A Review of the Transient Process and Control for a Hydropower Station with a Super Long Headrace Tunnel

Abstract

The hydropower station with a super long headrace tunnel is a significant development type for hydropower energy. By constructing a super long headrace tunnel, the huge natural water fall head can be utilized to generate more electricity. With the development of hydropower energy, a hydropower station with a super long headrace tunnel becomes more and more competitive. Compared with a hydropower station with a short headrace tunnel, the transient process and control for a hydropower station with a super long headrace tunnel is much more complicated and becomes an intractable challenge. It is well known that the transient process and control is the basis of the design and operation of a hydropower station. To overcome the challenge of the transient process and control, much research has been carried out. This paper provides a systematic review on the latest research progress of the transient process and control for hydropower stations with a super long headrace tunnel. Firstly, two key issues for the transient process and control, i.e., hydraulic design optimization of the surge tank and operation control of unit, are illuminated. Secondly, for both single surge tanks and surge tanks with special types or combinations, the hydraulic design optimization methods are described. The most disadvantageous design and advantageous operation of surge tanks under combined operating conditions are discussed. Thirdly, the stability and regulation quality of the hydro-turbine governing system under isolated and grid-connected operation conditions are presented. Finally, some trends and recommendations for future research directions are made. A research thought for establishing the complete theory and application system of the transient process and control for hydropower stations with a super long headrace tunnel from the perspective of multi-slice and multi-scale is proposed.

1. Introduction

Renewable energy sources, which include biomass, hydropower, geothermal, solar, wind, and marine energies, are considered to be clean, low risk, and inexhaustible [1]. Today, and in the near future, renewable energies are expected to be more widely implemented to help maintain the sustainable growth of human beings all around the world. Over recent years, significant attention has been devoted to the problem of integrating photovoltaics and wind generation into power systems. Solar–hydro hybrid power stations are a way to smooth power output and increase water retention [2]. Hybrid generation of large-scale photovoltaic power together with hydropower offers a promising option to promote the integration of photovoltaic power [3]. Improving the energy efficiency of water systems by hydraulic energy recovery is becoming an inevitable trend for energy conservation [4]. The worldwide water supply represents a significant portion of the global energy consumption [5].

Among the different renewable energy sources, hydropower energy acts in an important and unique role. Hydropower is predicted to take a greater part of the electricity generated from renewable energy. Some countries have shown that hydropower is the largest source of domestic electricity, including Canada (60%), Brazil (84%), Switzerland (55%), Iceland (80%), and Norway (98%) [6]. Asian countries, mainly China, are between countries, which have extensively expanded their hydro energy potential. The world’s hydroelectric systems added 157.8 GW in 2008, and nearly 83% of this expansion was placed in Asia [6].

With the development of hydropower energy in China, the development mode shows two trends [7,8]: (1) The development is increasingly difficult; and (2) the development focus has gradually shifted from the southwest region to the Tibet region. For hydropower stations with a surge tank, the general structure of the pipeline and power generating system is shown in Figure 1. The hydropower station contains the following subsystems: Upstream reservoir, headrace tunnel, surge tank, penstock, tailwater, hydro-turbine, unit (including a generator and governor), load, and power grid.

For the first trend, increasingly more hydropower stations with a super long headrace tunnel have been planned, designed, or constructed, which aim to make full use of hydropower resources. The reason is that the development of the easily developing hydropower energy has been nearly completed. The hydropower stations with a high water head (or large discharge) and short headrace tunnel become less frequent. To obtain a high water head and to increase the capacity of hydropower stations, a super long headrace tunnel needs to be constructed. A representative of that kind of hydropower station is the Jinping-II Hydropower Station in Southwest China. Jinping-II Hydropower Station has four headrace tunnels and the length of each headrace tunnel is 16.7 km. The scale of the headrace tunnel group is the largest in the world. The super long headrace tunnels of the Jinping-II Hydropower Station straighten the Jinping river bend, with a length of 150 km, and then utilize the hydropower resources produced by the huge natural water fall head. The construction of the Jinping-II Hydropower Station has overcome a series of challenges of hydropower development, which are caused by the huge flow inertia in a super long headrace tunnel, the complicated and huge surge tank, excavation of the super long headrace tunnel, and the complicated operation of the unit.

For the second trend, by 2030, the installed capacity of hydropower in China will exceed 450 GW and the development of hydropower energy, excluding the Tibet region, will be completed [9,10]. On that occasion, the Tibet region will become the main battlefield of the development of hydropower energy. Then, the project, “power transmission from Tibet to outside”, will be put on the schedule [11,12]. The Tibet region is rich in hydropower energy. The hydropower energy in the Big Bend of the Yarlung Zangbo River is the most concentrated. For the Big Bend of the Yarlung Zangbo River, the length and water fall head of the channel from the inlet to outlet are 213 km and 2.19 km, respectively, while the straight-line distance between the inlet and outlet is only 39 km. At present, the most competitive development schemes are the one-stage development scheme and three-stage development scheme. For the one-stage development scheme, eight to 10 super long headrace tunnels with a length of 39 km need to be constructed. For the three-stage development scheme, five three-stage super long headrace tunnels with a length of 43 km need to be constructed, and the lengths for the three stages of headrace tunnels are 32 km, 4 km, and 7 km, respectively. No matter which scheme is selected, hydropower stations with a super long headrace tunnel will always be constructed.

From the above analysis, we can see that the hydropower station with a super long headrace tunnel is a significant development type for hydropower energy. By constructing a super long headrace tunnel, the huge natural water fall head can be utilized to generate more electricity. Moreover, one super long headrace tunnel can supply water for several units. Those are the advantages of building hydropower stations with a super long headrace tunnel. Until now, the hydropower energy that is easy to develop has become less frequent. The above two development trends indicate that it is necessary to build hydropower stations with super long headrace tunnels if the development of hydropower energy is to be continually promoted, which is usually hard to develop. At the present stage and in the future, that hydropower station will always be a highly competitive development type for hydropower energy.

During the construction process of hydropower stations with super long headrace tunnels, several problems may emerge and become the main technical bottleneck. The super long headrace tunnel is created by a tunnel boring machine (TBM). Due to the great overburden plus the structural stress, the behavior of the rock masses under the high and complex in situ stress field is a key parameter influencing TBM construction [13]. For the throttled surge tank with a long headrace tunnel, superimposed mass oscillation often occurs and affects the safe and stable operation of the hydropower station [14]. In view of the topography and geological setting, the hydropower station with a super long headrace tunnel faces a series of huge technical issues related to the engineering geology and rock mechanics [15], including the safety of the underground caverns, predicting and managing the groundwater, and operation of the TBM under high geo-stresses and high groundwater pressures. Rockburst is the most prominent deep rock mechanics problem for deep and long tunnels. Rockburst not only destroys underground structures and equipment, but also threatens human safety [16].

