Abstract
In this paper, we focus on the stability of the hydropower plant with sharing tailrace surge tank. Considering the interaction between hydroturbines and timedelay of hydraulic servo system, a nolinear dynamic model of multihydroturbine governing timedelay systems with sharing tailrace surge tank is first established. The model can predict the transient characteristic of the hydroturbine governing system, promoting the development of transient analysis, and safe operation of the governing system. Then, bifurcation characteristics of the system with different governor parameters and timedelays are studied. Moreover, the effect of timedelay on the stability intervals of governor parameters are investigated and its optimal value with different timedelays are obtained. All of the above analytical results can provide theoretical guidance for the actual operation of a hydropower plant with sharing tailrace surge tank.
1. Introduction
Hydropower is a low cost, near zero pollutant emissions renewable energy source, and can respond to peak loads quickly [13]. With the rapid development of hydropower [47], the underground powerhouse of hydropower plant is widely used. In view of the overall design, excavation engineering and other reasons, it usually adopts the structure of multihydroelectric generating units with sharing tailrace surge tank [8, 9]. Due to this arrangement, hydroturbines are related to each other, which has a great impact on the hydroturbine governing system and may cause the abnormal vibration of the units [10]. Moreover, the governing system is directly controlled by the hydroturbine governor to realize different transient processes [1113]. In the actual operation, the displacement response of the servomotor piston has a lag in time and the dead zone of the main control valve causes the servomotor to remain stationary [14]. These affect the bifurcation characteristics and the stability of the hydroturbine governing system. Considering the above factors, it’s necessary to study the stability of multihydroturbine governing systems considering timedelay.
Fortunately, there are some contributions about the stability and timedelay effect of the hydroturbine governing system [1520]. For example, Zeng [15] integrated the generator equations and shafting lateral vibration into the generalized Hamiltonian system. Then the generalized Hamiltonian control model is proposed for hydroturbine generating sets in the shafting transient state. Guo [16] established a hydroturbine governing system with sloping ceiling tailrace tunnel, and the stability of the system considering different factors are analyzed based on stable domain. Nagode [17] assessed the relationship between random gate position and the hydroturbine deviated power on random water disturbance and load variation, the deviated power can tracks deviated power target signal accurately. Xu and Wang [21, 22] discussed the influence of fractionalorder and timedelay on the hydroturbine governing system with single unit, and their changing law was respectively identified. However, the influence of timedelay on the hydropower plant with sharing tailrace surge tank has not been discussed, considering the complexity of the above hydropower plant during the regulation process. It is necessary to study the effect of timedelay on the stability of the multihydroturbine governing systems with sharing tailrace surge tank.
Motivated by the above discussions, considering the continuous oscillation process that may occur during the operation of a multihydroelectric generating unit with sharing tailrace surge tank, and the displacement response of servo piston in hydraulic servo system has a lag in time, we first introduce the timedelay to the multihydroturbine governing systems with sharing tailrace surge tank. Second, the stable domain of the system without timedelay is presented. Third, the bifurcation characteristics of the system with different governor parameters and timedelays are analyzed. Finally, the stability intervals and optimal value of governor parameters with different timedelays are exhaustively investigated.
2. Multihydroturbine governing timedelay systems with sharing surge tank
To illustrate the interaction between hydroturbines in the hydropower plant, a general layout of a hydropower plant with sharing tailrace surge tank is shown in Fig. 1. From Fig. 1, each hydroelectric generating unit has independent water pipe and upstream surge tank. After the water flow through the rotating parts, it converges inside the tailrace surge tank. When it is in a steady state, the flow in the pipe is relatively stable, and each hydroelectric generating unit is in a transient relative equilibrium state.
Fig. 1The general layout of a hydropower plant with sharing tailrace surge tank
In the operation of a hydropower plant with sharing tailrace surge tank, the small disturbance of the system can be counteracted by the dead zone. However, if one of the units is subjected to a large dynamic adjustment, the flow of this unit will change accordingly. Subsequently, due to the inertial action of the fluid, the flow and pressure at the tailrace surge tank will also change. Other units will make the discharge changes in response to these changes, and thus have an impact on the unit, forming a process of repeated regulation eventually [23]. Therefore, the dynamic process of continuous oscillation is common for hydropower plant with sharing tailrace surge tank.
