Research on start-up transition process of high-head pumped storage units

1 Introduction

High-head pumped storage units play a critical role in modern power systems by providing peak shaving, frequency regulation, and emergency reserve services, and their operational flexibility directly affects the security and stability of the power grid. As new power systems demand faster response, frequent start-stop operations of pumped storage units have become routine. However, the transient process from start-up and no-load operation to grid synchronization is highly unstable (Mao et al., 2025; Chen H. et al., 2024; Liu et al., 2025a; Liu, 2025). During this process, due to the small guide-vane opening and rapid increase in rotational speed, the flow regime within the flow passage components is complex and turbulent, generating dynamic hydraulic loads with significant pulsation intensity; this can induce severe vibrations in both the unit and the plant structure (Ren et al., 2025; Chen, 2024; Deng, 2023; Liu, 2023). Therefore, the in-depth investigation of start-up transient dynamics and the development of rational start-up strategies is critical for safe, stable operation and extended service life.

Extensive research has been conducted on the start-up transient of pump-turbines, which can be broadly categorized into model tests, field measurements, one-dimensional simulations, and three-dimensional numerical simulations. In terms of experimental studies, Wang et al. (2024) conducted field tests on the start-up and shutdown processes of a pumped storage unit under turbine operating conditions. They found that stability is good at the point of start-up; during the runner acceleration phase, pressure pulsations manifest as broadband noise, whereas during the deceleration phase, they are dominated by the runner rotational frequency and its harmonics. Zhao et al. (2022) investigated the no-load characteristics and variable-speed evolution of doubly-fed pumped storage units based on dynamic experiments and provided experimental evidence for the start-up control of variable-speed units. In the realm of three-dimensional numerical simulations, Huang et al. (2023) established a one-dimensional characteristic-line hydraulic model for a pipeline system and a three-dimensional flow field model for a pump-turbine passage, and they performed coupled 1D–3D simulations. They found that during start-up to no-load conditions, a high-velocity water ring forms in the vaneless space, and the runner passage is filled with complex vortex structures, leading to abrupt drops in flow rate and torque. Jin et al. (2023) analyzed energy losses during the load rejection transient of a prototype reversible pump-turbine, identifying significant pressure pulsation amplitudes in the vaneless space and substantial energy dissipation in regions with high relative vortex stretching (RVS) and Coriolis force (CORF). Zhou et al. (2026) explored the improvement of S-shaped instability and the power performance of a reversible pump-turbine runner, discussing potential enhancements to start-up stability from the perspective of runner optimization.

Regarding the optimization of start-up control strategies, existing studies have mainly focused on guide vane movement patterns, governor parameters, and unit start-up modes. Hou et al. (2018) applied a multi-objective particle swarm optimization algorithm to optimize single- and two-stage guide vane control strategies for pumped storage units, demonstrating that the proposed strategy can significantly shorten acceleration time, reduce speed overshoot, and ensure rapid stabilization. Hou et al. (2019) further extended the multi-objective optimization approach to sequential start-up of two units sharing a common water conveyance system, revealing that the start-up interval significantly affects the transient of the first unit but has little influence on the second. Lei et al. (2021) developed a mathematical model that accounts for hydraulic coupling between the water conveyance pipeline, turbine, governor, and generator, and they analyzed optimal dual-unit start-up strategies under both symmetric and asymmetric branch-pipe configurations, quantifying the effects of start-up intervals and pipeline geometry on turbine head and rotational speed. Chen et al. (2025) investigated the influence of water pipelines on the transient process of variable-speed pump-turbines, showing that pipelines improve stability through inertia compensation and damping, increasing system efficiency by 2.6% and reducing flow gradient by 12%. Chen Y. et al. (2024) examined the effects of initial rotational speed and guide-vane control strategy on the pump power-off transient of variable-speed units, finding that time-domain response trends are similar in the early stages of forward-flow and reverse-flow phases and that entering the shutdown phase from the reverse-flow condition enhances system safety. Huang et al. (2025) analyzed the influence of hump characteristics on the pump power-off transient, concluding that a smaller hump deviation angle yields lower maximum spiral casing pressure and higher minimum draft tube pressure, thus favoring the extreme values of governing parameters.

