Hydraulic Instability Characteristics of Pumped-Storage Units During the Transition from Hot Standby to Power Generation

Abstract

Against the backdrop of the carbon peaking and neutrality (“dual-carbon”) goals and evolving new-type power system dispatch, the share of pumped-storage hydropower (PSH) in power systems continues to increase, imposing stricter requirements on units for higher cycling frequency, greater operational flexibility, and rapid, stable startup and shutdown. Focusing on the entire hot-standby-to-generation transition of a PSH plant, a full-flow-path three-dimensional transient numerical model encompassing kilometer-scale headrace/tailrace systems, meter-scale runner and casing passages, and millimeter-scale inter-component clearances is developed. Three-dimensional unsteady computational fluid dynamics are determined, while the surge tank free surface and gaseous phase are captured using a volume-of-fluid (VOF) two-phase formula. Grid independence is demonstrated, and time-resolved validation is performed against the experimental model–test operating data. Internal instability structures are diagnosed via pressure fluctuation spectral analysis and characteristic mode identification, complemented by entropy production analysis to quantify dissipative losses. The results indicate that hydraulic instabilities concentrate in the acceleration phase at small guide vane openings, where misalignment between inflow incidence and blade setting induces separation and vortical structures. Concurrently, an intensified adverse pressure gradient in the draft tube generates an axial recirculation core and a vortex rope, driving upstream propagation of low-frequency pressure pulsations. These findings deepen our mechanistic understanding of hydraulic transients during the hot-standby-to-generation transition of PSH units and provide a theoretical basis for improving transitional stability and optimizing control strategies.

1. Introduction

As a mature, large-scale mechanical energy storage technology, pumped-storage hydropower (PSH) has been widely deployed under high renewable-penetration conditions, and its share of installed capacity is continuously increasing [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. Conventional PSH plants were originally designed around a daily “one-pump–one-generation” cycle; however, escalating grid requirements for flexibility, frequent start–stop capabilities, rapid responses, and operational stability now compel PSH to undergo multi-cycle operations, switch operating modes rapidly, and participate in frequency regulation and reserve services [4,5,6,7]. Consequently, legacy stations struggle to satisfy the dispatch requirements of new-type power systems. Accordingly, this study considers the complete transition from shutdown hot-standby to generation mode in a PSH unit and performs full-flow-path three-dimensional transient simulations to elucidate hydraulic-instability mechanisms and provide a verifiable theoretical basis for plant optimization and retrofit.

Research on the pump–turbine transition from shutdown hot standby to generation has advanced from multiple perspectives. Lu et al. [8] conducted studies on the Computational Fluid Dynamics (CFD) of turbine startup under rapid guide vane opening (GVO) and, through analyses of hydraulic loss and pressure pulsation, elucidated their link to internal flow instability. Using continuous wavelet transform and variational mode decomposition, Jin et al. [9] characterized the time–frequency behavior of pressure fluctuations and applied entropy generation analysis to visualize the internal flow field. Under full-load conditions, Zhang D. et al. [10] employed CFD to examine pressure pulsation characteristics and to compute dynamic stresses induced by transient flow in a pump–turbine. Bantelay et al. [11] combined numerical simulation and experiments to investigate pump–turbine flow behavior and startup characteristics, demonstrating the role of vortical structures in instability during rapid switching. Guo et al. [12] employed user-defined functions and a dynamic mesh to simulate the startup-to-no-load evolution and, leveraging pulsation and vortex dynamics, identified contributors to runner instability under no-load conditions. To quantify flow-induced stresses on upper-cover bolts during startup, Wang Z. et al. [13] developed a CFD-to-structure map, pinpointing the locations and magnitudes of peak stresses in critical fasteners. Addressing low-head startup, Wang T. et al. [14] performed three-dimensional full-passage simulations, dissected the instability mechanisms, and proposed control/optimization strategies to improve synchronization success. Yin et al. [15] developed a fluid–structure interaction model for prototype high-head units during turbine startup, quantified pressure evolution and stress concentrations, and identified hot spots and safety margins on blades and stationary components. Jin et al. [16] proposed and evaluated a misaligned guide vane startup strategy, demonstrating improved startup stability at the cost of increased energy dissipation and thereby highlighting a stability–efficiency trade-off. Zhang M. et al. [17] developed a variable-step Euler algorithm for PSH hydraulic transients, used a staggered peak-shaving/valley-filling approach to set intervals for dual-unit startup and load regulation, and optimized the guide vane closure law during load-rejection transients. Although prior studies have analyzed hydraulic instability, energy dissipation, and structural stress during the shutdown-to-generation transition, many adopt simplified models that omit the upstream/downstream waterways and surge tank effects and do not explicitly identify the most instability-prone periods and locations; these limitations hinder quantitative improvements in grid synchronization reliability and stability margins.

To address these limitations, we perform high-fidelity numerical simulations of the hot-standby-to-generation transition in pumped-storage units. The modeled domain spans the full flow path—kilometer-scale headrace/tailrace conduits, meter-scale runner and casing passages, and millimeter-scale inter-component clearances—with the surge tank represented using a gas–liquid two-phase model. By analyzing macroscopic performance metrics, the evolution of internal flow structures, and pressure pulsation characteristics, we elucidate the mechanisms of hydraulic-instability formation and energy dissipation, thereby providing a theoretical basis for unit design optimization and operational control.

