# Adaptive Optimizing Control for Nonlinear Synchronous Generator System with Uncertain Disturbance

Authors: Guiyang Deng; Lianglun Cheng; Baojian Yang
Journal: Complexity
 (2019)

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## Abstract

In this paper, an adaptive optimizing control is proposed for nonlinear synchronous generator system with uncertain disturbance. Considering the issue of uncertain item, a low-order Extended State Observer (ESO) is constructed to track each order states. Then, combining Backstepping technology and proposed ESO, a dynamic compensation control is designed to compensate the uncertainty disturbance in each state quantity. The proposed method demonstrated that stability of the system can be ensured, and tracking burden of ESO can be reduced. Finally, simulation results can show a good tracking performance and anti-interference ability by the proposed control method.


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## Body

## 1. Introduction

At present, the energy development in the world is mostly hydraulic power generation, thermal power generation, wind power generation and so on. In modern power industry, synchronous generator is the most commonly used alternator, which is widely used in thermal power generation, diesel engine power generation, nuclear power generation, and so on [1–3]. During the operation of synchronous generators, there are many system uncertainties and serious external disturbances. It would lead to great energy loss and bring great negative impact on energy conversion efficiency. Therefore, how to effectively compensate for uncertain nonlinearity and improve energy conversion efficiency is an open issue. To achieve high-performance power system control, it is necessary to overcome the negative effects of uncertainties and nonlinear factors in the system. For the complex synchronous generator model, it is feasible to analyze the stability of synchronous generator directly by using Lyapunov function, but the calculation process is complex and the amount of calculation is large [4–6]. There are many uncertain disturbances in the actual application scenario of synchronous generator, which makes the design of control method greatly difficult.Since there are many uncertainties and unknown disturbances during the operation of the synchronous generator, it is very difficult to construct the Lyapunov function directly. At present, some scholars use Backstepping technology to construct Lyapunov functions effectively. They have completed the design of the whole system control law [7–9]. Although Backstepping technology effectively reduces the difficulty of constructing Lyapunov functions, it is still difficult to construct systems with multiple uncertainties. In this regard, some scholars use adaptive control based on the Backstepping technology to compensate for uncertainty interference [10–15]. However, this method increases the number of adaptive parameters, so that the amount of calculation increases.On the other hand, for systems with uncertainties, Extended State Observer (ESO) is used to achieve effective dynamic tracking [16–20]. The ESO is used to expand the uncertain disturbances in the system into a new state. By adopting ESO, the uncertain disturbances can be tracked efficiently. Moreover, the control accuracy of the system can be improved by using the tracking values of uncertain disturbances. However, the ESO designed for the whole system model will accumulate disturbances as the dimension of the system model increases. This will result in the performance of the observer greatly reduced. Furthermore, in practical application scenarios, synchronous generators are structure complex system. And uncertain disturbances are uncertain. In order to broaden the application scenario of ESO, it is necessary to optimize the construction method of ESO.To sum up, this paper designs a controller based on Backstepping technology and low-order ESO to enhance the anti-interference ability of the system and improve the efficiency of energy conversion. The specific mathematical model of this paper is described in Section2. The proposed control method is described in detail in Section 3. The corresponding simulation is shown in Section 4.


## 2. Mathematical Model of Synchronous Generator

In this paper, a mathematical model of synchronous generator based on rotord−q axis is considered. And L=Lq=Ld is one of the assumptions.(1)θ˙=ω,ω˙=1J3pϕf2iq−Bω−d,i˙d=1Lud+pωiq−RLid,i˙q=1Luq−pϕfLω−RLiq−pωid+f,whereθ is the rotor angle, B is the viscous friction coefficient, p is the pole logarithm, and J is the moment of inertia of the rotor. R and L are the stator resistance and phase inductance, respectively. ω the mechanical angular velocity. id and iq are current of d−q axis. ϕf is the permanent magnet flux. ud and uq are voltage of d−q axis. d represents the unknown load torque disturbances. f is the externally uncertain bounded disturbances term.In the magnetic field control of a synchronous generator, the direct currentid is generally set to zero. Therefore, the model (1) of the synchronous generator can be simplified as follows:(2)θ˙=ω,ω˙=−c1ω+c2iq−d,i˙q=−c3ω−c4iq+buq+f, where c1=B/J,c2=kT/J,c3=pϕf/L,c4=R/L,b=1/L,andkT=3pϕf/2.In order to facilitate the design of the controller,x1=θ, x2=ω, and x3=iq are the system status. Thus, the model (2) can be described as follows:(3)x˙1=x2,x˙2=−c1x2+c2x3−d,x˙3=−c3x2−c4x3+buq+f.Setting(4)uq=h+c3x2+c4x3b,whereh is the preset control amount.Then formula (3) is further represented as follows:(5)x˙1=x2,x˙2=−c1x2+c2x3−d,x˙3=h+f.The system model (5) is a system with uncertain disturbance terms. Based on the Backstepping technology, this paper combines the ESO method to design a control method. It improves the anti-interference ability of the system and further ensures the stability and tracking performance of the system. The design method is described in the next section.