With the increase of the water head, the water hammer pressure and other stresses on the equipment become increasingly larger. During transients, the Francis turbine experiences cyclic stresses and asymmetric forces on the runner, all of which reduce the operating life of the components. Transients create both steady and unsteady pressure loading on the runner blade, resulting in cyclic stresses and fatigue development in the runner. These effects shorten the runner life [17]. Xiao et al. [18] studied the dynamic stresses in a Francis turbine runner based on fluid-structure interaction analysis, and indicated that the dynamic stresses caused by the hydraulic forces during off-design operating points are one of the main reasons for the fatigue and cracks in the runner blade. Frunzaverde et al. [19] studied the failure of a Francis turbine runner. The investigations led to the conclusion that the cracking of the blade is caused by fatigue. The failure is accelerated by the hydrogen embrittlement of the filling material.

From the perspective of the transient process and control for hydropower stations, the facilities of pressure reduction need to be set on the headrace tunnel to achieve the security and stability of hydro-turbine unit operations [20,21,22]. The most commonly used facility of pressure reduction for hydropower stations is the surge tank [23,24,25]. Compared with the hydropower station with a short headrace tunnel, the transient process and control for hydropower stations with a super long headrace tunnel is much more complicated. The issues for the hydraulic design optimization of the surge tank and operation control of unit are particularly remarkable [26,27,28]. The details are embodied in the following two aspects.

(1)

During the transient processes, the super long headrace tunnel leads to extremely large flow inertia in the tunnel and an extremely large amplitude of water level oscillation in the surge tank. The flow inertia time constant of the super long headrace tunnel can reach tens of seconds. The extremely large flow inertia leads to conspicuous contradiction between the stability and regulation quality [29,30]. Because of the existence of the super long headrace tunnel, the size of the surge tank is extremely large, and the water level oscillation in the surge tank has the following characteristics: Long period, large amplitude, and slow attenuation. After the setting of the surge tank, the length of the penstock is decreased and the water hammer pressure is reduced. However, the water pressure caused by the extremely large amplitude of the water level oscillation in the surge tank always superimposes with the water hammer pressure. As a result, the influence factors of volute pressure become much more complicated. Moreover, the extremely large flow inertia leads to an extremely large critical stable sectional area of the surge tank [31]. In conclusion, the extremely large amplitude of the water level oscillation and extremely large critical stable sectional area bring enormous difficulties to the hydraulic design optimization of the surge tank.

(2)

The hydro-turbine governing system has strong nonlinear characteristics and complicated hydraulics-machinery-electrics coupling characteristics [32,33,34]. The hydro-turbine governing system is a nonlinear dynamic system and the subsystems of hydraulics, machinery, and electrics have typical nonlinear characteristics. Specifically, in the pipeline, the head loss is proportional to the square of the flow velocity [35,36]. The flow characteristics and energy characteristics of the turbine are complicated, and the model comprehensive characteristic curves reflect the nonlinear characteristics of the turbine [37]. The governor is the core component of the hydro-turbine governing system and contains the dead zone nonlinearity and saturation nonlinearity [38]. For the subsystem of hydraulics, the hydraulic nonlinearity becomes more significant with the increase of the length of the pipeline [39,40]. For instance, the nonlinearity of the head loss for the hydropower station with a long headrace tunnel would cause a nonlinear hydraulic gradient. Because of the nonlinear hydraulic gradient, the propagation and reflection of the water hammer wave are affected, and the transient processes of the hydro-turbine governing system become more complicated. Meanwhile, the low-frequency mass wave in the surge tank, high-frequency water hammer wave in the penstock, and low-frequency flow oscillation wave in the headrace tunnel superimpose with each other. The coupling effect among the superimposed hydraulic waves, machinery subsystem, and electrics subsystem appears and then makes the transient process for the hydropower station with a super long headrace tunnel much more complicated. As a result, the regulation of the governor and control of the unit operation face enormous challenges.

The above two issues restrict the popularization and application of the hydropower station with a super long headrace tunnel. At present, for the hydraulic design optimization of the surge tank and operation control of the unit for hydropower stations with a super long headrace tunnel, the complete theory and application system is deficient. There are significant differences between hydropower stations with super long headrace tunnels and hydropower stations with short headrace tunnels, such as the controlling factors of the transient processes and fluctuation types of the dynamic responses. The differences make it infeasible to apply the theories and methods of design optimization of the latter into the former indiscriminately. Because of the unique transient process and essential nonlinear characteristics of hydropower stations with super long headrace tunnels, it is necessary to study and propose the complete theory and application system for the hydraulic design optimization of the surge tank and operation control of the unit. By combining the above two key issues and carrying out the study on the transient process and control for hydropower stations with super long headrace tunnels, the above objectives can be achieved. The anticipated results would provide support for the development of hydropower stations with super long headrace tunnels.

Regarding the research on the transient process and control for hydropower stations with super long headrace tunnels, many achievements have been obtained in the world. Representative achievements are stated as follows. Yu et al. [14,41,42] studied the critical superposition instant of surge waves in surge tanks and the water hammer pressure under successive load rejections for long diversion-type hydropower stations. Martínez-Lucas et al. studied the frequency control support of a wind-solar isolated system by a hydropower plant with a long tail-race tunnel [43], and analyzed the power-frequency control of hydropower plants with long penstocks in isolated systems with wind generation [44]. Chen [45] and Murty et al. [46] studied the stability of hydro-turbine generating units with a long penstock or diversion tunnel. Ruud [47] studied the instability of a hydraulic turbine with a very long penstock. Sarasúa et al. [48] studied the dynamic response and governor tuning of a long penstock pumped-storage hydropower plant. Zhang et al. [49] and Wu et al. [50] studied the hydraulic transient simulation in a hydropower station with a long water diversion tunnel.

In the past several years, the authors and their research team have studied the transient process and control for hydropower stations with super long headrace tunnels. Many research results on the hydraulic design optimization of the surge tank and operation control of the unit have been obtained [51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79]. Because of the limited space, the present paper only covers the latest research results from the authors and their research team. In the text, Refs. [51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79] will be emphatically introduced.

Table 1. Comparison of the relevant work between the authors’ research team and others around the world.