The remainder of this paper is organized as follows: Section 2 introduces the governing equations, the volume-of-fluid (VOF) formula, and the entropy production theory. Section 3 describes the geometry, meshing strategy, and numerical implementation; establishes grid independence; validates the approach against model–test data; and outlines the transition control strategy and initial-condition setup. Section 4 systematically analyzes macroscopic performance, internal flow field structures, and pressure pulsation characteristics during the hot-standby-to-generation transition, elucidating the underlying instability mechanisms. Section 5 summarizes the work and presents the principal conclusions.

2. Basic Equations

2.1. Governing Equations

The pump–turbine startup is simulated using CFD, solving the Navier–Stokes and continuity equations as the governing equations [18,19,20]. Owing to the near-incompressibility of water and negligible temperature variations, density is treated as constant and the energy equation is omitted; accordingly, the governing system reduces to the continuity and momentum equations.

The continuity equation reads

$${\displaystyle \frac{\partial u_i}{\partial x_i}}=0$$

(1)

The momentum equation reads

$${\displaystyle \frac{\partial u_i}{\partial t}}+{\displaystyle \frac{\partial u_i{u}_j}{\partial x_j}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\mu }{\rho }}{\displaystyle \frac{{\partial }^2{u}_i}{\partial x_j\partial x_j}}-\mathrm{g}{\delta }_{i3}$$

(2)

where $u$ is the fluid velocity (m·s⁻¹), $\rho$ the fluid density (kg·m⁻³), $p$ the static pressure (Pa), $\mu$ the dynamic viscosity (kg·(m·s)⁻¹), t the time (s), $\mathrm{g}$ the gravitational acceleration vector (N·kg⁻¹), and ${\delta }_{i3}$ the Kronecker delta tensor.

2.2. Shear Stress Transport (SST) k-ω Turbulence

We employ the SST k-ω turbulence closure for Reynolds-Averaged Navier–Stokes (RANS), which improves the prediction of near-wall behavior and adverse pressure gradient flows.

The turbulent kinetic energy (TKE) $k$ equation is:

$${\displaystyle \frac{\partial \left(\rho k\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho {\overline{u}}_{j}k\right)}{\partial x_j}}=P_k-{\beta }^{\ast }\rho \omega k+{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\sigma }_k{\mu }_t\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]$$

(3)

where $k$ is the TKE (m²·s⁻²); $P_k$ the prototype of the TKE generation term (kg·(m·s²)⁻¹); ${\beta }^{\ast }$ the constant associated with TKE dissipation, with a standard value of 0.09; $\omega$ the turbulent dissipation (s⁻¹); and ${\sigma }_k$ the turbulence Prandtl number.

The dissipation rate $\omega$ equation is:

$${\displaystyle \frac{\partial \left(\rho \omega \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho {\overline{u}}_j\omega \right)}{\partial x_j}}=\alpha {\displaystyle \frac{\omega }{k}}{P}_k-\beta \rho {\omega }^2+{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\sigma }_{\omega }{\mu }_t\right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]+\left(1-F_1\right)2\rho {\sigma }_{\omega 2}{\displaystyle \frac{1}{\omega }}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}}$$

(4)

where $\alpha$ and $\beta$ are the model coefficients; $F_1$ the first and most important hybrid function; and ${\sigma }_{\omega 2}$ the diffusion coefficient, with a standard value of 0.856.

The turbulent viscosity coefficient ${\mu }_t$ and mixing function is:

$${\mu }_t={\displaystyle \frac{\rho a_{1}k}{\max \left(a_1\omega ,S{F}_2\right)}}$$

(5)

where $a_1$ is the constant limiting turbulent shear stress, with a standard value of 0.31; $S$ the modulus of the average strain rate tensor; and $F_2$ the second hybrid function.

2.3. VOF Formula

In the present computational model, the surge-chamber free surface is treated as a gas–liquid interface; a VOF two-phase approach is adopted to capture free-surface fluctuations. Within the VOF method, the free interface is tracked by solving a transported phase volume-fraction (indicator) equation during gas–liquid two-phase simulations [21,22,23,24]:

$$F={\displaystyle \frac{\mathrm{liquid} \mathrm{volume} \mathrm{within} \mathrm{the} \mathrm{control} \mathrm{volume}}{\mathrm{volume} \mathrm{of} \mathrm{the} \mathrm{control} \mathrm{volume}}}$$

(6)

When $F=1$, the control volume is completely filled with liquid;

When $F=0$, the control volume contains only gas;

When $0<F<1$, the control volume contains a gas–liquid interface.

In gas–liquid two-phase flow, the volume fraction (indicator) function satisfies the following transport equation:

$${\displaystyle \frac{\partial F}{\partial t}}+u·\nabla F=0$$

(7)

The foregoing equation constitutes the VOF formula for free-surface tracking; the volume fraction (indicator) field is a step function bounded in [0, 1].

At the interface, the volume fraction function $F$ of phase $i$ is governed by the continuity (transport) equation; the following relation holds:

$${\displaystyle \frac{\partial F_i}{\partial t}}+{\overrightarrow{v}}_i\cdot \nabla F_i={\displaystyle \frac{S_{F_i}}{{\rho }_i}}+{\displaystyle \frac{1}{{\rho }_i}}{\displaystyle \sum _{j=1}^n\left({\dot{m}}_{ji}-{\dot{m}}_{ij}\right)}$$

(8)

where $F_i$ is the volume fraction of phase $i$; ${\overrightarrow{v}}_i$ its velocity vector; $S_{F_i}$ the source term (taken as zero unless otherwise specified); ${\dot{m}}_{ji}$ the mass-transfer rate from phase $j$ to $i$; and ${\dot{m}}_{ij}$ the mass-transfer rate from phase $i$ to $j$.

