and u2 are the control input of the main valve and the fast valve, respectively; u1min⁡ and u1max⁡ are the min and max control input of the main valve, respectively; u2min⁡ and u2max⁡ are the min and max control input of the fast valve, respectively; ξ3 and ξ4 are the external disturbances of the main valve and the fast valve, respectively.According to (3) and (4), we can obtain the system equations of the main valve and the fast valve as follows.The main valve control equation is(6)δ˙=ω-ω0,ω˙=-DHω-ω0+ω0HPH+PML-Eq′VsXdΣ′sin⁡δ+ξ1,P˙m=P˙H=-1THΣPH-CHPm0+CHTHΣu1+ξ3,u1min⁡≤u1≤u1max⁡.The fast valve control equation is(7)δ˙=ω-ω0,ω˙=-DHω-ω0+ω0HPH+PML-Eq′VsXdΣ′sin⁡δ+ξ1,P˙m=P˙ML=-1TMLΣPML-CMLPm0+CMLTMLΣu2+ξ4,u2min⁡≤u2≤u2max⁡.The control model of the main valve and the fast valve contains the uncertain parametersD, external disturbances ξ1, ξ2, and ξ3, and input constraint bounds of the main valve and the fast valve control.In summary, the excitation and the main valve equations are as follows:(8)δ˙=ω-ω0,ω˙=-DHω-ω0+ω0HPm-Eq′VsxdΣ′sin⁡δ+ξ1,E˙q′=1Td0′-xdΣxdΣ′Eq′+xd-xd′xdΣ′Vscos⁡⁡δ+Efds+u1+ξ2,P˙m=P˙H=-1THΣPH-CHPm0+CHTHΣu2+ξ3.The excitation and the fast valve equations are as follows:(9)δ˙=ω-ω0,ω˙=-DHω-ω0+ω0HPm-Eq′VsxdΣ′sin⁡δ+ξ1,E˙q′=1Td0′-xdΣxdΣ′Eq′+xd-xd′xdΣ′Vscos⁡⁡δ+Efds+u1+ξ2,P˙m=P˙ML=-1TMHΣPML-CMLPm0+CMLTMLΣu2+ξ4.The two switching subsystems are the excitation and main valve control system and the excitation and fast valve control system. These systems are very similar and therefore we can only choose coordinated control of the excitation and main valve for instructing the controller design process.The coordinated control model of the excitation and main valve is as follows:(10)x˙1=x2,-kb<x1<kb,x˙2=θx2+k1x4+k1x5+a0+k2sin⁡x1+δ0x3+k9sin⁡x1+δ0+ξ1,x˙3=k3x3+k5+k4cos⁡⁡x1+δ0+k6u1+ξ2,x˙4=k7x4+k8u2+ξ3,x˙5=0,z=q1x1q2x2T,where kb and -kb are the upper and lower bounds of x1, respectively, -D/H=θ, ω0/H=k1, (ω0/H)Pm0=a0, -ω0Vs/HxdΣ′=k2, -xdΣ/Td0′xdΣ′=k3, (xd-xd′/Td0′xdΣ′)Vs=k4, -xdΣEq0′/Td0′xdΣ′+Efds/Td0′=k5, 1/Td0′=k6, -1/THΣ=k7, CH/THΣ=k8, -ω0Eq′Vs/(1+XdΣ′)=k9, z is the regulating output, q1 and q2 are nonnegative weighting coefficients that account for the weighted proportion between x1 and x2, with q12+q22≤1, D is the uncertain parameter that is difficult to accurately measure, and θ is an uncertain constant.