Work of the Authors’ Research Team	Work of Others around the World
(1) Hydraulic design optimization of surge tanks	(1) Surge waves in surge tank, Refs. [14,41,42]
Single surge tank, Refs. [51,52]	(2) Power-frequency control, Refs. [43,44]
Surge tank with special type or combination, Refs. [53,54,55,56,57,58,59,60,61,62]	(3) Stability and instability of turbine, Refs. [45,46,47,48]
Combined operating conditions, Refs. [63,64]	(4) Hydraulic transient simulation, Refs. [49,50]
(2) Operation control of unit
Isolated operation condition, Refs. [65,66,67,68,69,70,71,72,73]
Grid-connected operation condition, Refs. [74,75,76,77,78,79]

shows the comparison of the relevant work between the authors’ research team and others around the world.

This review paper presents a detailed literature survey focused on the transient process and control for hydropower stations with super long headrace tunnels. The paper is organized as follows. In Section 2, the hydraulic design optimization of the surge tank is described. In Section 3, the operation control of the unit is presented. In Section 4, the whole paper is summarized and the conclusions are given. Moreover, some trends and recommendations for future research directions are made.

2. Hydraulic Design Optimization of Surge Tanks

A surge tank is the most common and important pressure reduction facility in a hydropower station. The hydraulic design optimization of the surge tank is key for the design and optimization of the pipeline system of hydropower stations. Based on the unique transient process and essential characteristics of the hydropower station with super long headrace tunnels, research on the hydraulic design optimization of surge tanks is carried out. Many achievements have been obtained in three aspects, i.e., the hydraulic design optimization of single surge tanks, the hydraulic design optimization of surge tanks with a special type or combination, and the most disadvantageous design and the most advantageous operation of surge tanks under combined operating conditions based on a surge wave.

2.1. Hydraulic Design Optimization of Single Surge Tank

With the increase of the length of the headrace tunnel, the flow inertia in the tunnel and amplitude of the water level oscillation in the surge tank become greater, and the period of water level oscillation in the surge tank becomes longer. Ref. [51] indicated that: From the calculation results of the transient process for load rejection, i.e., Figure 2 and

Table 2. Length of headrace tunnel, the maximal VP, and highest WLST of hydropower stations A, B, and C during load rejection [52].

Hydropower Station	Length of Headrace Tunnel/m	Maximal VP/m (Occurrence Time/s)	Highest WLST/m (Occurrence Time/s)
A	16,737.0	390.77 (107.56)	1705.98 (115.98)
B	16,562.2	265.25 (174.3)	2138.32 (175.7)
C	10,037.8	89.82 (106.5)	877.74 (111.42)

, the maximal volute pressure (VP) of hydropower stations with super long headrace tunnels depends on the highest water level in the surge tank (WLST), but not the closing law of the guide vane. In a general case, the maximum and minimum pressures along the pipeline are determined by the highest and lowest water levels in the surge tank.

Based on the conclusion that the limitation value of the highest water level in the surge tank depends on the control value of the volute pressure, Ref. [52] derived the correlations for the hydraulic design parameters of a surge tank according to the explicit formulas of water level oscillations in a throttled surge tank, throttled surge tank with an upper chamber, and simple surge tank with an upper chamber during load rejection, respectively. The correlations are shown in Equations (1) and (2). Then, the applicable conditions of the above three types of surge tanks were proposed and shown in

Table 3. Applicable conditions of three types of surge tanks [52].

Premise	Criterion	Recommended Type of Surge Tank
Equation (1), $n_s=1$	$\gamma \le \left[{\gamma }_s\right]$	throttled surge tank, simple surge tank
	$\gamma >\left[{\gamma }_s\right]$	throttled surge tank with upper chamber, simple surge tank with upper chamber
Equation (2), $\eta =0$	$\gamma \le \left[{\gamma }_{\beta }\right]$	simple surge tank with upper chamber, simple surge tank
	$\gamma >\left[{\gamma }_{\beta }\right]$	throttled surge tank, throttled surge tank with upper chamber

. Moreover, the correlations were applied to engineering examples. The type of surge tank was selected and the hydraulic parameters of surge tanks were optimized. During the analysis process, the method for the hydraulic design optimization of surge tanks for hydropower stations with super long headrace tunnels was proposed.

Throttled surge tank [52]:

$$\beta ={\left\{k_1\gamma \cdot \left[\frac{3(1-k_2\gamma )}{n_{s}m}(1-\frac{\pi m+2k_2\gamma }{\pi \sqrt{{\left(k_2\gamma \right)}^2+2k_2\gamma (1-k_2\gamma )/n_s}+2k_2\gamma })-1\right]\right\}}^{-1/2}$$

(1)

Throttled surge tank with an upper chamber [52]:

$$\left[1-Dm\right]\sqrt{A{p}^2+Bp+C}=m\left[1+\frac{2D}{\pi }p\right]$$

(2)

Simple surge tank with an upper chamber [52]: The correlation is Equation (2) under $\eta =0$.

In Equations (1) and (2) and Table 3: $\beta =S/f$, $n_s=F_s/F_{th}$, $n_c=F_c/F_{th}$, $m=\left|z_{\max 1}\right|/H_0$, $p=Z_c/H_0$, $\gamma =l/H_0$, $k_1=2g{H}_0{\phi }^2{n}^2/R^{4/3}$, $k_2=n^2{v}_0^2/R^{4/3}$; n is the pipe roughness; R is the hydraulic radius of the headrace tunnel; $\phi$ is the discharge coefficient in the throttled orifice; $\eta =h_{c0}/h_{w0}$ is the coefficient of the throttled loss; $h_{w0}$ and $h_{c0}$ are the head losses when water goes through the headrace tunnel and throttled orifice with a discharge of $Q$, respectively; $l$, $f$, and $v_0$ are the length, sectional area, and flow velocity of the headrace tunnel, respectively; $Z_c$ is the distance between the bottom plate of the upper chamber and the initial water level; $z_{\max 1}$ is the highest water level in the surge tank; $S$ is the sectional area of the throttled orifice; $F_s$ is the sectional area of the surge tank; $F_c$ is the sectional area of the upper chamber; $F_{th}$ is the critical stable sectional area of the surge tank; $H_0$ is the rated head; and $k_1$, $k_2$, $A$, $B$, $C$, and $D$ are intermediate coefficients.

For the hydropower station with a super long headrace tunnel, the reasonable value of $\beta$ is about 0.2 if the throttled surge tank is adopted. $\beta =0.2$ is the small value among the recommended values in specification, i.e., 0.15–0.5 [53]. Compared with the $\beta$ for the hydropower station with a short headrace tunnel, i.e., 0.3–0.4, the sectional area of the throttled orifice for the hydropower station with a super long headrace tunnel is smaller. For the surge tank with an upper chamber, the initial values can be assumed as $n_s=1~1.2$ and $p=0~-0.02$. Then, the other hydraulic design parameters of the surge tank can be determined based on Equations (1) and (2).