2.4. Entropy Production Theory

For RANS flows, the total Entropy Production Rate (EPR) comprises a direct component associated with the mean flow and an indirect component arising from velocity fluctuations [25,26].

$${\dot{S}}_D^{‴}={\dot{S}}_{\overline{D}}^{‴}+{\dot{S}}_{D^{\prime }}^{‴}$$

(9)

$${\dot{S}}_{\overline{D}}^{‴}={\displaystyle \frac{2{\mu }_{eff}}{T}}\left[{\left({\displaystyle \frac{\partial {\overline{u}}_1}{\partial x_1}}\right)}^2+{\left({\displaystyle \frac{\partial {\overline{u}}_2}{\partial x_2}}\right)}^2+{\left({\displaystyle \frac{\partial {\overline{u}}_3}{\partial x_3}}\right)}^2\right]+{\displaystyle \frac{{\mu }_{eff}}{T}}\left[{\left({\displaystyle \frac{\partial {\overline{u}}_2}{\partial x_1}}+{\displaystyle \frac{\partial {\overline{u}}_1}{\partial x_2}}\right)}^2+{\left({\displaystyle \frac{\partial {\overline{u}}_3}{\partial x_1}}+{\displaystyle \frac{\partial {\overline{u}}_1}{\partial x_3}}\right)}^2+{\left({\displaystyle \frac{\partial {\overline{u}}_2}{\partial x_3}}+{\displaystyle \frac{\partial {\overline{u}}_3}{\partial x_2}}\right)}^2\right]$$

(10)

$${\dot{S}}_{D^{\prime }}^{‴}={\displaystyle \frac{2{\mu }_{eff}}{T}}\left[{\left({\displaystyle \frac{\partial u_1^{\prime }}{\partial x_1}}\right)}^2+{\left({\displaystyle \frac{\partial u_2^{\prime }}{\partial x_2}}\right)}^2+{\left({\displaystyle \frac{\partial u_3^{\prime }}{\partial x_3}}\right)}^2\right]+{\displaystyle \frac{{\mu }_{eff}}{T}}\left[{\left({\displaystyle \frac{\partial u_2^{\prime }}{\partial x_1}}+{\displaystyle \frac{\partial u_1^{\prime }}{\partial x_2}}\right)}^2+{\left({\displaystyle \frac{\partial u_3^{\prime }}{\partial x_1}}+{\displaystyle \frac{\partial u_1^{\prime }}{\partial x_3}}\right)}^2+{\left({\displaystyle \frac{\partial u_2^{\prime }}{\partial x_3}}+{\displaystyle \frac{\partial u_3^{\prime }}{\partial x_2}}\right)}^2\right]$$

(11)

where ${\dot{S}}_D^{‴}$, ${\dot{S}}_{\overline{D}}^{‴}$, and ${\dot{S}}_{D^{\prime }}^{‴}$ denote the total, direct, and indirect EPR (W·m⁻³·K⁻¹); ${\overline{u}}_1$, ${\overline{u}}_2$, and ${\overline{u}}_3$ are the mean velocity components (m·s⁻¹); $u_1^{\prime }$, $u_2^{\prime }$, and $u_3^{\prime }$ are the fluctuating velocity components (m·s⁻¹); $T$ is the absolute temperature (K); and ${\mu }_{eff}$ is the effective viscosity (Pa·s).

Since RANS does not provide the fluctuating velocity components explicitly, Kock et al. [27] and Mathieu et al. [28] derived indirect entropy production estimators expressed in terms of the turbulence model variables ε or ω. For the SST k-ω model, the indirect EPR is given by

$${\dot{S}}_{D^{\prime }}^{‴}=\beta {\displaystyle \frac{\rho \omega k}{T}}$$

(12)

where $\beta$ is an empirical constant (≈0.09); $k$ is the TKE (m²·s⁻²); and $\omega$ is the specific dissipation rate (s⁻¹).

In addition, the large velocity gradients in wall-bounded regions induce strong wall effects and associated entropy production. To evaluate near-wall entropy production, Duan et al. proposed a broadly applicable, accurate wall-function formula:

$${\dot{S}}_W^{″}={\displaystyle \frac{\overrightarrow{\tau }\cdot \overrightarrow{v}}{T}}$$

(13)

where ${\dot{S}}_W^{″}$ is the wall EPR due to shear (W·m⁻²·K⁻¹); $\overrightarrow{\tau }$ is the wall shear stress (Pa); and $\overrightarrow{v}$ is the velocity at the center of the first near-wall grid cell (m·s⁻¹).