In addition, multi-unit coordinated start-up and the start-up characteristics under different unit configurations have attracted increasing attention. Ma (2024) studied the start-up stability of a pumped storage system with different unit types sharing the same penstock, finding that when one fixed-speed unit is replaced by a variable-speed unit, the disturbance to the remaining fixed-speed unit decreases by 8.68 m under identical conditions. It was also noted that, besides selecting appropriate start-up strategies and parameters, the start-up interval and sequence are important factors that affect system behavior. Liu et al. (2025b) proposed a method for extracting electromagnetic vibration features and assessing unit condition based on the start-up transient, offering a new approach for condition monitoring during start-up.

Despite these valuable contributions, the following research gaps remain. First, most existing studies have focused on medium- or low-head pumped storage units or conventional hydroelectric units; systematic research on start-up transients of high-head (above 700 m) pumped storage units is relatively scarce. Second, under high-head conditions, the long water conveyance system exhibits significant water hammer effects, and the mechanisms by which the guide-vane actuation rate and dwell time affect pressure pulsations are not yet fully understood; quantitative comparisons and comprehensive evaluations across different start-up modes are lacking. Third, many studies concentrate on optimizing a single start-up strategy, with insufficient horizontal comparison and selection criteria among multiple typical start-up modes (open-loop, closed-loop, and their combination).

To address these gaps, this study takes a Francis reversible pump-turbine with a rated capacity of 350 MW and a rated head of 710 m as the research object. A one-dimensional transient flow model based on the method of characteristics (MOC) is employed to systematically compare four representative start-up modes: large-acceleration open-loop guide vane control, small-acceleration open-loop control, acceleration-controlled closed-loop control, and combined open-loop plus closed-loop control. The start-up time, rotational speed variation, and pressure pulsation characteristics at the spiral casing and draft tube are analyzed for each mode. Furthermore, the preferred scheme is further validated by adjusting guide vane motion parameters, and a comprehensive multi-indicator evaluation is performed using the entropy weight method. The results are intended to provide a theoretical basis and engineering reference for the optimal selection of start-up schemes for high-head pumped storage units.

2 Theoretical model for pipeline water hammer calculation

The fundamental equations for water hammer calculation in pipelines as Equation 1 include the motion equation and the continuity equation, which can be described by the following pair of partial differential equations:

∂ H ∂ x + 1 g A ∂ Q ∂ t + V ∂ Q ∂ x + f Q Q 2 g D A 2 = 0 ∂ H ∂ t + V ∂ H ∂ x + V ⁡ sin ⁡ θ + a 2 g A ∂ Q ∂ x = 0 , (1)

where: H is the piezometric head, m; Q is the pipeline flow rate, m³/s; A is the pipeline cross-sectional area, m²; V is the average flow velocity in the pipeline, m/s; g is the gravitational acceleration, m/s²; x is the distance, m; t is time, s; f is the friction loss coefficient; D is the pipeline diameter, m; θ is the pipeline inclination angle; and a is the water hammer wave speed, m/s.

For circular pipes, the water hammer wave speed a is calculated using Equation 2:

a = K / ρ 1 + K / E D / e , (2)

where: K is the bulk modulus of elasticity of water; ρ is the density of water, approximately 1,000 kg/m³; E is the elastic modulus of the pipe material, N/m²; D is the pipe inner diameter, m; and e is the pipe wall thickness, m.