2.2. Hydraulic Design Optimization of the Surge Tank with Special Type or Combination

2.2.1. Air Cushion Surge Chamber

An air cushion surge chamber is one feasible and important type of surge tank for hydropower stations with super long headrace tunnels. For the hydropower station with a super long headrace tunnel and an air cushion surge chamber, Refs. [54,55,56,57,58] studied the characteristics of the transient process and hydraulic design optimization of the surge chamber. Four aspects, i.e., steady flow condition, large fluctuation condition, small fluctuation condition, and critical stable sectional area, were included.

Under the steady flow condition, the air pressure and water level in the surge chamber changes with the variation of the operating conditions. The change law depends on not only the upstream reservoir level and unit output, but also the set value of air pressure in the surge chamber. If the variation amplitude of the upstream reservoir level is too large, the set value of air pressure is directly related to the safe operation of the surge chamber. Therefore, Ref. [54] firstly established the mathematical model of the steady flow for the pipeline system with an air cushion surge chamber. Then, the analytical solutions for the water level and air pressure in the surge chamber, i.e., Equations (3) and (4), were derived. The effects of the upstream reservoir level, initial set value of the air pressure, top elevation of the surge chamber, and head loss on the water level and air pressure in the surge chamber were analyzed. Finally, the matching relationship between the variation amplitude of the water level in the surge chamber and variation amplitude of the upstream reservoir level was revealed. The range for the set value of the air pressure and water level in the surge tank was provided.

If the air pressure (or water level) in the surge chamber under one operating condition is known, the air pressure and water level in the surge chamber under any other operating condition can be determined by Equations (3) and (4) [54]. Note that the known operating condition and unknown operating condition are denoted by the subscripts, 1 and 2, respectively.

$$z_{w2}=\frac{1}{2}\left[z_2+z_m+H_2-\sqrt{{(z_2-z_m+H_2)}^2+4(z_1+H_1-z_{w1})(z_m-z_{w1})}\right]$$

(3)

$$\frac{p_2}{\gamma }=\frac{1}{2}\left[z_2-z_m+H_2+\sqrt{{(z_2-z_m+H_2)}^2+4(z_1+H_1-z_{w1})(z_m-z_{w1})}\right]$$

(4)

where $z_i$ is the upstream reservoir level, $z_{wi}$ is the water level in the surge chamber, $p_i$ is the air pressure in the surge chamber, $z_m$ is the top elevation of the surge chamber, $\gamma$ is the unit weight of water, $H_i=H_a-\varsigma Q_i^2$, $H_a$ is the atmospheric pressure, $\varsigma$ is the coefficient of the head loss, and $Q_i$ is the discharge.

For the hydropower station with a super long headrace tunnel, the setting of the air cushion surge chamber can effectively inhibit the amplitude of the water level oscillation in the surge chamber during transient processes. However, the change law of the volute pressure becomes more complicated because of the effect of the air cushion surge chamber. According to the numerical simulation of the large fluctuation transient process, Ref. [55] analyzed the effect of the closing time of the guide vane; flow inertia time constant of the headrace tunnel, T_wy; flow inertia time constant of the penstock; and parameters of the surge tank on the maximal volute pressure (as shown in Figure 3). By comparison with the conventional surge tank, the effect of the air cushion surge chamber on reflecting the water hammer wave was discussed. The results indicated that the reflection effect of the air cushion surge chamber on the water hammer wave was worse than that of the conventional surge tank. The sum of air pressure and surge pressure in the air cushion surge chamber was greater than that in the conventional surge tank. For the hydropower station with a super long headrace tunnel and air cushion surge chamber, the condition that the maximal volute pressure is determined by the highest water level in the surge chamber was more likely to appear.

For the hydropower station with a super long headrace tunnel and air cushion surge chamber, Ref. [56] broke through the Thoma assumption and then established the complete mathematical model of a hydro-turbine governing system. The Thoma assumption refers that the governor is absolutely sensitive and can maintain the power output of the turbine generator constant. The overall transfer function and free vibration equation that describe the dynamic characteristics of the system were derived. Then, the stable domain was drawn by using the stability criterion. By the stable domain, the effects of the flow inertia of the pipeline, parameters of the air cushion surge chamber, turbine characteristics, generator characteristics, and regulation modes of the governor on the system stability were analyzed (as shown in Figure 4). The results indicated that the turbine characteristics and flow inertia of the pipeline are unfavorable for the system stability, while the generator characteristics are favorable for the system stability. With the increase of air chamber height and sectional area, and the decrease of the air chamber pressure and air polytropic index, the system stability became better. The stability of the system under the power regulation mode was better than that under the frequency regulation mode.

In Ref. [57], the effect of nonlinear throttling orifice head loss on the dynamic behavior of a hydro-turbine governing system with an air cushion surge chamber was studied by using the Hopf bifurcation theory. The results indicate that the throttling orifice cannot affect the system stability through its head loss. However, the throttling orifice can affect the system dynamic response through its head loss (as shown in Figure 5). The throttling orifice head loss was favorable for the damping of water level oscillations in the surge chamber. In Figure 4 and Figure 5, b_t is the temporary droop; T_d is the damping device time constant; q_P, x, and y are the relative deviations of discharge in the penstock, unit frequency, and guide vane opening, respectively.

Based on the rigid water hammer theory, Ref. [58] derived the formula of the critical stable sectional area of the air cushion surge chamber, considering the flow inertia of the penstock, turbine characteristics, and the governor (as shown in Equation (5)). Based on the obtained formula, the value for the most unfavorable stable sectional area was analyzed. According to the data of 10 hydropower stations in China, the statistical analysis for each term of the obtained formula was carried out. The results indicated that the formula of the critical stable sectional area of the air cushion surge chamber consists of three terms, i.e., the headrace tunnel term considering the turbine characteristics, F_th1; the penstock term considering the turbine characteristics, F_th2; and the governor term, F_th3. The form of the formula of the critical stable sectional area of the air cushion surge chamber was the same with that of the conventional surge tank, and the former is (1 + mp₀/l₀) times of the latter. For the air cushion surge chamber, the most unfavorable stable sectional area depended on a head between the maximum head and rated head. That head makes (1 + mp₀/l₀) great and also satisfies e > 1. With the increase of the rated head, (1 + mp₀/l₀) increased linearly. It cannot be simply regarded that the air cushion surge chamber is suitable for a hydropower station with a high head.