The structure diagram of the hydroturbine governing system is shown in Fig. 2 [24] and its fundamental equations are listed in Section 2.1 to Section 2.4. In Section 2.5, the nonlinear dynamic model of multihydroturbine governing timedelay systems with sharing tailrace surge tank is established.
Fig. 2The structure diagram of Francis hydroturbine governing system
2.1. Pipe network
For the hydropower plant with sharing tailrace surge tank, suppose there are altogether $n$ units of hydroturbine. In this paper, the momentum equation of the $i$th diversion tunnel can be described as [10]:
The continuity equation of the $i$th surge tank:
The momentum equation of the $i$th penstock:
2.2. Hydroturbine
The dynamic model of the $i$th hydroturbine can be described as [19]:
where ${m}_{ti}$, ${q}_{ti}$, ${h}_{i}$, ${x}_{i}$ and ${y}_{i}$, the operation parameters of the $i$th hydroturbine, denote the relative deviations of the hydroturbine torque, the hydroturbine flow, the hydroturbine head, the rotate speed and the guide vane opening, respectively. ${e}_{mhi}=\partial {m}_{ti}/\partial {h}_{i}$, ${e}_{mxi}=\partial {m}_{ti}/\partial {x}_{i}$ and ${e}_{myi}=\partial {m}_{ti}/\partial {y}_{i}$ are the partial derivatives of the hydroturbine torque with respect to the hydroturbine head, the hydroturbine speed and the guide vane opening. Similarly, ${e}_{qhi}=\partial {q}_{ti}/\partial {h}_{i}$, ${e}_{qxi}=\partial {q}_{ti}/\partial {x}_{i}$ and ${e}_{qyi}=\partial {q}_{ti}/\partial {y}_{i}$ denote the partial derivatives of the hydroturbine flow with regard to the hydroturbine head, the hydroturbine speed and the guide vane opening, respectively.
2.3. Generator system
The firstorder generator model is introduced for the sake of simplicity, and its mathematical equation is presented as [25]:
where ${T}_{abi}={T}_{ai}+{T}_{bi}$, ${T}_{ai}$ and ${T}_{bi}$ denote the inertia time constant of generator and load, respectively. ${e}_{ni}$ is the synthetic selfregulation coefficient, ${m}_{g0i}$ is the load disturbance of the $i$th hydroturbine.
2.4. Hydraulic servo system with timedelay
The servomotor is used to amplify the control signal and supply power for operating the guide vane. The mathematical model of hydraulic servo system can be written as [26]:
where ${T}_{yi}$ is the major relay connecter response time of hydraulic servo system. ${u}_{i}$ is the output signal of the hydraulic servo model, and its equation can be written as:
where ${r}_{i}$ is the frequency disturbance. ${x}_{6i}$ is the intermediate variable, ${\dot{x}}_{6i}={r}_{i}{x}_{i}$.
In actual operation of hydropower station, considering the fact that the hydraulic servo system exists time lag, a discrete timedelay is introduced to the mathematical model of the hydraulic servo system and it can be expressed as [21]:
2.5. System mathematical model
From Eqs. (1)(8), the nonlinear dynamic model of multihydroturbine governing timedelay systems with sharing tailrace surge tank is:
and:
3. Stability analysis
Taking a hydropower plant with sharing tailrace surge tank as an example [27], the stability of the system are researched in the subsequent. The hydropower plant has two hydroelectric generating units of the same type ($n=$ 2), and the corresponding parameters of pipeline and upstream surge tank are also same. In the following, we assume that unit 1 is the object of the study. Specifically, the basic data of the hydropower plant are listed in Table. 1. The fixed step of numerical experiments is 0.01 and the iteration step is 15000. Moreover, the initial value of the model are all considered as 0.001.
Table 1System parameters of the hydroturbine governing system
Parameters  Values  Units  Parameters  Values  Units 
${H}_{0}$  419  m  ${e}_{myi}$  1  p.u. 
${Q}_{0i}$  81.56  m^{3}/s  ${e}_{qhi}$  0.5  p.u. 
${T}_{abi}$  9.864  s  ${e}_{qxi}$  0  p.u. 
${T}_{w1i}$  1.718  s  ${e}_{qyi}$  1  p.u. 
${T}_{w2}$  2.308  s  ${m}_{g01}$  0.05  p.u. 
${T}_{wti}$  2.586  s  ${m}_{g02}$  0.05  p.u. 