Since the water hammer wave speed a is much greater than the flow velocity V in the pipe, the convective term V ∂ ∂ x in Equation 1 can be ignored relative to the transmission term ∂ ∂ t. In addition, ignoring the influence of the pipe axis inclination and denoting the simplified motion and continuity equations as L ₁ and L ₂, respectively, we have Equations 3, 4:

L 1 = ∂ H ∂ x + 1 g A ∂ Q ∂ t + f Q Q 2 g D A 2 = 0 , (3)

L 2 = ∂ H ∂ t + a 2 g A ∂ Q ∂ x = 0 . (4)

Using the method of characteristics, the system of partial differential Equations 3, 4 can be transformed along the characteristic lines C ⁺ and C ⁻ into the following two pairs of ordinary differential equations as Equations 5, 6:

C + 1 a d H d t + 1 g A d Q d t + f Q Q 2 g D A 2 = 0 d x d t = + a , (5)

C − 1 a d H d t − 1 g A d Q d t − f Q Q 2 g D A 2 = 0 d x d t = − a . (6)

For a given pipeline, a is usually constant. Taking the pipeline axis along the flow direction as the x-axis and time t as the y-axis, two straight lines can be drawn on the x–t plane according to Equations 5, 6—the positive and negative characteristic lines, respectively.

Dividing a pipe of length L into n equal segments, each of length Δ x = L / n, with a computational time step Δ t = Δ x / a, the characteristic grid shown in Figures 1, 2 can be plotted.

Assuming the flow rate and head at points A and B in the pipe are known, integrating along the positive characteristic line AP and the negative characteristic line BP transforms Equations 5, 6 into the following finite difference equations, also called compatibility equations as Equations 7, 8:

C + H p i = C p − B p Q p i , (7)

C − H p i = C m + B m Q p i , (8)

where C p, C m, B p, and B m are known quantities at time t − Δ t, this can be represented by Equations 9–12:

C p = H i − 1 + B Q i − 1 , (9)

C m = H i + 1 − B Q i + 1 , (10)

B p = B + R Q i − 1 , (11)

B m = B + R Q i + 1 , (12)

where constants B = a g A , R = f Δ x 2 g D A 2.

Solving Equations 7, 8 simultaneously yields, we have Equation 13:

Q p i = C p − C m B p + B m H p i = B m C p + B p C m B p + B m . (13)

It should be noted that Section I is any grid intersection point in the x direction; the subscripted H and Q at each section are always known from the previous time step, either given as initial conditions or as results calculated from the previous time step; the unknown head H and flow Q at the current time are denoted with the subscript p.

When solving hydraulic transient flow problems, the calculation usually starts from the steady flow state at time zero. Therefore, the initial values of H and Q at each computational section of the pipeline are known. The solution begins by calculating H and Q at the grid points corresponding to intermediate sections at time t = Δ t using Equation 13. Then, the boundary conditions at both ends of the pipeline are combined with Equations 7, 8 (the upstream end with Equation 8, the downstream with Equation 7) to obtain H and Q at the grid points corresponding to every section of the pipeline at time t = Δ t. The calculation then proceeds on the t = 2 Δ t grid, and so on, until the desired calculation time is reached.

3 Engineering application

3.1 Project overview

A first-class large (Type 1) hydropower project, equipped with six reversible mixed-flow turbine-generator units, each has a capacity of 350 MW. A schematic diagram of the water conveyance system is shown in Figures 1 and 2, and parameters of the single unit are shown in Table 1. The units have a rated speed of 500 r/min, a rated head of 710 m, and a rated flow of 56.85 m³/s. The water diversion and power generation system is located within the mountain between the upper and lower reservoirs, with a total length of approximately 2,750 m, including a 2,128 m-long water intake system and a 637 m-long tailwater system. It adopts a conventional three-tunnel, six-unit layout, divided into three hydraulic units.

TABLE 1. Parameters of the single unit.

3.2 Start-up methods

This study conducted numerical simulations for four start-up methods: guide-vane large-acceleration open-loop start-up (Method 1), guide-vane small-acceleration open-loop start-up (Method 2.1), acceleration-controlled closed-loop start-up (Method 3), and open-loop + closed-loop start-up (Method 4).

3.2.1 Guide-vane large-acceleration open-loop start-up (Method 1)

The start-up pattern for the guide-vane large-acceleration open-loop start-up is the actual operational pattern currently adopted by the power station, detailed as follows. The first start-up opening is the no-load opening limit (no-load opening 120% + 1%, with the guide vane control output change rate v limited to approximately 2.8%/s). When the unit speed exceeds 95% of the rated speed, it is reduced to the second start-up opening (no-load opening 110%). When the unit speed exceeds 96% of the rated speed, the unit transitions to no-load operation, and the guide vane opening is reduced to the no-load opening.