## 3. Excitation and Valve Controller Design of State and Input Constraints

Lemma 1 (see [12]).
A Barrier Lyapunov Function is a scalar functionVx, defined with respect to the system x˙=fx on an open region D containing the origin, that is continuous, positive definite, has continuous first-order partial derivatives at every point of D, has the property Vx→∞ as x approaches the boundary of D, and satisfies Vx≤b∀t>0 along the solution of x˙=fx for x0∈D and some positive constant b.Lemma 2 (see [12]).
For any positive constantskb, let Z1≔z1∈R:-kb<z1<kb⊂R and w⊂Rl-1 be open sets. Consider the system μ˙=h(t,μ), where μ=[z1,w]T and h⊂Rl is piecewise continuous in t. Suppose that there exist functions U: w→R+ and V1: Z1→R+, continuously differentiable and positive definite in their respective domains, such that(11)V1z1⟶∞asz1⟶-kborz1⟶kb,λ1w≤Uw≤λ2w,where λ1 and λ2 are class K∞ functions. Let Vμ=V1(z1)+U(w), and z1(0) belong to the set Z1. If the inequality holds V˙=(∂V/∂μ)h≤0, then z1(t) remains in the open set Z1.Step 1.
Consideringx2 to be a virtual controller and choosing x2∗=-c1(kb2-e12)e1, c1>0, let e1=x1, e2=x2-x2∗; we obtain the common Barrier Lyapunov Function of V1 in the first step as follows:(12)V1=12log⁡⁡kb2kb2-e12.
Then(13)V˙1=12kb2-e12kb2·2e1kb2e˙1kb2-e122=e1e2kb2-e12-c1e12.Clearly, if e2=0, we have V˙1<0.Step 2.
We consider the second equation of (10), our task in this step is to make e2→0. By augmenting V1, the Lyapunov function V2 is choosen as V2=V1+(1/2)e22. Satisfying dissipation inequality for arbitrary disturbance, we define the energy function as (14)H1=V˙2+12z2-γ2ξ12.
The performance function is(15)J1=∫0∞z2-γ2ξ12dt,because(16)V˙2=e1e2kb2-e12-c1e12+e2θx2+k1z4+a0+k2sin⁡x1+δ0x3+k9sin⁡x1+δ0+ξ1+c1kb2x2-3c1e12x2.
By substituting (16) into (14), we get(17)H1=-c1e12+e2e1kb2-e12+θx2+k1z4+a0+k2sin⁡x1+δ0x3+k9sin⁡x1+δ0+ξ1+c1kb2x2-3c1e12x2+12q12x12+12q22x22-12γ2ξ12.
Take the first partial derivative ofH1 for ξ1 and let H1˙=0; then ∂H1/∂ξ1=e2-γ2ξ1=0. Therefore, we can obtain (18)ξ1∗=e2γ2.
Take the second partial derivative ofH1 for ξ1; then we get ∂2H1/∂ξ12=-(1/2)γ2<0. We know that H1 has a maximum value for ξ1. According to the proof presented in the previous chapter, we know that H1 can get the maximum for ξ1∗. Thus, we can conclude that the performance J1 reaches the maximum. Clearly, ξ1∗ is the greatest degree of disturbance impact on the system.