The total entropy production (TEP) is obtained by integrating ${\dot{S}}_{\overline{D}}^{‴}$ and ${\dot{S}}_{D^{\prime }}^{‴}$ over the flow volume and ${\dot{S}}_W^{″}$ over the wall surfaces and then summing the contributions [29,30].

$$S_{pro,\overline{D}}={\displaystyle \underset{V}{\int }{\dot{S}}_{\overline{D}}^{‴}dV}$$

(14)

$$S_{pro,D^{\prime }}={\displaystyle \underset{V}{\int }{\dot{S}}_{D^{\prime }}^{‴}dV}$$

(15)

$$S_{pro,W}={\displaystyle \underset{A}{\int }\frac{{\overrightarrow{\tau }}_w\cdot {\overrightarrow{v}}_w}{T}}dA$$

(16)

$$S_{pro}=S_{pro,\overline{D}}+S_{pro,D^{\prime }}+S_{pro,W}$$

(17)

where $S_{pro}$, $S_{pro,\overline{D}}$, $S_{pro,D^{\prime }}$, and $S_{pro,W}$ denote TEP, the EPR caused by direct dissipation (EPDD), EPR caused by turbulence dissipation (EPTD), and EPR caused by wall shear stress (EPWS), with units W·K⁻¹.

$$h_{\mathrm{e}p}={\displaystyle \frac{T\cdot S_{pro}}{\dot{m}g}}$$

(18)

Energy dissipation is expressed in terms of hydraulic loss:

• where $\dot{m}$ is the mass flow rate (kg·s⁻¹) and $S_{pro}$ is the entropy production term (W·K⁻¹).

3. Numerical Algorithm for the 3D Transient Process

3.1. Geometric Model

This study considers the full flow-passage system of a PSH unit. The complete prototype domain comprises the upstream and downstream conduits and the machine section, which includes the spherical valve, spiral casing, vanes, runner, draft tube, upper-crown clearance, lower-ring clearance, and the equalizing pipe. Figure 1 shows a schematic of the unit’s full flow-passage model, and

Table 1. Main parameters of the pump–turbine unit.

Parameter	Unit	Value
Runner outlet diameter	D1 (m)	5.259
Runner outlet diameter	D2 (m)	3.57
Rated rotational speed	nr (rpm)	250
Rated head	Hr (m)	195
Rated discharge	Qr (m3·s−1)	176.1
Rated GVO	θ (°)	37.4
Number of runner blades	Zg	9
Number of guide vanes	Zg	20
Number of stay vanes	Zs	20

summarizes the principal parameters of the pump–turbine.

3.2. Mesh Generation and Grid Independence Verification

The model is meshed in ICEM. Grid independence is assessed using Richardson extrapolation, and the grid convergence index (GCI) is employed to quantify discretization uncertainty; the governing expressions are as follows [31,32,33]:

$$GC{I}^{21}={\displaystyle \frac{F_s\cdot e_a^{21}}{r_{21}^p-1}}$$

(19)

$$e_a^{21}=\left|{\displaystyle \frac{{\varphi }_1-{\varphi }_2}{{\varphi }_1}}\right|$$

(20)

$$r_{21}=\sqrt[3]{{\displaystyle \frac{N_1}{N_2}}}$$

(21)

$$p={\displaystyle \frac{1}{\ln (r_{21})}}\left|\ln \left|{\displaystyle \frac{{\epsilon }_{32}}{{\epsilon }_{21}}}\right|+q(p)\right|$$

(22)

$$q(p)=\ln \left({\displaystyle \frac{r_{21}^p-s}{r_{32}^p-s}}\right)$$

(23)

$$s=1\cdot \mathrm{sgn}\left({\displaystyle \frac{{\epsilon }_{32}}{{\epsilon }_{21}}}\right)$$

(24)

where $F_s$ is the grid safety factor (taken as 1.25); $r$ the grid refinement ratio; $p$ the observed order of convergence; $e_a$ the relative error between the numerical solutions ${\varphi }_1$ and ${\varphi }_2$ on two grids; and ${\epsilon }_{21}$ and ${\epsilon }_{32}$ the solution differences between successive grid levels.

Three meshes generated with the same strategy were employed, with the cell count and characteristic length scale progressively reduced. Following ASME guidance that the refinement ratio r exceed 1.3, the three meshes contained 36.20, 15.74, and 6.34 million cells. Head and efficiency were used as convergence metrics, and all simulations were conducted at the rated operating condition.

As shown in

Table 2. Grid independence verification.

Parameter	φ = H (m)	φ = η (%)
number of cells N1	36,202,635
number of cells N2	15,740,276
number of cells N3	6,743,598
grid refinement ratio r21	1.3200
grid refinement ratio r32	1.3265
computed value φ1	198.0891	94.4047
computed value φ2	197.8946	94.4360
computed value φ3	189.0851	97.9472
Richardson extrapolated value φext21	198.0938	94.4044
approximate relative error ea21	0.0982%	0.0332%
extrapolation error eext21	0.0024%	0.0003%
grid convergence index on the fine pair GCIfine21	0.2969%	0.0405%

, the GCI for head and efficiency are 0.2969% and 0.0405%, respectively, satisfying the acceptance criterion (<3%). Considering both accuracy and computational cost, the mesh with 15.74 million cells was adopted; the mesh model is shown in Figure 2.

3.3. Numerical Method and Boundary Conditions

Three-dimensional simulations of the transition process were carried out in the commercial solver ANSYS Fluent 2022 R1, solving the RANS equations using a finite-volume discretization. Fluid dynamics were modeled with a VOF formula, and the SST k-ω turbulence closure was used to simulate pump–turbine startup while explicitly accounting for surge tank effects. The VOF formula effectively captures liquid–gas interface motion and is suitable for the surge tank’s complex flow. The initial condition for startup specified a closed spherical-valve opening (SVO), a near-zero GVO, and zero rotational speed. Water levels were prescribed as 307.472 m upstream and 97.394 m downstream. Following an all-liquid (single-phase) surge tank run, the VOF simulation was initialized with a surge tank level of 307.472 m; a convergence tolerance of 10⁻⁵ was enforced, yielding the initial field shown in Figure 3. The steady solution was used to initialize the transient computation, with a time step of 0.002 s (equivalent to a 3° rotation of the runner at the rated speed). Moreover, gravity was prescribed as g = 9.8 m·s⁻².