3.2.2 Guide-vane small-acceleration open-loop start-up (Method 2.1)

The start-up pattern for guide-vane small-acceleration open-loop start-up is as follows. The start-up opening Yjs1 is approximately 60% of the no-load opening Y₀ (with the guide vane control output increase rate v limited to approximately 0.7%/s). After waiting approximately 6 s, it switches to the second start-up opening Yjs2, which is the no-load opening Y₀ plus 5% (with the guide-vane control output increase rate v limited to approximately 0.6%/s). When the unit speed reaches approximately 90% of the rated speed, it is reduced to the third start-up opening Yjs3, which is 2% larger than the no-load opening (with the guide vane control output decrease rate v limited to approximately 1.1%/s). When the unit speed reaches approximately 95% of the rated speed, the governor engages the frequency regulation PID control.

3.2.3 Acceleration-controlled closed-loop start-up (Method 3)

The startup procedure for the guide-vane acceleration closed-loop is as follows. The regulator always adjusts according to the frequency PID closed-loop control. From the moment the unit starts rotating until it reaches 20% of the rated speed, the frequency setpoint follows the unit frequency. Before the unit speed reaches approximately 85% of the rated speed, the unit acceleration is set to 2 Hz/s. Before the unit speed reaches approximately 95% of the rated speed, the unit acceleration is set to 1 Hz/s. After the unit speed reaches 95% of the rated speed, the unit acceleration is set to 0.5 Hz/s until the frequency setpoint reaches the rated value. If the frequency setpoint has already exceeded 49.7 Hz, it should be maintained for more than 5 s; if the rated frequency of 50 Hz is still not reached, then it should be directly set to the rated frequency or the grid frequency.

3.2.4 Open-loop + closed-loop start-up (Method 4)

The rules for starting with the guide-vane open-loop and closed-loop are as follows. After the governor receives the start command, it sets the opening control to the start opening (no-load opening Y0 3%), sets the frequency to 40 Hz, and the opening tracking control quickly opens to the start opening. When the frequency increases to 40 Hz, PID regulation is automatically activated, the no-load opening limit is released, and the frequency setpoint increases automatically from 40 Hz to 50 Hz according to the set ramp rate (0.4 Hz/s) (in frequency-tracking mode, it matches the grid frequency). The unit frequency will then enter 50 Hz relatively smoothly.

3.3 Comparison of different start-up methods

Using one-dimensional numerical simulation, the operational characteristics of starting Unit 4 alone and simultaneously starting Units 3 and 4 were calculated; the specific results are shown in Table 2. Due to space limitations, only the transient curves for Methods 1 and 2.1 are listed (Figures 3–6).