$$F_{th}=(F_{th1}+F_{th2}+F_{th3})(1+m{p}_0/l_0)$$

(5)

where $F_{th1}=\frac{e{L}_y{f}_y}{2{\alpha }_{y}g(H_0-2e{h}_{t0})}$, $F_{th2}=\frac{e{L}_t{f}_t}{2{\alpha }_{t}g(H_0-2e{h}_{t0})}$,

$$F_{th3}=\left\{1-e\frac{2(h_{y0}+h_{t0})}{H_0}+\left[\frac{(e_g-e_x)}{e_y}+\frac{e_{qh}(e_g-e_x)+e_{qx}{e}_h}{e_y}\frac{2(h_{y0}+h_{t0})}{H_0}\right]b_t\right\}\frac{Q_0{H}_0{T}_d}{2h_{t0}(H_0-2e{h}_{t0})}\left[0,1\right],$$

m is the air polytropic index; p₀ is the air chamber pressure; l₀ is the air chamber height; L_y, f_y, h_y0, and α_y are the length, sectional area, head loss, and head loss coefficient of the headrace tunnel, respectively; L_t, f_t, h_t0, and α_t are the length, sectional area, head loss, and head loss coefficient of the penstock, respectively; e_h, e_x, and e_y are the moment transfer coefficients of the turbine; e_qh, e_qx, and e_qy are the discharge transfer coefficients of the turbine; e is the comprehensive characteristic coefficient of the turbine; Q₀ is the rated discharge of the turbine; and e_g is the load self-regulation coefficient.

2.2.2. Upstream and Downstream Double Surge Tanks

Upstream and downstream double surge tanks are a common combination layout type of surge tanks. When the middle development way is adopted, that type of surge tank is usually applied to hydropower stations with super long headrace tunnels. For the hydropower station with an upstream and downstream double surge tank, Ref. [59] established the complete mathematical model of the hydro-turbine governing system. The overall transfer functions and free vibration equations of the system under both the frequency regulation mode and power regulation mode of the governor were derived. Based on the Hurwitz stability criterion, the stable domain was drawn, as shown in Figure 6. By using the stable domain, the effects of the Thoma assumption, flow inertia of the pipeline, regulation modes of the governor, and parameters of the governor on the system stability were analyzed. The results indicated that the Thoma assumption reduces the stable domain. Under both the frequency regulation mode and power regulation mode of the governor, the flow inertia of the pipeline was unfavorable for the system stability. Under the same condition, the stability under the power regulation mode of the governor was obviously better than that under the frequency regulation mode of the governor. With the decrease of b_p and the increase of b_t and T_d, the system stability improved.

For the hydropower station with upstream and downstream double surge tanks, Ref. [60] established the nonlinear mathematical model of the hydro-turbine governing system considering the saturation nonlinearity of the governor. Then, the existence condition and direction of Hopf bifurcation for the nonlinear dynamic system were analyzed, and the algebraic criterion for the appearance of Hopf bifurcation was obtained. Based on the algebraic criterion, the stable domain for the system was drawn. Finally, the stability of the system under different states was analyzed by the stable domain. The results indicated that the Hopf bifurcation for the nonlinear hydro-turbine governing system was supercritical. The state points that far away from the bifurcation line was favorable for system stability. The stability of the hydro-turbine governing system considering the saturation nonlinearity of the governor was worse than that not considering the saturation nonlinearity of the governor (as shown in Figure 7). In Figure 6 and Figure 7, n₁ is the ratio of the sectional area of the upstream surge tank and its own critical stable sectional area, and n₂ is the ratio of the sectional area of the downstream surge tank and its own critical stable sectional area. K_p is the proportional gain, and K_i is the integral gain.

2.2.3. Upstream Series Double Surge Tanks

Upstream series double surge tanks are also one feasible and important type of surge tank for hydropower stations with super long headrace tunnels. For that combination layout type of surge tank, Refs. [61,62] studied and revealed the stability of the hydro-turbine governing system and the water level oscillation characteristics in surge tanks during the load rejection transient process.

Ref. [61] established the mathematical model of the hydro-turbine governing system considering the pipeline, surge tanks, turbine, governor, and generator. The overall transfer function and free vibration equation of the system were derived. The stable domain was drawn and is shown in Figure 8. Then, the effects of the sectional area of the surge tank and flow inertia of the pipeline on the system stability were analyzed. The results indicated that, when the hydro-turbine governing system for the hydropower station with upstream series double surge tanks and the hydropower station with one upstream surge tank reached the same stable domain, the sum of the two surge tanks of the former area was larger than the latter. The proportion that the flow inertia of the pipeline between upstream series double surge tanks held in the whole pipeline from the upstream reservoir to the second surge tank became smaller, and the differences between the sum of the two surge tanks of the former area and the latter was smaller. In Figure 8, T_F is the time constant of the surge tank.

In Ref. [62], the concept of the transfer function from the perspective of the control system was introduced. The water level oscillations in upstream series double surge tanks are regarded as an open-loop system. The change of the discharge in the penstock is regarded as the downstream boundary condition. By using the Laplace transform and inverse Laplace transform, the explicit formulas of water level oscillations in upstream series double surge tanks were obtained and are shown in Equations (6) and (7). The formulas were verified by using an example calculation. The effect of head loss on the highest water levels in surge tanks was analyzed and the results are shown in Figure 9. In Equations (6) and (7) and Figure 9, z₁ and z₂ are the water levels in upstream series double surge tanks; α₁ is the head loss coefficient between the upstream reservoir and the first surge tank; t is time; and the other parameters are intermediate variables.

$$z_1(t)=-M_1\sin \frac{t}{\sqrt{X}}+M_2\sin \sqrt{\frac{X}{c_1}}t$$

(6)

$$z_2(t)=N_1\sin \frac{t}{\sqrt{X}}-N_2\sin \sqrt{\frac{X}{c_1}}t$$

(7)

2.3. The Most Disadvantageous Design and the Most Advantageous Operation of the Surge Tank under Combined Operating Conditions Based on the Surge Wave

For the hydropower station with a super long headrace tunnel, wave superposition must exist in the surge tank. Therefore, the extreme water levels in the surge tank during the load acceptance condition and load rejection condition are not the controlling water levels. The surge wave superposition should be studied. Combined operating conditions (as shown in Figure 10) are usually the controlling operating conditions of the highest and lowest water levels in the surge tank. The study on the wave superposition characteristics of the transient process for the hydropower station with a super long headrace tunnel includes two aspects, i.e., the most disadvantageous design and the most advantageous operation of the surge tank. The above two issues were studied in Refs. [63,64].