## 3. Compensation Control Design

According to the aforementioned model, the design steps of the control method are as follows.Step 1.
Define tracking errore1 as follows:(6)e1=x1−x1r,
wherex1r is the expected value, and the second order is derivable.
The Lyapunov function is defined as follows:(7)V1=12e12.
Thus, it can be obtained as follows:(8)V˙1=e1e˙1=e1x2−x˙1r.
Define the tracking errore2 as follows:(9)e2=x2−α1,
whereα1 is the virtual controller.
From formulas (8) and (9), it can be obtained as follows:(10)V˙1=e1e2+α1−x˙1r.
To ensure thatV˙1 is negative, the virtual controller is designed as α1=−k1e1+x˙1r, where k1>0. Then, the following result can be obtained:(11)V˙1=−k1e12+e1e2.
In the above formula, the inequality−k1e12≤0 is always established. The processing of the item e1e2 is described in the next step.Step 2.
The Lyapunov functionV2 is chosen as:(12)V2=V1+12e22.
Thus, it can be obtained that:(13)V˙2=−k1e12+e2e1−c1x2+c2x3−d−α˙1.
Define tracking errore3 as follows:(14)e3=x3−α2.
Thus, the system status is described as follows:(15)x3=e3+α2.
From formulas (13) and (15), it can be obtained as follows:(16)V˙2=−k1e12+e2e1−c1x2+c2e3+α2−d−α˙1.
In order to satisfy the inequalityV˙2≤0, the virtual controller α2 is designed as follows:(17)α2=1c2α˙1+c1c2x2+1c2d−1c2e1−k2c2e2.
Then, it can be obtained the following formula:(18)V˙2=−k1e12−k2e22+c2e2e3,
wherek1>0 and k2>0. Therefore, −k1e12−k2e22≤0 is established. Then, further design is needed to eliminate the impact of the error e3 on system stability.
Formula (16) contains an externally bounded disturbance d. In order to ensure the stability of the system, the second-order ESO-1 is constructed to estimate the external uncertain, which is bounded interference. The construction method will be introduced in the subsection.
The ESO-1 is established as follows:(19)ed1=zd1−x2,z˙d1=zd2−βd1ed1−c1x2+c2x3,z˙d2=−βd2faled1,εd,δd.
The functionfaled1,εd,δd is designed as follows:(20)faled1,εd,δd=ed1δdεd−1,ed1≤δd,ed1εdsigned1,ed1>δd,
whereed1 is the systematic error, εd is a constant in the interval 0,1. And δd is the length of the interval of the linear segment.
The stability of the second order ESO-1 of formula (19) is demonstrated as follows:
Setting(21)ed1=zd1−x2,ed2=zd2−d.
Thus, it can be obtained as follows:(22)e˙d1=ed2−βd1ed1,e˙d2=−βd2faled1,εd,δd−d˙.
Case 1. The error ed1 is satisfied with ed1>εd, and then it can be obtained that faled1,εd,δd=ed1εdsigned1.(23)e˙d1=ed2−βd1ed1,e˙d2=∓βd2ed1εd−d˙.
When the system is stable, it can be obtained thate˙d1→0 and e˙d2→0(24)e˙d1=ed2−βd1ed1=0,e˙d2=∓βd2ed1εd−d˙=0,
and(25)ed1=∓d˙βd2εd−1,ed2=±βd1d˙βd2εd−1.
Therefore, if the inequalityβd2>d˙ was established with the true parameters, it could be obtained zd1→x2,zd2→d.
Case 2. The error ed1 is satisfied by ed1≤εd, it can be obtained faled1,εd,δd=ed1/δdεd−1.
Then,(26)e˙d1=ed2−βd1ed1,e˙d2=−βd2edδdεd−1−d˙.
When the system is stable, it can be obtained thate˙d1→0 and e˙d2→0. Thus(27)e˙d1=ed2−βd1ed1=0,e˙d2=−βd2edδdεd−1−d˙=0.
and(28)ed1=−d˙·δdεd−1βd2,ed2=−βd1d˙·δdεd−1βd2.
Similarly, if the inequalityβd2>d˙·δdεd−1 was established with the true parameters, it could be obtained zd1→x2 and zd2→d.
From case 1 and case 2, the stability of the second-order ESO-1 can be ensured when the appropriate parameters are satisfied withβd2>d˙ and βd2>d˙·δdεd−1.Step 3.
The Lyapunov functionV3 is chosen as:(29)V3=V2+12e32.Thus, it can be obtained that:(30)V˙3=−k1e12−k2e22+c2e2e3+e3e˙3.From formula (14), it can be obtained that e˙3=x˙3−α˙2. Thus, V˙3 can be written as:(31)V˙3=−k1e12−k2e22+c2e2e3+e3h+f−α˙2.In order to satisfy the inequalityV˙2≤0, it can be designed as follows:(32)h=−f+α˙2−k3e3−c2e2,wherek3>0. Formula (30) contains an externally bounded disturbance f. Similarly, the second-order ESO-2 is designed to estimate the uncertainty disturbance.Thus, the ESO-2 is established as follows:(33)ef1=zf1−x3,z˙f1=zf2−βf1ef1−c3x2−c4x3+buq,z˙f2=−βf2falef1,εf,δf.The functionfalef1,εf,δf is designed as follows:(34)falef1,εf,δf=ef1δfεf−1,ef1≤δf,ef1εfsignef1,ef1>δf,whereef1 is the systematic error, εf is a constant in the interval 0,1, and δf is the length of the interval of the linear segment.Same as ESO-1 stability proof process, the stability of ESO-2 can be guaranteed when bothβf2>f˙ and βf2>f˙·δfεf−1 are satisfied.According to the aforementioned analysis, ESO-2 can accurately estimate the uncertainty disturbancef in real time. Therefore, formula (30) can be rewritten as:(35)V˙3=−k1e12−k2e22−k3e32≤0.In summary, the system model (1) is stablity.