• For Method 1, the unit’s start-up duration is the shortest at 40.59 s. However, when Units 3 and 4 start simultaneously, the peak-to-peak pressures at the spiral case inlet and end of Unit 4 reach 193.81 m and 205.86 m, respectively. The peak-to-peak pressures at the draft tube inlet and outlet of Unit 4 are 75.18 m and 73.92 m, respectively. When only Unit 4 starts, the peak-to-peak pressures at the spiral case inlet and end of Unit 4 decrease to 108.89 m and 142.69 m, respectively. The peak-to-peak pressures at the draft tube inlet and outlet of Unit 4 decrease to 73.00 m and 69.91 m, respectively.
• For Method 2.1, the unit’s start-up duration increases by nearly 12 s compared to Method 1, reaching approximately 52.64 s. However, when Units 3 and 4 start simultaneously, the peak-to-peak pressure at the spiral case inlet of Unit 4 decreases by 2.6% compared to Method 1. The peak-to-peak pressures at the draft tube inlet and outlet decrease by 7.2% and 8.7%, respectively, compared to Method 1. The peak-to-peak pressure at the spiral case end shows minimal change compared to Method 1. When only Unit 4 starts, the peak-to-peak pressure at the spiral case inlet of Unit 4 decreases by 5.7% compared to Method 1. The peak-to-peak pressures at the draft tube inlet and outlet decrease by 17.2% and 25.3%, respectively, compared to Method 1. The peak-to-peak pressure at the spiral case end shows minimal change compared to Method 1.
• For Method 3 (i.e., unit acceleration-controlled closed-loop start-up), during simultaneous start-up of Units 3 and 4, the peak-to-peak pressures at the spiral case inlet and end of Unit 4 decrease by 12.3% and 12.0%, respectively, compared to Method 1. The peak-to-peak pressures at the draft tube inlet and outlet show minimal change compared to Method 1. When only Unit 4 starts, the peak-to-peak pressure at the spiral case inlet of Unit 4 decreases by 7.2% compared to Method 1. The peak-to-peak pressures at the draft tube inlet and outlet decrease by 9.6% and 8.6%, respectively, compared to Method 1. The peak-to-peak pressure at the spiral case end shows relatively small change compared to Method 1. However, the start-up duration for Method 3 increases by nearly 29 s compared to Method 1, reaching 69.38 s.
• For Method 4 (open-loop + closed-loop start-up), during simultaneous start-up of Units 3 and 4, the peak-to-peak pressures at the spiral case inlet and end of Unit 4 decrease by 25.0% and 25.1%, respectively, compared to Method 1. The peak-to-peak pressures at the draft tube inlet and outlet increase slightly by 3.5% and 3.4%, respectively, compared to Method 1. When only Unit 4 starts, the peak-to-peak pressures at the spiral case inlet and end of Unit 4 decrease by 23.1% and 8.4%, respectively, compared to Method 1. The peak-to-peak pressures at the draft tube inlet and outlet decrease by 8.2% and 6.2%, respectively, compared to Method 1. However, the unit’s start-up duration under this method is excessively long, close to 1.8-times that of Method 1.

TABLE 2. Characteristic values of hydraulic transients under different start-up methods.

Methods 2.1, 3, and 4 can all ensure good unit operational stability. However, the start-up duration of Method 2.1 is much shorter than for Method 4, and the reduction in peak-to-peak pressures at the draft tube inlet and outlet for Method 2.1 is greater than that for Method 3. Therefore, considering both start-up duration and unit stability comprehensively, Method 2.1 is more advantageous.

The rotational speed settling time is defined as the duration from the issuance of the start-up command to when the speed fluctuation remains within ±0.2% of the rated speed (500 r/min) for over 10 consecutive seconds. Under the dual-unit simultaneous start-up scenario of Method 1, the rotational speed reaches 95% of the rated value at 38.6 s, achieves full stabilization at 42.3 s, and exhibits a final fluctuation amplitude of ±0.8 r/min. Regarding pressure pulsation dominant frequencies, the spiral case pressure exhibits a dominant frequency of 83.3 Hz (corresponding to the runner rotational frequency) along with its harmonics, induced by guide vane wake interference and rotor-stator interaction; the draft tube pressure displays a dominant frequency of from 42–76.8 Hz (five to ten times the rotational frequency), corresponding to the precessing vortex rope in the draft tube cone. The peak pulsation amplitudes consistently occur during the guide vane acceleration phase (15–25 s after start-up), when the flow velocity change rate is maximal and the flow regime is most unstable. Compared with Method 1, Method 2.1 under dual-unit simultaneous start-up exhibits an 18.7% slower pressure pulsation attenuation rate but a 7.2% lower peak amplitude, indicating that the more gradual guide vane opening rate extends the flow adjustment duration, effectively mitigating transient flow-induced disturbances and yielding superior flow regime stability.