${e}_{ni}$  0  p.u.  ${k}_{p1}$  2  p.u. 
${h}_{10i}$  2.329  m  ${k}_{p2}$  2  p.u. 
${h}_{t0i}$  6.236  m  ${k}_{i1}$  0.5  s^{1} 
${h}_{20}$  6.732  m  ${k}_{i2}$  0.5  s^{1} 
${e}_{mhi}$  1.5  p.u.  ${k}_{d2}$  2  s 
${e}_{mxi}$  –1  p.u. 
3.1. The stable domain of the system without considering timedelay
When the timedelay of hydraulic servo system is not considered, The Jacobian matrix of the hydroturbine governing system can be obtained as:
Then, the characteristic equation is:
From Ref. [22], the system can remain stable if $\left\mathrm{a}\mathrm{r}\mathrm{g}\left({\chi}_{i}\right)\right\ge \pi /2$. Moreover, in the operation of hydropower plant, the differential adjustment coefficient ${k}_{d1}$ generally ranges from 0 to 5 [28]. In view of these, ${k}_{i1}$ is selected as the bifurcation parameter, then the Jacobi matrix of the system with different ${k}_{d1}$ (${k}_{d1}=$ 0, 1, 2, 3, 4 and 5) values are solved. The curves of bifurcation points and the stable domain of ${k}_{p1}$ and ${k}_{i1}$ are shown in Fig. 3.
Fig. 3The stable domain of kp1 and ki1 with different kd1 values
From Fig. 3, the PID parameter plane is divided into two parts by the curves of bifurcation points, i.e. stable domain and unstable domain. Specifically, the PID parameters of ${k}_{p1}$ and ${k}_{i1}$ can be combined freely in the stable domain. Moreover, ${k}_{p1}$ and ${k}_{i1}$ do not exhibit simple linear variation rules on the bifurcation line. For the area of stable domain, the larger the ${k}_{d1}$ value, the greater the stable domain of the system. However, when ${k}_{d1}=$ 5, the stable domain is not significantly increased.
3.2. Bifurcation characteristics of the system with different timedelay
In order to study the influence of timedelay on bifurcation characteristics of the system, the following study is carried out with referring to the results of Fig. 3. The hydraulic servo system of the two units has the same timedelay $\tau $ and the range is 00.4 s, then the bifurcation diagrams of the hydroturbine speed $x$ with different ${k}_{p1}$, ${k}_{d1}$ and timedelay $\tau $ are shown in Fig. 4.
Fig. 4Bifurcation diagrams of the hydroturbine speed xwith with different kp1, kd1 and timedelay τ values
a)${k}_{d1}=$ 0
b)${k}_{d1}=$ 1
c)${k}_{d1}=$ 2
d)${k}_{d1}=$ 3
e)${k}_{d1}=$ 4
f)${k}_{d1}=$ 5
From Fig. 4, the bifurcation characteristics of the system can be clearly observed. Fig. 4(a) shows, when ${k}_{d1}=$ 0 and $\tau $ ≤ 0.3 s, the stability interval of the ${k}_{p1}$ decreases with the increase of timedelay $\tau \text{.}$ Specifically, when the value of ${k}_{p1}$ is small (for example ${k}_{p1}<$ 1.2), the hydroturbine speed x will continue to shock and cannot reach the steady state. For the larger value of ${k}_{p1}$ (for example 1.2 < ${k}_{p1}$ < 3.3), the hydroturbine speed $x$ can always be stabilized at the rated speed, and the system meets the requirements of stable operation. Subsequently, there will be severe oscillation in the hydroturbine speed $x$ as the value of ${k}_{p1}$ is larger than its bifurcation point, which is extremely adverse for the hydropower plant and power grids. Therefore, the above unfavorable circumstance should be avoided in the actual operation of hydropower plants. It is worth noting that when timedelay $\tau $ > 0.3 s, no matter how to adjust the value of ${k}_{p1}$, the system cannot meet the requirements of stable operation.