By substituting (18) into (17), we get(19)H1=-αe12+e2e1kb2-e12+θx2+k1z4+a0+k2sin⁡x1+δ0x3+k9sin⁡x1+δ0+h1x2-3c1e12x2+h2e2-q22c1kb2-e12e1,where α=c1-(1/2)q12-(1/2)q22c12(kb2-e12)2, h1=c1kb2, and h2=1/2γ2+1/2q22.Let(20)h2e2+k2sin⁡x1+δ0x3+k9sin⁡x1+δ0=-e2,e1kb2-e12+a0+θx2+k1z4+h1x2-q22c1kb2-e12e1-3c1e12x2=-e2.We can get the virtual control(21)x3∗=-1k2sin⁡x1+δ0h3e2+k9sin⁡x1+δ0,z4∗=-1k1e1kb2-e12+a0+θ^x2+h1x2-q22c1kb2-e12e1-3c1e12x2+e2,where h3=h2+1, and we use the estimated θ^ as θ. Then(22)H1=-αe12-2e22+θ-θ^x2e2.Step 3.
Considering the unknown parameterθ in the second equation of (10), θ∈θmin⁡θmax⁡; θmin⁡, and θmax⁡ are, respectively, the upper and lower bounds of θ. Augmented V2, the function of a common Lyapunov function of subsystem 1 and subsystem 2, is (23)V3=V2+12e32+12e42+12ρθ--θ2-θ--θ^2,where θ- is a mapping mechanism, θ^ is the estimate of θ,  ρ is the adaptive law of gain, ρ>0, e3=x3-x3∗, and e4=z4-z4∗.
We select the energy function(24)H2=V˙3+12z2-γ2ξ2.
The performance function is(25)J2=∫0∞z2-γ2ξ2dt,where ξ=ξ1ξ2ξ3T and the derivative of formula (23) is(26)V˙3=V˙2+e3e˙3+1ρθ--θθ-˙-θ--θ^θ-˙-θ^˙.
The derivatives of formula (21) are (27)x˙3∗=-h3θx2+k1z4+a0+k2sin⁡x1+δ0x3+k9sin⁡x1+δ0+e2/γ2k2sin⁡x1+δ0-h3c1kb2x2-3c1e12x2+k9cos⁡⁡x1+δ0x2k2sin⁡x1+δ0-h3e2+k9sin⁡x1+δ0k2cos⁡⁡x1+δ0x2k22sin2⁡x1+δ0,(28)z˙4∗=-1k1x2kb2+e12kb2-e122+θ^˙x2+θ^x˙2+h1+1x˙2+3q22c1-3c1e12x2-6c1e1x22-3c1e12x˙2+c1kb2-q22c1kb2x2.
By substituting from (26) to (28) into (24), we get (29)H2=H1+e3k3x3+k5+k4cos⁡⁡x1+δ0+k6u1+ξ2-x˙3+e4k7x4+k8u2+ξ3-z˙4-12γ2ξ22-12γ2ξ32-1ρθ-θ^θ-˙-θ--θ^θ^˙.
Take the first partial derivative ofH2 for ξ2 and ξ3 and make them equal to 0; then(30)∂H2∂ξ2=e3-γ2ξ2=0,∂H2∂ξ3=e4-γ2ξ3=0.
Therefore(31)ξ2∗=e3γ2,ξ3∗=e4γ2.
Take the second partial derivative ofH2 for ξ2 and ξ3; then we can get ∂2H2/∂ξ22=-γ2<0 and ∂2H2/∂ξ32=-γ2<0. We know that H2 has a respective maximum value for ξ2 and ξ3. Therefore, because ξ2∗ and ξ3∗ can make H2 reach the maximum value, we can also make the performance J2 reach the maximum. Clearly, ξ2∗ and ξ3∗ are really the greatest degrees of disturbance impact on the system.
By substituting (31) into (29), we get(32)H2=H1+e3k3x3+k5+k4cos⁡⁡x1+δ0+k6u1+e32γ2+h3θx2+k1z4+a0+k2sin⁡x1+δ0x3+k9sin⁡x1+δ0+e2/γ2k2sin⁡x1+δ0+h3c1kb2x2-3c1e12x2+k9cos⁡⁡x1+δ0x2k2sin⁡x1+δ0+h3e2+k9sin⁡x1+δ0k2cos⁡⁡x1+δ0x2k22sin2⁡x1+δ0+e4k7x4+k8u2+e42γ2+1k1x2kb2+e12kb2-e122+θ^˙x2+θ^x˙2+h1+1x˙2+3q22c1-3c1e12x2-6c1e1x22-3c1e12x˙2+c1kb2-q22c1kb2x2-1ρθ-θ^θ-˙-θ--θ^θ^˙.