## 4. Simulation Results

In order to verify the feasibility of the above method, the following verification simulation is performed for the synchronous generator control system model. The parameters are shown in Table1.Table 1
Parameters of synchronous generator.
Parameters
Ratings
Torque constantkT
0.14
N
·
m
·
A
−
1
Number of pole pairsp
4
Stator phase resistanceR
2
Ω
Stator phase inductanceL
25
m
H
Coefficient of frictionB
0.0001
N
·
m
·
s
·
r
a
d
−
1
Moment of inertiaJ
0.0086
k
g
·
m
2The parameters of the system model are derived from the data in Table1. And choosing c1=0.5, c2=16.28, c3=26, c4=80, and b=40. In the simulation, the initial value of the equation of state is set as x10x20x30=100. The parameters in the Backstepping technology are selected as: k1=30, k2=40, and k3=0.1. The parameters in ESO are selected as: βd1=βf1=100, βf1=βf2=100, εd=εf=0.5, and δd=δf=0.01.In order to track the rotor angle and verify the stability of the system, the expected value of the given rotor angle in the simulation example isθ1r=x1r=sintrad/s. And assume that the system’s uncertain disturbance terms are d=cost and f=2∗sint/2. The simulation results are shown in Figures 1–6.It can be seen from Figures1 and 2 that both the mechanical angular velocity and the q-axis current are bounded, which guarantees the stability of the system. It can be seen from Figures 3 and 4 that the established ESO can perform effective dynamic compensation for the uncertain interference term and good real-time tracking. Figure 5 is a comparison of the real rotor angle control with ESO and without ESO. The figure indicates that the system without ESO will more easily affect by the uncertain disturbances and generate the system jitter. This must not be allowed to exist in the practical engineering application. By using the proposed method, the system has stronger anti-interference ability and satisfies the practical engineering requirements. Figure 6 is the error between the two. The error curve is basically stable at 0, that is, the error is close to zero. It can be seen that the method proposed in this paper can improve the stability of the system and has great practical application value.Figure 1
Mechanical angular velocityω.Figure 2
q axis current iq.Figure 3
Uncertain disturbanced and predicted disturbance zd2.Figure 4
Uncertain disturbancef and predicted disturbance zf2.Figure 5
Rotor angle tracking curve.Figure 6
Error between expected rotor angleθ1r and real rotor angle θ.


## 5. Conclusions

Based on the model of synchronous generator system, this paper proposes a new control method by combining Backstepping design method and ESO method. Firstly, a low-order ESO is constructed, which can estimate the uncertain disturbances in real-time. Secondly, the Lyapunov function of the synchronous generator system model is constructed effectively by using Backstepping method. On the premise of reducing the amount of calculation, a control method is designed to achieve effective tracking and strong antijamming performance of the system. The simulation results show that this method can improve the antijamming ability and tracking performance of the system, and has great application value.



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*Source: 7604320-2019-10-16.xml*