The distinct variations in pressure pulsation among the four methods can be attributed to the coupling effect between guide vane motion and flow dynamics. In Method 1, the high opening acceleration (v ≈ 2.8%/s) causes a rapid surge in discharge, which induces severe flow separation and vortex shedding in the draft tube, leading to the highest peak-to-peak pressure values. Conversely, Method 2.1 employs a significantly reduced acceleration (v ≈ 0.7%/s initially). This gradual increase in discharge allows the flow field to adjust smoothly, mitigating the formation of low-pressure zones and turbulent vortices in the draft tube and thereby reducing pressure fluctuations. With Method 3, the acceleration-controlled closed-loop logic maintains a stable rotational speed trajectory, preventing abrupt changes in the relative velocity between the runner blades and the fluid. Although Method 4 achieves a smooth frequency transition via open- to closed-loop switching, the extended duration of large opening angles sustains high-velocity flow in the penstock, resulting in prolonged hydraulic disturbances. Therefore, the reduced pressure pulsation in Method 2.1 is fundamentally due to the optimized balance between the flow inertia and the mechanical inertia of the unit facilitated by the small acceleration strategy.

3.4 Verification of optimal start-up method

The entropy weight method is an objective weighting method. It primarily utilizes objective data collected from the real environment to obtain the original information of the objects to be evaluated, analyzing the intrinsic relationships between various evaluation indicators. It can effectively avoid the shortcomings of subjective weighting methods that rely on human subjective factors.

3.4.1 Basic process

The calculation steps for determining the weights of different indicators under various startup methods using the entropy weight method are as follows.

• Data Standardization. Based on the experimental data obtained, construct the evaluation indicator matrix X = x i j m × n, where each row represents the values of different indicators corresponding to the same startup method, and each column represents the values of a certain indicator under different startup methods. The evaluation indicator matrix is shown in Equation 14.

X = x 11 x 12 … x 1 n x 21 x 22 … x 2 n ⋮ ⋮ ⋮ ⋮ x m 1 x m 2 … x m n . (14)

To eliminate the influence of different dimensions and trends of evaluation indicators on the final analysis results, standardize the elements in the evaluation indicator matrix to obtain the standardized evaluation indicator matrix. Evaluation indicators can be divided into positive indicators (where larger values are better) and negative indicators (where smaller values are better). For positive indicators, the standardization formula is shown in Equation 15; for negative indicators, the standardization formula is shown in Equation 16.

After standardization, the original experimental data becomes benefit-type indicators within the interval [0,1], as shown in Equation 17, denoted as Y = y i j m × n.

y i j = x i j − min x j max x j − min x j , (15)

y i j = max x j − x i j max x j − min x j , (16)

Y = y 11 y 12 … y 1 n y 21 y 22 … y 2 n ⋮ ⋮ ⋮ ⋮ y m 1 y m 2 … y m n , (17)

where y i j is the standardized value of x i j; max x j and min x j are the maximum and minimum values of the jth indicator, respectively.

Calculate the information entropy e _*j* of indicator j. After standardizing the original data, use the standardized data to calculate the probability p _*ij* of y _*ij* appearing, as shown in Equation 18. Then calculate the information entropy e _*j* of indicator j, as shown in Equation 19.

P i j = y i j ∑ j = 1 n y i j , (18)

e j = − 1 ln ⁡ m ∑ j = 1 n p i j · ⁡ ln ⁡ p i j , (19)

where m is the number of startup methods.

Calculate indicator weights. The calculation formula for the weight of each indicator is shown in Equation 20:

w j = 1 − e j ∑ j = 1 n 1 − e j . (20)

Calculate the comprehensive score of different schemes, as shown in Equation 21:

S = ∑ j = 1 n w j · y i j . (21)

3.4.2 Evaluation results

The evaluation results are shown in Table 3. The entropy weight method was used to determine the weights of various parameters in the stability test, and the scores of different startup schemes were evaluated overall, along with ranking. Method 2.1 is the optimal solution, indicating that the guide-vane small-acceleration open-loop startup scheme is the most friendly for unit startup.

TABLE 3. Entropy weight method evaluation results.

The distinct variations in pressure pulsation and water hammer characteristics among these four methods can be attributed to the coupling between guide vane kinematics and hydrodynamic inertia.

In Method 1, the large acceleration rate (approximately 2.8%/s) causes a drastic and rapid increase in discharge. This abrupt change in flow velocity excites severe water hammer waves and induces intense flow separation in the draft tube, resulting in the highest peak-to-peak pressure values observed among all methods.