Let(33)-c3e3=k3x3+k5+k4cos⁡⁡x1+δ0+k6u1+x3+e32γ2+k9sin⁡x1+δ0+e2/γ2k2sin⁡x1+δ0+h3c1kb2x2-3c1e12x2+k9cos⁡⁡x1+δ0x2+h3θx2+k1z4+a0x3k2sin⁡x1+δ0+h3e2+k9sin⁡x1+δ0k2cos⁡⁡x1+δ0x2k22sin2⁡x1+δ0,-c3e4=k7x4+k8u2+e42γ2+1k1x2kb2+e12kb2-e122+θ^˙x2+θ^x˙2+h1+1x˙2+3q22c1-3c1e12x2-6c1e1x22-3c1e12x˙2+c1kb2-q22c1kb2x2.
The controller functions are(34)u1=-1k6k3x3+k5+k4cos⁡⁡x1+δ0+e32γ2+c3e3+h3θ^x2+k1z4+a0+k2sin⁡x1+δ0x3+k9sin⁡x1+δ0+e2/γ2k2sin⁡x1+δ0+h3c1kb2x2-3c1e12x2+k9cos⁡⁡x1+δ0x2k2sin⁡x1+δ0+h3e2+k9sin⁡x1+δ0k2cos⁡⁡x1+δ0x2k22sin2⁡x1+δ0,u2=-1k8k7x4+e42γ2+c4e4+1k1x2kb2+e12kb2-e122+θ^˙x2+θ^x˙2+h1+1-3c1e12θ^x2+k1z4+a0+k2sin⁡x1+δ0x3+k9sin⁡x1+δ0+e2γ2+3q22c1-3c1e12x2-6c1e1x22+c1kb2-q22c1kb2x2.
By substituting (34) into (32), we get(35)H2=-αe12-2e22-c2e32-c3e42+θ-θ^x2e2+e4k1θ^-3c1e12θ-θ^x2+1k1h1+1θ-θ^e4x2+h3θ-θ^x2k2sin⁡x1+δ0e3-1ρθ-θ^θ-˙-θ--θ^θ^˙.
The adaptive law follows(36)θ-˙=ρx2e2+h3x2k2sin⁡x1+δ0e3+e4k1θ^-3c1e12x2+1k1h1+1e4x2-θ^˙ρθ-θ^θ--θ^,where (37)θ^=θ-,θmin⁡≤θ-≤θmax⁡,θmin⁡,θ-<θmin⁡,θmax⁡,θ->θmax⁡.
Then, we can obtain(38)H2=-αe12-2e22-c2e32-c3e42<0.
LetV=2V3; we get V˙=2V˙3≤(γ2ξ2-z2), taking integral from both sides as follows:(39)V∞-V0≤∫0∞γ2ξ2-z2.Satisfying the dissipation inequality, the closed-loop system (10) is stable asymptotically.
To avoid duplication, we can design coordinated excitation and valve control and just provide the design results as follows:(40)u1=-1k6k3x3+k5+k4cos⁡⁡x1+δ0+e32γ2+c3e3+h3θ^x2+k1z4+a0+k2sin⁡x1+δ0x3+k9sin⁡x1+δ0+e2/γ2k2sin⁡x1+δ0+h3c1kb2x2-3c1e12x2+k9cos⁡⁡x1+δ0x2k2sin⁡x1+δ0+h3e2+k9sin⁡x1+δ0k2cos⁡⁡x1+δ0x2k22sin2⁡x1+δ0,u3=-1k11k10x5+e42γ2+c4e4+1k1x2kb2+e12kb2-e122+θ^˙x2+θ^x˙2+h1+1-3c1e12θ^x2+k1z4+a0+k2sin⁡x1+δ0x3+k9sin⁡x1+δ0+e2γ2+3q22c1-3c1e12x2-6c1e1x22+c1kb2-q22c1kb2x2,where k10=-1/TMLΣ and k11=CML/TMLΣ. We design the adaptive law as follows(41)θ-˙=ρx2e2+h3x2k2sin⁡x1+δ0e3+e4k1θ^-3c1e12x2+1k1h1+1e4x2-θ^˙ρθ-θ^θ--θ^,where (42)θ^=θ-,θmin⁡≤θ-≤θmax⁡,θmin⁡,θ-<θmin⁡,θmax⁡,θ->θmax⁡.Theorem 3.
For the closed-loop system (10), under the condition of Lemma 1, satisfying the initial condition as follows x1(0)∈Ωx10={x1(0)∈R3and x1(0)<kb}, the error variable x1 is always in the compact set Ωx1 as follows:(43)Ωx1=x1∈R3:x1≤kb1-e-2V30.Proof.
Due toV˙3≤0, we get V3(t)≤V3(0). For x1(0)<kb, the state variable x1(t) satisfies x1(t)<kb, acoording to Lemma 2. Because of the inequality (1/2)log⁡⁡(kb2/(kb2-x12))≤V3(t)≤V30, we can get x1≤kb1-e-2V3(0).