By using the nonlinear oscillatory evolutionary method, Ref. [63] derived the analytical expressions of the most disadvantageous superimposition time under the four typical combined operating conditions. Then, the analytical solutions of the extreme water levels in the surge tank were obtained and are shown in Equations (8)–(11). The analytical solutions were verified by numerical simulation. The results indicated that, with the increase of the throttled loss coefficient of the surge tank, the controlling operating condition for the highest water level in the surge tank changed from the load-acceptance-then-rejection condition to the successive load rejection condition, while the controlling operating condition for the lowest water level in the surge tank changed from the load-rejection-then-acceptance condition to the successive load acceptance condition.

The analytical solutions for the water level oscillation of the superimposition operating condition under the four typical combined operating conditions are as follows [63].

$${Z_1}^{\prime }=A_1^{\prime }\cos {\phi }_1^{\prime };A_1^{\prime }=\frac{{A_{10}}^{\prime }}{1+\frac{4\omega }{3\pi }\left(1+\eta \right)C_3{A_{10}}^{\prime }t};{\phi }_1^{\prime }=\omega t+{{\phi }_{10}}^{\prime }$$

(8)

$${Z_2}^{\prime }=A_2^{\prime }\cos {\phi }_2^{\prime };A_2^{\prime }=\frac{{A_{20}}^{\prime }}{1+\frac{4\omega }{3\pi }\left(1+\eta \right)C_3{A_{20}}^{\prime }t};{\phi }_2^{\prime }=\omega t+{{\phi }_{20}}^{\prime }$$

(9)

$${Z_3}^{\prime }=A_3^{\prime }\cos {{\phi }_3}^{\prime }-4h_{w0};A_3^{\prime }=\frac{{C_1}^{\prime }{A_{30}}^{\prime }}{-C_2{A_{30}}^{\prime }+\left({C_1}^{\prime }+C_2{A_{30}}^{\prime }\right)e^{{C_1}^{\prime }{C}_{3}t}};{\phi }_3^{\prime }=\omega t+{{\phi }_{30}}^{\prime }$$

(10)

$${Z_4}^{\prime }=A_4^{\prime }\cos {\phi }_4^{\prime }-\alpha {(\frac{q+Q_0}{f})}^2;A_4^{\prime }=\frac{{C_q}^{\prime }{A_{40}}^{\prime }}{-C_2{A_{40}}^{\prime }+\left({C_q}^{\prime }+C_2{A_{40}}^{\prime }\right)e^{{C_q}^{\prime }{C}_{3}t}};{\phi }_4^{\prime }=\omega t+{{\phi }_{40}}^{\prime }$$

(11)

where h_w0 and V₀ are the head loss and flow velocity in the headrace tunnel under Q₀ of turbine discharge, respectively; $\eta =h_{r0}/h_{w0}$; h_r0 is the throttled loss under Q₀ through the throttled orifice; ${Z_i}^{\prime }$, ${A_i}^{\prime }$, and ${{\phi }_i}^{\prime }$ are the level, amplitude, and phase of the surge wave in the surge tank under the superimposition operating condition; $\omega$ is the angular frequency; the subscript, “0”, represents the initial operating condition; q is the change of the discharge; f is the sectional area of the pipeline; and the other parameters are intermediate variables.

For the hydropower station with a super long headrace tunnel, the water level oscillation in the surge tank has the following characteristics: Long period, large amplitude, and slow attenuation. Moreover, the wave superposition in the surge tank appears when the different units have different operating conditions, which affect the safe, stable, and flexible operation of the hydropower station. In order to reduce the amplitude of the surge wave and quicken the attenuation of water level oscillation, Ref. [64] derived the analytical expressions of the most advantageous superimposition time under the four typical combined operating conditions. Based on the most advantageous control principle, the cascaded load adjustment mode of the hydropower unit was proposed and is shown in Figure 11. Based on the cascaded load adjustment mode, the most advantageous operation of the surge tank under the combined operating conditions based on the surge wave was achieved.

3. Operation Control of the Unit

The operation control of the unit relates to the safe and stable operation of the hydropower station. For the hydropower station, the operation control of the unit is realized by the hydro-turbine governing system. Based on the hydraulics-machinery-electrics coupling effect of the hydropower station with a super long headrace tunnel, the study on the stability and regulation quality of the hydro-turbine governing system under isolated and grid-connected operation conditions is carried out.

3.1. Stability and Regulation Quality of the Hydro-Turbine Governing System under the Isolated Operation Condition

3.1.1. Regulation Quality and Critical Stable Sectional Area of the Surge Tank Based on Regulation Modes

The previous study on the stability and regulation quality of the hydro-turbine governing system with the surge tank has usually been aimed at the frequency regulation (FR) mode. However, for the hydropower station with a long headrace tunnel, complex arrangement, and important peak modulation and frequency modulation tasks, the contradiction between the stability and regulation quality is prominent. The single frequency regulation mode cannot coordinate the contradiction effectively. Therefore, the regulation mode should be changed and the study on the power regulation (PR) mode and servomotor stroke regulation (SSR) mode should be carried out. According to the above considerations, Refs. [65,66,67] studied the regulation quality and critical stable sectional area of the surge tank based on regulation modes.

Ref. [65] established the mathematical models of the hydro-turbine governing system with the surge tank under the FR mode, PR mode, and SSR mode. The overall transfer functions and free vibration equations of the system were derived. According to the free vibration equations and Hurwitz stability criterion, the stable domains under the three regulation modes were drawn and are shown in Figure 12. By contrastive analysis, it can be known that the systems under FR and PR were both conditionally stable, and the stable domain of the former was far smaller than that of the latter. The system under SSR was absolutely stable. In Figure 12 and Figure 13, n_f is the ratio of the sectional area of the surge tank and its critical stable sectional area.

Based on the overall transfer functions of the hydro-turbine governing system under the three regulation modes, Ref. [66] derived the Laplace transform of the unit frequency under step disturbance of the load. Then, by using the method of retaining dominant poles and deleting nondominant poles, the order reduction of the high order system was realized. The obtained low order equivalent system could be solved analytically. The dominant poles produced the tail wave and reflected the response of the unit frequency caused by the water level oscillation in the surge tank. The tail wave has the characteristics of a long period and slow attenuation, and is the main body of the unit frequency oscillation. The regulation quality of the hydro-turbine governing system is mainly determined by the tail wave. The nondominant poles produce the head wave and reflect the response of the unit frequency caused by the water hammer wave in the penstock. The head wave has the characteristics of a short period and fast attenuation, and can be regulated by the governor effectively. Based on the obtained low order equivalent system, the wave equation of the unit frequency was derived. Then, the expressions of the characteristic parameters for the regulation quality were obtained. By comparing the characteristic parameters for regulation quality (as shown in Figure 13), the differences and connections of the three regulation modes were revealed.