Method 2.1 significantly mitigates these effects by limiting the initial acceleration to approximately 0.7%/s. This gradual opening allows the water column in the long penstock to adjust progressively, reducing the inertial pressure head. Consequently, the flow regime remains more stable, effectively suppressing vortex shedding in the draft tube and lowering pressure fluctuations compared to Method 1.

In Method 3 (acceleration-controlled closed-loop start-up), the logic prioritizes maintaining a preset rotational acceleration (e.g., 2 Hz/s initially). By actively regulating the guide vane to stabilize the unit’s acceleration, the relative velocity between the runner blades and the fluid remains consistent. This closed-loop regulation prevents abrupt torque changes, thereby reducing the amplitude of pressure oscillations in the spiral casing. However, the strict adherence to acceleration setpoints extends the start-up duration.

In Method 4 (open-loop + closed-loop) utilizes a hybrid strategy. The open-loop phase rapidly positions the guide vanes to a predetermined starting point, while the subsequent closed-loop PID control ensures a smooth frequency capture. This approach effectively restrains the overshoot of rotational speed and minimizes the impact of water hammer in the upstream pipelines. Nevertheless, the transition between open and closed-loop states and the relatively slow frequency ramp rate (0.4 Hz/s) result in a significantly prolonged start-up time of nearly 1.8 times that of Method 1.

Based on preliminary calculations of the four start-up methods, Method 2.1 (i.e., guide-vane small-acceleration open-loop start-up) demonstrates optimal performance in terms of adaptability to start-up duration and unit service stability. It was preliminarily identified as the optimal method. To verify this, modifications were made to Method 2.1 by adjusting different guide vane movement rates, resulting in Methods 2.2 and 2.3.

3.4.3 Method 2.2 (guide-vane small-acceleration open-loop start-up)

The start-up opening Yjs1 is approximately 60% of the no-load opening Y₀ (with the guide vane control output increase rate v limited to approximately 1.4%/s). After waiting approximately 8 s, it switches to the second start-up opening Yjs2, which is the no-load opening Y₀ plus 3% (with the guide vane control output increase rate v limited to approximately 0.9%/s). When the unit speed reaches approximately 90% of the rated speed, it is reduced to the third start-up opening Yjs3∼, which is 2% larger than the no-load opening (with the guide vane control output decrease rate v limited to approximately 0.4%/s). When the unit speed reaches approximately 95% of the rated speed, the governor engages the frequency regulation PID control.

3.4.4 Method 2.3 (guide-vane small-acceleration open-loop start-up)

Method 2.3 (guide-vane small-acceleration open-loop start-up): the start-up opening Yjs1 is approximately 60% of the no-load opening Y₀ (with the guide vane control output increase rate v limited to approximately 1.6%/s). After waiting approximately 5 s, it switches to the second start-up opening Yjs2, which is the no-load opening Y₀ plus 3% (with the guide vane control output increase rate v limited to approximately 0.6%/s). When the unit speed reaches approximately 90% of the rated speed, it is reduced to the third start-up opening Yjs3, which is 2% larger than the no-load opening (with the guide vane control output decrease rate v limited to approximately 0.4%/s). When the unit speed reaches approximately 95% of the rated speed, the governor engages the frequency regulation PID control.

In Methods 2.1, 2.2, and 2.3, a dwell time of approximately 6 s, 8 s, and 5 s, respectively, is introduced after the first-stage guide vane opening. This dwell time serves three physical purposes. First, after the guide vanes are initially opened to a small position, water hammer pressure waves are excited within the water conveyance system. The waiting period allows these pressure waves to complete a propagation–reflection cycle through the long penstock, enabling the pressures at the spiral casing and draft tube to approach relative stability, thereby preventing further pressure peak escalation due to wave superposition. Second, under high-head conditions, the substantial inertia of the water column requires a finite response time for flow acceleration. During the dwell period, the flow rate gradually builds up and the velocity circulation at the runner inlet progressively matches the rotating runner, establishing stable hydraulic initial conditions for the subsequent stage of guide vane opening and avoiding severe pressure pulsations induced by abrupt flow rate changes. Third, the specific dwell time values (5–8 s) are determined based on the theoretical wave period estimated from the pipeline length and water hammer wave speed (approximately 1000–1200 m/s) and are further optimized through multiple computational trials to achieve a balance between start-up speed and pressure control.