3.4. Control Strategies and Algorithm Implementation

In the simulation of the hot-standby-to-generation transition, actuation timing is constrained such that the spherical valve governs inlet discharge and the guide vanes regulate rotor speed; dynamic balance conditions include speed control, excitation build-up, and load ramping. Based on plant measurements, the simulation control sequence is as follows: The entire transition adopts open-loop control. Initially, the unit is stopped, the spherical valve is closed, the GVO is near zero, and the flow field is approximately hydrostatic. At t = 0 s, a transition command is issued; the SVOs open linearly, and water enters the unit. At t = 56 s, the spherical valve is fully open. Over t = 56–60 s, the mechanical brake is released to enable rotor acceleration. From t = 60 to 184 s, the unit operates at no load, allowing for speed and excitation build-up. At t = 60 s, the guide vanes begin to experience staged, progressive opening. The runner begins to rotate under hydraulic torque. At t = 105 s, the GVO reaches the no-load setting of 11.05°. At t = 150 s, the runner reaches the rated speed. At t = 184 s, the unit synchronizes to the grid; speed locks to the synchronous value, the GVO continues to increase to 22.28°, and active power rises from 0 to 200 MW. Thus, the hot-standby-to-generation transition is complete. The operating sequence of the guide vanes and spherical valve are shown in Figure 4.

We introduce the angular momentum (torque) balance and exploit the time-marching nature of the unsteady solver at discrete time steps to sequentially update the runner speed at each time level [34,35,36]. At the start of the transition, the unit is at rest with zero rotational speed. The torque balance equation thus reads

$$M_0-M_1-M_2-M_3=J{\displaystyle \frac{d\omega }{dt}}={\displaystyle \frac{\pi J}{30}}{\displaystyle \frac{dn}{dt}}$$

(25)

where $M_0$ is the motor electromagnetic torque; $M_1$ the hydrodynamic resisting torque on the runner (read in real time via a user-defined function, UDF); $M_2$ the bearing-friction torque; $M_3$ the motor windage torque (small and neglected here); $J$ the unit’s rotational moment of inertia; $\omega$ the runner angular velocity; and $n$ the runner rotational speed.

4. Results and Discussion

4.1. Validation of Model Accuracy

For both the pumping and generating modes, two distinct GVOs were selected for validation; the experimental–numerical comparisons are shown in Figure 5. Under pumping operation, simulated head and efficiency deviate by less than 2%, meeting the acceptance criterion (≤5%). In the generating mode, the relative errors for flow rate and efficiency are likewise below 2%, satisfying the ≤5% accuracy requirement. Overall, the simulations exhibit good agreement with the measurements, confirming the reliability of the computed results.

4.2. External Characteristics During the Hot-Standby-to-Generation Transition

4.2.1. Evolution of External Characteristic Parameters

Figure 6 shows the evolution of rotational speed and discharge over the hot-standby-to-generation transition. As indicated, the transition decomposes into several stages: 0–56 s (valve-opening stage), when passage filling dominates, Q ≈ 0, n ≈ 0; 60–150 s (no-load acceleration), as the GVO ramps to 11.05°, Q responds promptly—first surging and then exhibiting step-like fluctuations about the no-load level—while n rises quasi-linearly to 250 r·min⁻¹ under the effective inertia; 105–150 s, with vane opening held constant and speed increasing, local separation and recirculation develop in the vaneless region, and concurrently, stronger outlet swirl and an enhanced adverse pressure gradient in the draft tube further restrict through-flow, yielding a modest decrease in Q; at 150 s, synchronous speed is reached and held, and control switches from guide vane/torque–speed regulation to a constant-speed lock; 150–184 s (steady no-load), no-load operation at the rated speed is steady (n = 250 r·min⁻¹), and Q = 59 m³·s⁻¹, showing minor oscillations; and 184–205 s (grid-connected ramp), when speed remains locked, the GVO increases to 22.28°, active power reaches 200 MW, and Q rises to 112 m³·s⁻¹—i.e., the required output is supplied by higher discharge.

4.2.2. Evolution of Rotational Speed and Discharge

Figure 7 illustrates how the runner torque and axial force vary throughout the hot-standby-to-generation transition. The figure shows a clear stage-wise correspondence between torque and axial force during startup. At t = 0 (valve opening), torque is nearly zero, while the axial force is dominated by hydrostatic pressure (F_z = 4.44 × 10⁵ N). After t = 60 s, progressive GVO drives the torque to negative values with oscillations; the axial force rises sharply—signatures of S-shaped characteristic region (S-region) hydraulic instability [37]. The extrema are M = −8.502 × 10⁶ N·m and F_z = −3.065 × 10⁶ N. As speed increases and the S-region is traversed, torque switches from negative to positive and becomes steadier; axial-force oscillations attenuate and settle into a narrower band, indicating progressive recovery of flow stability. During the grid-connected ramp (GVO ≈ 22°, increasing discharge), the runner torque reaches a stable power-producing level (M = −6.608 × 10⁶ N·m) after departing from near zero. F_z increases in magnitude concurrently and may exhibit a spike at synchronization, reaching 5.828 × 10⁶ N.