Ref. [67] proved that the criterion for the critical stable sectional area of the surge tank is the coefficient of the first-order term of the free vibration equation. When the coefficient of the first-order term of the free vibration equation equaled zero, the formula of the critical stable sectional area of the surge tank considering the flow inertia of the penstock and governor parameters were derived and are shown in Equation (12). The effects of the headrace tunnel, penstock, and governor are all included in the formula of the critical stable sectional area of the surge tank.

$$F_{th}=F_{th1}+F_{th2}+F_{th3}$$

(12)

where $F_{th1}=e\frac{L_y{f}_y}{2{\alpha }_{y}g(H_0-2e{h}_{t0})}$, $F_{th2}=e\prime \frac{L_t{f}_t}{2{\alpha }_{t}g(H_0-2e{h}_{t0})}$,

$$F_{th3}=\frac{[H_0+2e_{qh}(h_{y0}+h_{t0})]\left\{(1+b_t\frac{(e_g-e_x)}{e_y})H_0+2\left[b_t\frac{(e_g-e_x)e_{qh}+e_h{e}_{qx}}{e_y}-e\right]h_{t0}\right\}+2\frac{e_h}{e_y}(b_t{e}_{qx}-e_{qv})h_{y0}{H}_0}{(H_0-2e{h}_{t0})(H_0+2e_{qh}{h}_{t0})}\frac{Q_{y0}}{2h_{y0}}{T}_d$$

3.1.2. Transient Process and Control of the Hydro-Turbine Governing System under the Effect of the Surge Tank

Under small load disturbance, the linear mathematical model of water flow in the surge tank can be established. By combining with the models of other subsystems, the linear mathematical model of the hydro-turbine governing system with the surge tank is obtained. However, the introduction of the surge tank increases the order of the mathematical model of the hydro-turbine governing system. As a result, the hydro-turbine governing system becomes a high order dynamic system with an order higher than four. For the high order system, the theoretical analysis for the transient process and control is difficult. Moreover, the design and realization of the control strategy for the high order system are also a great challenge. How to reduce the order of the system and construct a low order equivalent model is the key problem to realize the transient process and control of the hydro-turbine governing system with the surge tank.

On the premise of the isolated operation condition and rigid water hammer, Ref. [68] studied the effect mechanism of flow inertia and head loss of the penstock on the stability and regulation quality of the hydro-turbine governing system with (or without) a surge tank. Firstly, the linear mathematical model of the hydro-turbine governing system with (or without) a surge tank was established. The overall transfer functions and free vibration equations of the system were derived. Then, the stable domain and dynamic response of the unit frequency were determined, and the effect mechanism of flow inertia and head loss of the penstock on the stability and regulation quality of the hydro-turbine governing system with (or without) a surge tank was revealed. Finally, by using the obtained effect mechanism, the method for improving the stability and regulation quality of the hydro-turbine governing system was proposed. The construction method for the low order equivalent model of the original system was also proposed and is shown in Figure 14.

For the hydropower station with a surge tank, Refs. [69,70] proposed a novel research thought for the operation control of the unit. In that novel research thought, a given sine wave of water level in the surge tank was used to describe the unsteady flow characteristics in the headrace tunnel and surge tank (as shown in Figure 15). The hydraulic parameters and dynamic characteristics of the headrace tunnel and surge tank were reflected in the characteristic parameters of the given sine wave. By using the assumption of the sine wave of the water level in the surge tank and its mathematical description, the stability, dynamic response performance, and transient process control of the hydro-turbine governing system under primary frequency regulation were studied. The stability of the hydro-turbine governing system under the opening control mode and power control mode was analyzed. The concept of the critical stable sectional area of the surge tank for primary frequency regulation was proposed. The joint setting method for the critical stable sectional area of the surge tank and governor parameters was provided. The analytical expression for the dynamic response of the power output under primary frequency regulation was derived. According to the control requirement of the dynamic response of power output under primary frequency regulation, the concept of the domain of primary frequency regulation was proposed and is shown in Figure 16. The effect of characteristic parameters on the domain of primary frequency regulation was analyzed.

With the development of the theory of the nonlinear dynamic system and control, some advanced, complex, and nonlinear control methods have been adopted to hydro-turbine governing systems with surge tanks.

In Refs. [71,72], the Hopf bifurcation theory was applied to analyze the nonlinear dynamical behaviors of a hydro-turbine governing system with a surge tank. The nonlinear dynamical behaviors of the system were presented, including bifurcation diagrams, time waveforms, phase orbits, and Poincare maps with the frequency fluctuations of the power network. From the analysis of nonlinear control for the governing system, the value of the differential adjustment coefficient should be between two critical places and the farther the better. To control the undesirable chaotic behaviors in the system, the fuzzy sliding mode governor based on the sliding mode control and the fuzzy logic were designed. Series of numerical simulations proved that the hydro-turbine governing system can maintain a better operation station under the designed governor.

In Ref. [73], the nonlinear dynamic analysis and robust controller design for the hydro-turbine governing system with a straight-tube surge tank were carried out. In the hybrid controller, the sliding mode control law makes full use of the proposed model to guarantee the robust control in the presence of system uncertainties, while the fuzzy system was applied to approximate the proper gains of the switching control in the sliding mode technique to reduce the chattering effect, and particle swarm optimization was developed to search the optimal gains of the controller. Numerical simulations showed that the performances of the nonlinear system assisted with the proposed controller were much better than that with the commonly used optimal proportional-integral-derivative (PID) controller.

3.2. Stability and Regulation Quality of the Hydro-Turbine Governing System under the Grid-Connected Operation Condition

For the hydropower station with a super long headrace tunnel, the flow inertia in the headrace tunnel is extremely large. As a result, the water level oscillation in the surge tank has the following characteristics: Long period, large amplitude, and slow attenuation. Under the isolated operation condition, the regulation quality of the hydro-turbine governing system is poor. In the engineering practice, the sectional area of the surge tank needs to be increased to improve the regulation quality. However, that method would increase the investment significantly. It is necessary to study the stability and regulation quality of the hydro-turbine governing system under the grid-connected operation condition.

By using the toolbox of Sim Power Systems in Simulink, Ref. [74] established the detailed simulation model for the hydro-turbine governing system under the grid-connected operation condition. The stability and regulation quality of the hydro-turbine governing system under the grid-connected operation condition was analyzed (as shown in Figure 17). The obtained results provide guidance to the operation control of units.

From the perspective of improving the stability and regulation quality of the hydro-turbine governing system, the hydropower station with a super long headrace tunnel is suitable for the grid-connected operation condition.