When the values of ${k}_{d1}$ vary from 1 to 5, the bifurcation diagrams of the system at different timedelay $\tau $ are shown in Fig. 4(b)(f). It is not difficult to find that with the increase of timedelay $\tau $, the bifurcation characteristics of the system are similar to that of ${k}_{d1}=$ 0. From Fig. 4(e) and Fig. 4(f), when $\tau $ > 0.25 s, timedelay $\tau $ has a significant effect on the reduction of the stability interval of ${k}_{p1}$. Moreover, the hydroturbine speed $x$ is in a state of intense fluctuations and cannot reach the rated speed when ${k}_{d1}=$ 5 and $\tau $ > 0.25 s. It is worth noting that timedelay $\tau $ has little effect on the ${k}_{p1}$ stability interval at ${k}_{d1}=$ 2 and 3.
In order to clearly and intuitively explain the variation of the stability interval in Fig. 5, and further analyze the bifurcation characteristics of the system, the bifurcation points of the system are listed in Table 2.
Fig. 5Change law of stability interval of kp1 with different kd1 and timedelay τ values
From Fig. 5, we can easily get that no matter what the value of ${k}_{d1}$ is, the stability interval decreases with the increase of $\tau $. Moreover, the stability interval increases with the increase of ${k}_{d1}$ when ${k}_{d1}=$ 0, 1, 2, 3, 4. It is worthwhile to note that when ${k}_{d1}=$ 4 and $\tau $ > 0.3 s, the stability interval is greatly affected by $\tau $. Furthermore, there is no stability interval when ${k}_{d1}=$ 0. For ${k}_{d1}=$ 5, the stability interval decreases rapidly with the increase of $\tau $. When $\tau $ is greater than 0.15, there is no stability interval of ${k}_{p1}$, and the system cannot maintain stable operation.
From the above analysis, when ${k}_{d1}=$ 0, 1, 2, 3, 4, and the timedelay $\tau $ of hydraulic servo system is small (for example $\tau $ < 0.3 s), the larger value of ${k}_{d1}$ is beneficial to stable operation. When $\tau $ is greater than 0.3 s, ${k}_{d1}$ takes 2 or 3 is more conducive to stable operation, while ${k}_{d1}=$ 5 is not conducive to the actual operation. Hence, selecting appropriate PID parameters based on the value of $\tau $ is critical and the above analyses can provide some theoretical guidance for the actual operation.
Table 2The bifurcation points of the system with different τ
${\mathrm{k}}_{\mathrm{d}1}$  Timedelay  Bifurcation points  ${\mathrm{k}}_{\mathrm{d}1}$  Timedelay  Bifurcation points 
0 s  $\mathrm{\tau}=$ 0 s  (1.174, 3.302)  3 s  $\mathrm{\tau}=\text{0 s}$  (0.550, 4.679) 
$\mathrm{\tau}=$ 0.05 s  (1.175, 3.292)  $\mathrm{\tau}=\text{0.05 s}$  (0.550, 4.654)  
$\mathrm{\tau}=$ 0.10 s  (1.176, 3.282)  $\mathrm{\tau}=\text{0.10 s}$  (0.551, 4.628)  
$\mathrm{\tau}=$ 0.15 s  (1.177, 3.272)  $\mathrm{\tau}=\text{0.15 s}$  (0.551, 4.602)  
$\mathrm{\tau}=\text{0.20 s}$  (1.178, 3.262)  $\mathrm{\tau}=\text{0.20 s}$  (0.551, 4.577)  
$\mathrm{\tau}=\text{0.25 s}$  (1.179, 3.252)  $\mathrm{\tau}=\text{0.25 s}$  (0.552, 4.552)  
$\mathrm{\tau}=\text{0.30 s}$  (1.180, 3.243)  $\mathrm{\tau}=\text{0.30 s}$  (0.552, 4.528)  
$\mathrm{\tau}=\text{0.35 s}$  /  $\mathrm{\tau}=\text{0.35 s}$  (0.552, 4.504)  
$\mathrm{\tau}=\text{0.40 s}$  /  $\mathrm{\tau}=\text{0.40 s}$  (0.553, 4.480)  
1 s  $\mathrm{\tau}=\text{0 s}$  (0.890, 3.885)  4 s  $\mathrm{\tau}=\text{0 s}$  (0.438, 4.825) 
$\mathrm{\tau}=\text{0.05 s}$  (0.891, 3.872)  $\mathrm{\tau}=\text{0.05 s}$  (0.438, 4.773)  
$\mathrm{\tau}=\text{0.10 s}$  (0.891, 3.860)  $\mathrm{\tau}=\text{0.10 s}$  (0.438, 4.722)  