4.3. Internal-Flow Characteristics During the Hot-Standby-to-Generation Transition

For the internal-flow analysis, the following characteristic instants are selected: t = 60 s (onset of GVO), t = 105 s (GVO at the no-load setting of 11.05°), t = 150 s (rotational speed reaches the rated 250 r·min⁻¹), t = 184 s (grid synchronization), and t = 205 s (active power stabilized at 200 MW). At t = 60 s, guide vane actuation initiates runner rotation; at 105 s, flow modifications induced by vane opening are assessed; at t = 150 s, passage through the S-region is evaluated; and by t = 205 s, the system is in stable operation for assessing flow-field stability and energy dissipation. Taken together, these time points provide a comprehensive picture of the internal-flow features and their temporal evolution during startup.

During the hot-standby-to-generation transition, the internal flow undergoes pronounced variations; informed by prior studies, we focus on instability-sensitive zones—namely, the guide vane region, vaneless region, runner domain, upper-crown clearance, lower-ring clearance, and draft tube.

4.3.1. Flow-Field Analysis of the Guide Vanes, Inter-Component Clearances, and Runner Passages

From Figure 8 and Figure 9, at t = 60 s (startup onset), the guide vanes remain closed; high-speed recirculation develops in the vaneless region, and high-velocity jets impinge the runner passages. This arises because, once the spherical valve is fully open, substantial inflow enters the unit; even with the vanes “closed”, imperfect sealing in the modeled geometry permits gap leakage into the guide vane passages. The leakage proceeds into the runner as high-velocity jets. These jets yield localized peaks of TKE and TEP around the guide vane clearances and at the runner exit; the stage remains quasi-hydrostatically dominated. At t = 105 s, under a small GVO, a high-speed annular stream adheres to the vanes and primarily strikes the nose of the blade pressure side. A fraction diverts to neighboring channels, producing a local high-pressure pocket; the remainder convects downstream along the blade pressure side, forming a high-speed band. Concurrent tangency with counter-flow on the suction side induces large-scale vortices within the passage, where entropy production is predominantly concentrated. From 105 s to 184 s, as the speed approaches the rated value, a low-pressure zone develops at the runner outlet, while extensive large vortices appear at the inlet and between blades, intensifying shear. Consequently, the inter-blade TKE footprint expands, and TEP shifts toward the vaneless region, exhibiting a band-like distribution. The pairing of high rotational speed and small vane opening magnifies inflow-incidence error, which triggers the observed features. After t = 184 s (entering generating mode), additional vane opening and increased discharge shift the low-pressure pocket into the draft tube region. The flow re-smooths with vortex suppression; elevated TKE contracts to the blade pressure side at the runner inlet, coinciding with the primary loss region. Throughout the transition, abundant vortical structures persist upstream of the labyrinth rings at the upper-crown and lower-ring clearances, extracting jet kinetic energy and manifesting as edge-wall dissipation. By comparison, the lower-ring clearance is more prone to recirculation and vortex formation and thus exhibits poorer hydraulic stability.

4.3.2. Draft Tube Flow-Field Analysis

As shown in Figure 10, at t = 60 s (onset of GVO), the bulk velocity is very low; only weak recirculation appears near the inner wall of the straight-cone section and the elbow. TKE and TEP are nearly zero, indicating a quasi-stagnant, fully flooded state.

By t = 105 s, a continuous centerline low-pressure core develops in the straight-cone section together with an upstream-oriented recirculation core; a closed shear vortex forms between this core and the outer descending main stream. On the elbow inner wall, curvature and diffusive expansion impose an adverse pressure gradient, causing boundary layer separation and localized backflow, followed by downstream reattachment. TKE increases sharply within the shear layer and inner separation region, while TEP appears as banded streaks near the straight-cone inlet wall and around the elbow bend. At t = 150 s, the centerline low pressure intensifies and reaches the elbow inlet; the recirculation core persists, encased by a strong shear envelope. Separation strengthens on the straight-cone outer wall and elbow inner wall; TKE and TEP both grow in magnitude and extent, marking a high-risk window for instability and cavitation. After synchronization at t = 184 s, the discharge increases, the axial core weakens markedly, the straight-cone core flow strengthens and attaches to the elbow’s outer (convex) side, and shear-layer energy subsides. The TKE hot spot shifts from the inlet to the elbow bend; TEP decreases but remains as thin bands along the inner elbow and diffuser wall. By t = 205 s, the centerline low pressure has essentially vanished and no continuous recirculation core remains; the through-flow attaches stably to the outer bend. Only a thin near-wall high-shear layer persists; TKE is localized, and TEP is filamentary along the wall, indicating good pressure recovery and substantially reduced loss. Overall, draft tube instability is driven by an axial recirculation core coupled with swirl-induced shear, strongest at 105–150 s; after synchronization, swirl diminishes and separations contract, shifting the instability hotspot to the elbow’s near-wall shear band with markedly reduced severity.

4.4. Pressure Pulsations During the Hot-Standby-to-Generation Transition

To elucidate the origins of hydraulic instability during this transition, multiple pressure-monitoring points were arranged throughout the PSH unit. For the pressure pulsation analysis, monitoring points were deployed in the machine section, as shown in Figure 11.