## 4. Design of Switching Law

The switching law is designed to ensure that the amplitude controller constraints are not destroyed in this section. Given the existence of a common Lyapunov function in subsystems, subsystems 1 and 2 are stable, and switching systems are stable under arbitrary switching laws. Thus, switching law can be designed in any form to avoid frequent switching systems. Thus, we design the hysteresis switching law as follows [13].Hysteresis Switching Law. Consider σ(0)=1, for t>0; if σ(t-)=i∈{1,2}, uimin⁡≤ui≤uimax⁡, then holding σ(t)=i. If σ(t-)=1, but u1<u1min⁡ or u1>u1max⁡, then we get σ(t)=2; similarly, if σ(t-)=2, but u2<u2min⁡ or u2>u2max⁡, then we obtain σ(t)=1, where σ(t) is the switching signal σ(t)=1, showing that system runs subsystem 1, the main valve control system; consider σ(t)=2, showing that system runs subsystem 2, the fast valve control system. uimin⁡ and uimax⁡ are, respectively, the opening minimum and maximum of the main valve or the quick steam valve. The hysteresis switching law ensures that the system switches subsystem 2, after the system runs for some time, even if the control input satisfies subsystem 1, to avoid frequent switching and continue running subsystem 2, until the constraints are reached.


## 5. Simulation Analysis

The system parameters are selected asVs=1, THΣ=0.2, TMLΣ=0.35, CH=0.3, CML=0.7, H=10, u1min⁡=u2min⁡=-3, and u1max⁡=u2max⁡=6. We choose the stable operation points of the system as δ0=57.3∘, ω0=314.159rad/s, Pm0=0.852, Eq0′=0.3913, u1min⁡=u2min⁡=-4, u1max⁡=u2max⁡=6, and kb=0.2.The following are simulations for two kinds of stem valve opening degree conditions: within the limit and exceeding the limit.(
1
) Mechanical power only appears 10% of the perturbation in a certain period of time (6 s to 6.1 s) and then returns to the known initial value, that is, P+ΔP(t), and the main valve control amplitude does not exceed the limit, (44)ΔPt=0,0≤t<6s,0.1,6s≤t≤6.1s0,6.1s<t.,Figures3–6 show the state response curves of the system. The solid line denotes the response curve of the controller designed by Barrier Lyapunov, and the dotted line represents the response curve without considering state constraints. Figure 3 shows the condition of main valve control within the limit, in which the state response curve of x1 is constrained. Visibly, the state is within the constraints. Moreover, the oscillation of the amplitude is also small for the other states. As shown in Figures 7, 9, and 10, which show the control input curve, the designed controller gain becomes larger with Barrier Lyapunov Function. The main valve control input does not exceed the limited amplitude, and the switching signal is keeping the value 1, as shown in Figure 8.Figure 3
Transient response curves of the angle.Figure 4
Transient response curves of the speed.Figure 5
Transient response curve of the generator potential.Figure 6
Transient response curve of the mechanical power.Figure 7
The excitation control input curves.Figure 8
Switching signal.Figure 9
The main steam valve control input.Figure 10
The fast steam valve control input.(
2
) Mechanical power only appears 20% of the perturbation in a certain period of time (6 s to 6.1 s) and then returns to the known initial value, that is, P+ΔP(t), and the main valve control exceeds limited amplitude. Consider(45)ΔPt=0,0≤t<6s,0.2,6s≤t≤6.1s,0,6.1s<t.Figures11 to 14 show the state response curve of the system. The solid line denotes the response curve of the controller designed by Barrier Lyapunov, and the dotted line represents the response curve without considering state constraints. Figure 11 shows the condition of main valve control exceeding the limit, in which the state response curve of x1 is constrained. Visibly, the state is within the constraints. Moreover, the oscillation of amplitude is also small for the other states. As shown in Figures 15, 17, and 18, which show the control input curve, the designed controller gain becomes larger with Barrier Lyapunov Function. When the main valve control input exceeds the limited amplitude, the switching signal changes from 1 to 2, as shown in Figure 16. That is, when the main steam valve is overrun, its control input is 0, and the fast valve starts to work. The valve control inputs are within the set range, without constraints destroyed.Figure 11
Transient response curves of the angle.Figure 12
Transient response curves of the speed.Figure 13
Transient response curve of the generator potential.Figure 14
Transient response curve of the mechanical power.Figure 15
The excitation control input curves.Figure 16
Switching signal.Figure 17
The main steam valve control input curves.Figure 18
The fast steam valve control input curves.


## 6. Conclusion

In this paper, the coordinated excitation and valve controller of the steam turbine generator is designed by Barrier Lyapunov Function and backstepping method. Simulation results show that the generator power angle did not exceed the allowable value and the excitation and steam valve control gain is limited and within a reasonable range. With just requiring the initial power angle to satisfy certain conditions, we can conclude that the constrained state does not exceed the limit set by the constraint in the system. Thus, the design of the controller reduces the extent strictly required by the initial conditions. Furthermore, we can design a more effective controller and improve the efficiency of the power system.



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*Source: 560191-2015-05-14.xml*