$\mathrm{\tau}=\text{0.15 s}$  (0.892, 3.848)  $\mathrm{\tau}=\text{0.15 s}$  (0.438, 4.668)  
$\mathrm{\tau}=\text{0.20 s}$  (0.892, 3.836)  $\mathrm{\tau}=\text{0.20 s}$  (0.438, 4.613)  
$\mathrm{\tau}=\text{0.25 s}$  (0.893, 3.824)  $\mathrm{\tau}=\text{0.25 s}$  (0.438, 4.555)  
$\mathrm{\tau}=\text{0.30 s}$  (0.893, 3.812)  $\mathrm{\tau}=\text{0.30 s}$  (0.438, 4.493)  
$\mathrm{\tau}=\text{0.35 s}$  (0.894, 3.801)  $\mathrm{\tau}=\text{0.35 s}$  (1.097, 4.425)  
$\mathrm{\tau}=\text{0.40 s}$  (1.460, 3.790)  $\mathrm{\tau}=\text{0.40 s}$  (1.863, 4.349)  
2 s  $\mathrm{\tau}=\text{0 s}$  (0.698, 4.340)  5 s  $\mathrm{\tau}=\text{0 s}$  (0.340, 4.375) 
$\mathrm{\tau}=\text{0.05 s}$  (0.699, 4.323)  $\mathrm{\tau}=\text{0.05 s}$  (0.340, 4.135)  
$\mathrm{\tau}=\text{0.10 s}$  (0.699, 4.305)  $\mathrm{\tau}=\text{0.10 s}$  (0.340, 3.814)  
$\mathrm{\tau}=\text{0.15 s}$  (0.700, 4.289)  $\mathrm{\tau}=\text{0.15 s}$  (1.100, 3.100)  
$\mathrm{\tau}=\text{0.20 s}$  (0.700, 4.272)  $\mathrm{\tau}=\text{0.20 s}$  /  
$\mathrm{\tau}=\text{0.25 s}$  (0.701, 4.257)  $\mathrm{\tau}=\text{0.25 s}$  /  
$\mathrm{\tau}=\text{0.30 s}$  (0.701, 4.241)  $\mathrm{\tau}=\text{0.30 s}$  /  
$\mathrm{\tau}=\text{0.35 s}$  (0.701, 4.226)  $\mathrm{\tau}=\text{0.35 s}$  /  
$\mathrm{\tau}=\text{0.40 s}$  (0.701, 4.211)  $\mathrm{\tau}=\text{0.40 s}$  / 
4. Conclusions
In this paper, considering the hydropower plant with sharing tailrace surge tank and timedelay of hydraulic servo system, a nolinear dynamic model of multihydroturbine governing timedelay systems with sharing tailrace surge tank is established and its stability is analyzed. First, the stable domain of the hydroturbine governing system is presented. Moreover, the bifurcation characteristics of the system are investigated with considering different $\tau $ and ${k}_{d1}$. When ${k}_{d1}$ is constant, the stability interval of ${k}_{p1}$ decreases with the increase of $\tau $. When $\tau $ < 0.3 s and ${k}_{d1}=$ 0, 1, 2, 3, 4, the larger value of ${k}_{d1}$ is beneficial to the stable operation. However, when $\tau $ > 0.3 s, ${k}_{d1}$ takes 2 or 3 is more conducive to the stable operation. Therefore, in order to reduce the vibration of the unit in actual operations, we select the appropriate governor parameters for different timedelays.
In this paper, we assumed that the two units go through the same load disturbance during operation. For the hydropower plant with sharing tailrace surge tank, units may be influenced by stochastic disturbances with different attributes at the same time. Moreover, units may have different operating conditions, for example, one unit in the load rejection transient, another one in the sudden load increase transient. Hence, the stochastic transient stability of coupling multihydroturbine will be studied in our future work.
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About this article
This study was supported by the scientific research foundation of National Natural Science Foundation – Outstanding Youth Foundation (51622906); National Science Foundation (51479173); Fundamental Research Funds for the Central Universities (201304030577); Scientific research funds of Northwest A&F University (2013BSJJ095); the scientific research foundation on water engineering of Shaanxi Province (2013slkj12); the Science Fund for Excellent Young Scholars from Northwest A&F University (Z109021515); and Shaanxi Nova program (2016KJXX55).