In Ref. [75], the mathematical model of the hydro-turbine governing system under the grid-connected operation condition was established by introducing the power grid model. Using the Laplace transform, the overall transfer function, for which the input signal is external disturbance and the output signal is unit frequency, was obtained. Then, the numerical simulation of the transient process of the hydro-turbine governing system under the joint effect of the power grid and surge tank was carried out by using Simulink. The effects of the scale of the power grid, different generator models, and different regulation modes on the transient process of the system were analyzed. The results indicated that the surge tank only affects the tail wave of the unit frequency response, x. With the increase of the sectional area of the surge tank, the system becomes more stable. The power grid can affect both the head wave and tail wave (as shown in Figure 18). With the increase of the scale of the power grid, the head wave and tail wave decay more rapidly. The unit frequency response, x, under the third order generator model and fifth order generator model only have a small difference, while they have a significant difference with that under the first order generator model.

In Ref. [76], the state-of-the-art in variable-speed operation of hydropower plants was reviewed. Variable-speed operation of pumped-storage hydropower plants can bring additional flexibility to the power system, offering a variety of valuable ancillary services. In addition to the power system, the hydropower plant itself could substantially benefit from variable-speed operation. The flexibility and stabilization of the power system can be greatly improved with ancillary services that hydropower plants with variable speed are able to provide.

In Ref. [77], salient issues associated with the anti-islanding protection and islanding operation of grid-connected hydropower distributed generation were discussed. Design criteria and solutions for effective operation and control were developed. A technique was developed to use islanding detection and anti-islanding protection. The authors have also proposed guidelines for islanding operation of hydropower distributed generation to improve the quality and reliability of power supply.

MATLAB is an effective tool to simulate and analyze the stability and regulation quality of hydro-turbine governing systems under the grid-connected operation condition and is widely used. Some other software, such as PSASP [78] and TOPSYS [79], can also be applied.

4. Conclusions

With the development of hydropower energy, hydropower stations with super long headrace tunnels become increasingly more competitive. Compared with hydropower stations with short headrace tunnels, the transient process and control for hydropower stations with super long headrace tunnels is much more complicated and becomes an intractable challenge. The study on the transient process and control is the key to overcoming the challenge of the design and operation of hydropower stations with super long headrace tunnels.

This review paper presented a detailed literature survey focused on the transient process and control for hydropower stations with super long headrace tunnels. Firstly, two key issues for the transient process and control, i.e., the hydraulic design optimization of the surge tank and operation control of the unit, were illuminated. Secondly, for both single surge tanks and surge tanks with special type or combination, the hydraulic design optimization methods were described. The most disadvantageous design and advantageous operation of surge tanks under combined operating conditions were discussed. Thirdly, the stability and regulation quality of hydro-turbine governing systems under isolated and grid-connected operation conditions were presented. Finally, some trends and recommendations for future research directions were made.

The main conclusions drawn from the existing research are summarized as follows:

(1)

The maximal volute pressure of hydropower stations with super long headrace tunnels depends on the highest water level in the surge tank, but not the closing law of the guide vane. For the hydropower station with a super long headrace tunnel, the reasonable value of $\beta$ is about 0.2 when the throttled surge tank is adopted.

(2)

For the hydropower station with a super long headrace tunnel and air cushion surge chamber, the condition that the maximal volute pressure is determined by the highest water level in the surge chamber is more likely to appear. The throttling orifice cannot affect the system stability through its head loss, but can affect the system dynamic response. The formula of the critical stable sectional area of the air cushion surge chamber consists of three terms, i.e., the headrace tunnel term considering the turbine characteristics, F_th1 the penstock term considering the turbine characteristics, F_th2; and the governor term, F_th3.

(3)

For the hydropower station with upstream and downstream double surge tanks, the Thoma assumption reduces the stable domain. The stability of the hydro-turbine governing system considering the saturation nonlinearity of the governor is worse than that not considering the saturation nonlinearity of the governor. For hydropower stations with upstream series double surge tanks, the explicit formulas of water level oscillations in surge tanks are Equations (6) and (7).

(4)

With the increase of the throttled loss coefficient of the surge tank, the controlling operating condition for the highest water level in the surge tank changes from the load-acceptance-then-rejection condition to the successive load rejection condition, meanwhile, the controlling operating condition for the lowest water level in the surge tank changes from the load-rejection-then-acceptance condition to the successive load acceptance condition. Based on the cascaded load adjustment mode, the most advantageous operation of the surge tank under combined operating conditions based on the surge wave can be achieved.

(5)

The systems under FR and PR are both conditionally stable, and the stable domain of the former is far smaller than that of the latter. The system under SSR is absolutely stable. The regulation quality of hydro-turbine governing system is mainly determined by the tail wave. The criterion for the critical stable sectional area of the surge tank is the coefficient of the first-order term of the free vibration equation.

(6)

From the perspective of improving the stability and regulation quality of the hydro-turbine governing system, the hydropower station with a super long headrace tunnel is suitable for the grid-connected operation condition. The power grid can affect both the head wave and tail wave. With the increase of the scale of the power grid, the head wave and tail wave decay more rapidly.

The existing research achievements fill the gaps in the field of hydraulic design optimization of surge tanks and operation control of units for hydropower stations with super long headrace tunnels. Progress of the transient process and control for hydropower stations with super long headrace tunnels is achieved. This review can help in understanding the key science and technology problems of the transient process and control for hydropower stations with super long headrace tunnels.

From the above literature review, it can been found that although a lot of research works on the transient process and control for hydropower stations with super long headrace tunnels have been performed and many valuable results have been obtained, there are still much work that needs to be performed. Some of the future development areas in this topic are listed as follows:

(1)

Free surface-pressurized flow can effectively reduce the flow inertia in super long headrace tunnels. The applicability and hydraulic design theory of headrace tunnels with free surface-pressurized flow to hydropower stations with super long headrace tunnels needs to studied.

(2)

The control requirements of hydropower stations with super long headrace tunnels are high, and the linear control strategies cannot be met in some cases. It is necessary to study the nonlinear control strategies for the operation of hydropower stations with super long headrace tunnels.

(3)

The hydropower station with a super long headrace tunnel usually operates under complex conditions and undertakes the tasks of power modulation and frequency modulation. The transient process and control under stochastic or uncertain external disturbance should be explored.

(4)

The transient process and control for hydropower stations with super long headrace tunnels under the complementary operation of wind power, solar power, and hydropower is a novel research direction.

The authors suggest that the research thoughts for establishing the complete theory and application system of the transient process and control for hydropower stations with super long headrace tunnels from the perspective of multi-slice and multi-scale can be adopted.