Four monitoring points were installed in the volute region. SC01 is located near the upstream conduit and serves as the reference inlet-pressure tap; SC04 lies near the tongue; and SC02–SC04 are circumferentially spaced at 120°. Six monitoring points were set in the guide vane region, each centered in an inter-vane passage. SV01 is at the stay vane inlet near the tongue; SV03 is at the stay vane outlet; and SV02 is centered in the stay vane passage. GV01 is placed between the stay vane and guide vane rows; GV02 at mid-channel between guide vanes; and GV03 at the guide vane exit. In the vaneless region, eight points (VL01–VL08) were arranged circumferentially every 45°. Seven points were installed in the upper-crown clearance: HC01 near the vaneless region; HC02 and HC07 where the clearance area changes; HC03 and HC06 at the labyrinth ring inlet and outlet; and HC04 and HC05 where the labyrinth ring flow turns. Seven points were arranged in the lower-ring clearance: BR01 near the vaneless region; BR07 near the draft tube; and BR02–BR06 at sections with changes in area and/or flow direction. Twelve monitoring points were mounted near the draft tube wall. DT11–DT14 are at the draft tube inlet with 90° circumferential spacing; DT21–DT24 are in the straight-cone section with 90° spacing; and DT31–DT34 are in the elbow section, likewise spaced by 90°.

The measured pressures at the monitoring points were converted to a nondimensional pressure pulsation coefficient using the following relation [38,39]:

$$C_p={\displaystyle \frac{P-\overline{P}}{\rho g{H}_r}}$$

(26)

where $P$ is the instantaneous pressure (Pa), $\overline{P}$ the time-averaged pressure (Pa), and $H_r$ the rated head of the pump–turbine (m).

A Savitzky–Golay filter (second-order polynomial, 50-sample window) was applied to extract the running-mean pressure during startup; the pulsation pressure is defined as the deviation of the measured total pressure from this running mean. For frequency-domain analysis, the pulsation signals are processed via short-time Fourier transform (STFT) to obtain time-resolved power spectra over the startup process [40].

Pressure Pulsation Analysis at Monitoring Points

Figure 12(a1) shows the pressure pulsation characteristics at volute points SC01–SC04. The pressure traces exhibit similar trends and amplitudes across the points, with higher levels closer to the volute inlet. The volute measurements record a maximum pressure head of 364 m and a minimum of 97 m. Based on Figure 12(a2), SC02 is selected as the representative point for subsequent analysis. At SC02, pulsation amplitudes lie between −1.5 × 10⁻⁵ and 1.5 × 10⁻⁵, remaining small during the SVO stage. At ≈62 s, a pronounced peak (1.24 × 10⁻⁵) appears—attributable to the initial GVO (water hammer) and local incidence mismatch—after which the signal rapidly returns to low amplitude. Similar behavior is observed at other monitoring points across the unit. Figure 12(a3) shows the SC02 power spectrum, dominated by 9fn and 18fn (the blade-passing frequency, BPF, and its second harmonic), chiefly arising from runner–guide vane rotor–stator interaction (RSI) [41]. The near-vertical energy ridge around 60 s marks the transient impulse from vane opening; thereafter, low-frequency content reflects unsteadiness driven by vane motion.

Figure 12(b1) shows the pressure fluctuations at the stay vane probes SV01–SV04. The signals track the variation in unit head, with comparable amplitudes across the probes. Pressure is relatively higher near the volute inlet; the measured pressure head ranges from a maximum of 360 m to a minimum of 97 m. Figure 12(b2) takes SV02 as the representative point; its pressure pulsation amplitude spans −1.0 × 10⁻⁵ to 1.5 × 10⁻⁵, peaking at 1.39 × 10⁻⁵. Figure 12(b3) shows dominant components at 9fn (BPF) and 18fn (2 × BPF), chiefly due to runner–guide vane RSI. Because SV02 is closer to the runner, the dominant-frequency content is more pronounced there than at the volute locations.

Figure 12(c1) presents the pressure pulsation behavior measured at GV01–GV03 near the guide vanes. Across the three probes, the pressure head peaks at 360 m and bottoms out at 75 m. During early SVO, the flow path is established without pronounced swirl or periodic forcing; the mean pressure rises quickly with filling and stabilizes, while fluctuations remain small. Mean pressures follow GV01 > GV02 > GV03, evidencing higher static head upstream. Once the guide vanes begin to open, pulsations are most sensitive near the movable row—especially at GV02—primarily driven by runner–guide vane RSI. At GV03, the mean pressure increases markedly during the power ramp, whereas rectification effects keep its pulsation amplitudes below those at GV02. Figure 12(c2) selects GV02 as the representative point for detailed processing and analysis. At GV02, pulsation amplitudes span −1.0 × 10⁻⁵ to 1.0 × 10⁻⁵, peaking at 9.82 × 10⁻⁶. Figure 12(c3) shows a comparatively quiescent flow over 0–56 s. Around 60 s, initial vane motion induces a transient impulse. During 60–105 s (small-opening spin-up), the pulsation envelope attains its maximum; STFT reveals two dominant energy ridges that track increasing speed at BPF = 9fn and its second harmonic 18fn. Between 105 and 150 s, further acceleration to rated speed stabilizes the dominant spectral components. Post synchronization (≥184 s), overall pulsations weaken, accompanied by faint amplitude-modulation sidebands. The dominant mechanisms are runner–stator interference and high-frequency amplification due to passage jetting and incidence misalignment at small openings. The 20fn line is the third harmonic, attributed to nonlinear distortion of the BPF within the guide vane channel.