3.4.5 Verification results

Table 4 lists the transition process characteristic values for Methods 2.2 and 2.3. The following are apparent.

• When using Method 2.2 for start-up, if Units 3 and 4 start simultaneously, the start-up duration shortens to 44.50 s. The peak-to-peak pressures at the spiral case inlet and end decrease by 10.2% and 8.2%, respectively, compared to Method 2.1; however, the peak-to-peak pressures at the draft tube inlet and outlet increase by 24.6% and 26.3%, respectively, compared to Method 2.1. If only Unit 4 starts, the peak-to-peak pressure at the spiral case inlet decreases by 1.6% compared to Method 2.1, but the peak-to-peak pressures at the spiral case end, draft tube inlet, and outlet increase by 2.9%, 13.4%, and 27.6%, respectively, compared to Method 2.1.
• When using Method 2.3 for start-up, if Units 3 and 4 start simultaneously, the start-up duration shortens to 43.59 s. The peak-to-peak pressures at the spiral case inlet and end decrease by 7.5% and 4.9%, respectively, compared to Method 2.1; however, the peak-to-peak pressures at the draft tube inlet and outlet increase by 23.1% and 24.0%, respectively, compared to Method 2.1. If only Unit 4 starts, the peak-to-peak pressures at the spiral case inlet and end, and the draft tube inlet and outlet, increase by 1.7%, 2.7%, 10.6%, and 23.8%, respectively, compared to Method 2.1.

TABLE 4. Characteristic values of hydraulic transients under different guide-vane small-acceleration start-up methods.

Table 5 presents the entropy-based evaluation results for Methods 2.1, 2.2, and 2.3, indicating that Method 2.1 is superior.

TABLE 5. Entropy weight method evaluation results.

In summary, under the condition of guide-vane small-acceleration open-loop start-up, different methods have their own advantages and disadvantages. However, overall, Methods 2.1, 2.2, and 2.3 are still superior to Methods 1, 3, and 4.

4 Conclusion

Taking a Class I Large (1) hydraulic project as the case study—specifically its 350 MW mixed-flow reversible pump-turbine generating unit—this study conducted one-dimensional numerical simulations to calculate the hydraulic transient processes for two scenarios: simultaneous start-up of Units 3 and 4 within the same hydraulic unit, and start-up of Unit 4 alone. For the power generation start-up of Unit 4, the guide vane small-acceleration open-loop start-up scheme effectively balances start-up duration with operational stability, making it a viable option for the power plant. However, key parameters significantly influence the start-up transient, including the initial opening for acceleration, no-load opening limit, rate of change of guide-vane control output, frequency setpoint ramp profile, and dwell time at each intermediate opening. Based on entropy weight method evaluation, Method 2.1 outperformed Methods 2.2 and 2.3. Therefore, Method 2.1 is recommended as the preferred start-up strategy, pending further validation through prototype testing.

A qualitative cost-benefit analysis demonstrates that Method 2.1 delivers substantial net benefits. The incremental costs are limited to a modest ∼12 s extension in start-up time (negligible in routine dispatch) and a one-time governor software modification. In return, the 7.2–25.3% reduction in pressure pulsation amplitude effectively mitigates runner cracking and penstock fatigue, extends overhaul intervals, reduces unplanned outages, and enhances grid frequency regulation capability. The approach offers considerable long-term operational and grid reliability gains at minimal cost, confirming its strong engineering feasibility.

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Footnotes

Edited by: Zhigang Zuo, Tsinghua University, China

Reviewed by: Zhiyi Yu, Beijing Institute of Technology, China

Wangxu Li, Lanzhou University of Technology, China