The vaneless region exhibits the most intense pressure pulsations and the most complex spectral content during the transition, making it especially susceptible to hydraulic instability. Figure 12(d1) presents pressure fluctuation curves for VL01–VL08 with broadly similar trends and magnitudes across points. The measured pressure head in this area ranges from a maximum of 205 m to a minimum of 83 m. Using VL02 for nondimensional processing, Figure 12(d2) shows that during 0–56 s (SVO), the coefficient C_p remains near zero, with only weak disturbances. At 60 s, vane initiation produces narrow time-domain spikes and a vertical energy stripe in the spectrum. During 60–105 s (small-opening acceleration), the pressure envelope peaks, and pulsation amplitudes span −2.11 × 10⁻⁵ to 1.57 × 10⁻⁵. Spectral ridges at the BPF and its second harmonic (2 × BPF) rise with speed, and faint higher-order harmonics appear, indicating nonlinear distortion of the BPF signal. Between 105 and 150 s, the oscillation amplitude contracts and the principal spectral components stabilize. Within 184–205 s (grid ramp), the variance decreases while mild amplitude modulation generates sidebands around the BPF. Mechanistically, the vaneless zone—between the guide vane exit and runner inlet—is primarily excited by runner–stator interaction. At small openings, guide vane jetting and outlet swirl induce incidence errors and strong circumferential nonuniformity, amplifying the BPF and its harmonics. As speed stabilizes and discharge increases, separation weakens and the strength of runner–stator coupling decreases. Following synchronization, low-frequency fluctuations from the power/electromagnetic torque control loop modulate the BPF component, producing spectral sidebands. Overall, pressure pulsations in the vaneless space are governed by the BPF and its multiples, strongest during the small-opening spin-up; once synchronized, flow rectifies, pulsations abate, and slight AM sidebands emerge.

Figure 12(e1) presents the pressure pulsation traces for DT11–DT31 in the draft tube. At a given axial station, probes exhibit similar fluctuation patterns and magnitudes. During 60–150 s (small-opening spin-up), the order of amplitudes is DT11 (inlet) > DT31 (elbow) > DT21 (straight cone). In the 150–184 s rated-speed/no-load interval, DT21 and DT31 remain the principal forced-response zones, dominated by a low-frequency vortex rope. Post synchronization, amplitudes decrease at all three locations, and flow-field instability is clearly mitigated. DT11 is chosen as the representative probe for subsequent analysis (see Figure 12(e2)). Its pulsation envelope spans −1.0 × 10⁻⁵ to 1.0 × 10⁻⁵, peaking at 6.99 × 10⁻⁶ near 62 s. Figure 12(e3) indicates dominant components at 9fn and 18fn (BPF and its 2 × BPF), attributable to runner–guide vane RSI. Low-frequency content after 60 s originates from a vortex rope extending from the draft tube entry into the straight-cone section.

Figure 12(f,g) depicts pressure responses at probes in the upper-crown clearance, lower-ring clearance, and equalizing pipe. Both gaps show broadly similar trends; pressure head increases with proximity to the runner. Pulsations downstream of the labyrinth rings are effectively attenuated. The nondimensional pressure coefficient C_p follows the unit-wide trend but with slightly smaller amplitudes. Spectra reveal no distinct dominant tone in the upper-crown gap, while BPF and its 2× component persist in the lower-ring gap. Overall, the clearance regions are characterized by broadband pulsations with weak BPF content, making strong resonance unlikely.

5. Conclusions

We numerically simulated the hot-standby-to-generation transition of a pumped-storage unit, analyzing macroscopic performance, internal-flow structures, and pressure pulsation behavior. This study pinpoints the temporal windows and spatial locations of hydraulic instabilities and explains their underlying mechanisms, thus informing stability enhancement and control-strategy optimization for the transition process.

1.

Instability window. Instability is most conspicuous from vane opening to attainment of the rated speed, featuring sharp increases in Q, M, and axial force, together with amplified radial-force oscillations. In particular, during acceleration at a small opening (11.05°), Q and M partly roll back, axial force peaks, and the high-frequency content of radial load intensifies.

2.

Internal-flow mechanisms and localization. The dominant instability arises during 105–150 s at small openings, localized near the guide vane leading edge/exit, in the vaneless region, and along the inner surfaces of the draft tube cone and elbow. Mechanistically, incidence mismatch, high-speed swirl in the vaneless space, and an adverse pressure gradient at the draft tube inlet trigger runner–inlet separation, strong shear in the vaneless region, and an axial recirculation core with a vortex rope in the draft tube.

3.

Pressure pulsation signatures. Time domain: staged vane opening (60–105 s) raises the pulsation envelopes concurrently in the volute, vane cascades, vaneless zone, and draft tube before moving on to the interval with the strongest pressure pulses. During 105–150 s, incidence error and a strengthened draft tube adverse gradient sustain high variance, defining the main instability window. Spectrum: in the vaneless zone, the dominant peak tracks the rising speed during acceleration and settles into a horizontal ridge near rated conditions. The draft tube spectrum combines BPF components with a low-frequency vortex rope signature that reflects instability severity. In the clearance gaps and equalizing passage, broadband content dominates and blade frequency tones are weak, underscoring the damping provided by the labyrinth rings.

Although this study identified the fundamental periods of hydraulic instability during this transition process, the specific locations prone to instability remain somewhat ambiguous. Subsequent research will focus on pinpointing the exact vulnerable areas. We will further analyze the evolution of internal flow using third-generation vortex identification methods to validate the conclusions. Based on this analysis, we will then propose and validate an optimization method for control strategies.