Time‐ and frequency‐domain analyses of the hydraulic system using a discrete numerical scheme based on equivalent circuit modelling

Abstract

Mathematical modeling is widely used for the time‐and frequency‐domain analysis of hydraulic systems. To explore the complicated hydraulic transients of the piping system under various operating conditions, different formulations of the equivalent circuit model (ECM) are systematically discussed in this study. In particular, comparative studies are performed regarding the restrictions on the tempo‐spatial discretization of the time‐domain ECM and the state‐of‐the‐art numerical schemes. In addition, the eigenanalysis of the free oscillation response and frequency features of the forced oscillation response were discussed in detail, respectively. Furthermore, the influence of spatial discretization on the frequency response precision for ECM is also quantified. Analytical results indicate that the ECM can achieve satisfactory performances in both numerical simulation and frequency analysis with affordable computational effort and relatively loose tempo‐spatial discretization restriction.

INTRODUCTION

Numerical simulation has long been recognized as one of the most efficient and economical techniques to predict the dynamic behavior and to reveal the stability of hydraulic systems including the hydraulic structures,¹ the pipelines² and the water passages in hydropower plants,³ and so forth. The common numerical tools for the transient simulation or stability analysis of hydraulic systems can be categorized into frequency-domain methods and time-domain methods.⁴ Over the past few decades, there are two technically mature frequency-domain schemes in applied hydraulic transients, that is, the hydraulic impedance method^5,6 and the transfer matrix method.^7–9 The hydraulic impedance, defined as the complex head deviation divided by the complex flow rate deviation, is always a function of the complex frequency s. The relationship between the piezometric head and the discharge at an arbitrary longitudinal position along a pipeline or a channel can be explicitly described with the hydraulic impedance method. The transfer matrix illustrates the transfer characteristics between the heads and discharges at two ends of the hydraulic components. It excels in determining the fluctuations of head and discharge at either end of a pipe or pipe series caused by the sudden or continuous excitation at another end. Although the two aforementioned frequency-domain modelling approaches had been extensively studied and widely applied in many piping systems, it is still difficult to precisely reflect the hydraulic transients of complex pipeline networks with multiple loops or branches.¹⁰ Frequency-domain stability analysis is suitable for linear systems or nonlinear systems operating within a limited range around the reference state.^5,11 However, for significant disturbance scenarios with large-scale operating condition conversions, it's usually rather difficult to quantitatively reveal the developing trends of various system states with frequency-domain methods. Therefore, time-domain numerical tools such as the method of characteristics (MOC)^12,13 and the finite difference method,^5,10 and so forth, had also been extensively studied worldwide due to their capability of characterizing the wave propagation and retrogradation phenomena along the pressurized pipelines. The aforementioned time-domain numerical schemes excel in transforming the distributed-parameter continuous systems into lumped-parameter discrete systems, thus reducing the hydraulic system model's degree of freedom and to calculate the time evolutions of system states with affordable computational cost and sufficient numerical precision. However, these models have too many tunable parameters due to their structural complexity, so parameter identification is challenging. Besides this, the strict tempo-spatial restrictions resulting from the well-known Courant–Friederichs–Lewy (CFL) stability criteria^14,15 prohibit the coordination of the time step of the discrete-time simulation and the spatial steps of different pipes, especially for hydraulic systems with complex network structure.

To overcome the defect, the methodology of equivalent circuit model (ECM) in the nonlinear differential equation formalism was proposed to solve the hyperbolic partial differential equations of the pressurized piping systems. This resolution was first inspired by an analogy between the hydraulic circuit and the electrical conductor by Paynter¹⁶ and Jaeger.¹⁷ In 1999, Souza et al.¹⁸ applied the equivalent electrical circuit to simulate the hydraulic transient process of a real hydropower plant. In 2007, Nicolet et al.¹⁹ extended ECM to the mathematical description of various typical hydraulic facilities and systematically revealed the transient phenomena in hydroelectric power plants. Recently, Zhao et al.²⁰ introduced ECM to the numerical simulation of pumped-storage plant and achieved satisfactory results. Zheng, et al.²¹ expanded the ECM to the modeling of pressurized pipe and open channel combination systems and performed numerical simulation of the hydraulic dynamic behaviors of a large hydropower plant with mixed tailrace flows. Similar to the MOC, ECM can also calculate the time evolutions of piezometric heads and discharges at the given locations along the pipeline. In addition, the relationship between the time step and spatial step of ECM is subject to much looser restriction. This feature makes it possible for ECM to choose a relatively larger spatial step with a certain sampling time.

To the best of the authors' knowledge, most of the existing literature about ECM only focused on its privilege in time-domain numerical simulation.^16–21 Seldom had the ECM-based frequency-domain analysis been reported. Since the frequency response characteristics of the dynamic system is as important as its time-domain transient behaviors, it's urgent to explore the features of ECM in the frequency domain.

In this paper, both time- and frequency-domain mathematical formulations of the ECM that precisely describe the hydraulic characteristics were presented in detail. In addition, the relationship between the hydraulic impedance and the equivalent resistance of the important valve boundary was also deduced. Furthermore, we investigated the superiority of the time-domain formulation of ECM in the tempo-spatial discretization for numerical simulation and its capability in determining the natural frequencies of the system and the oscillation modes under different conditions. A typical pressurized water delivery and control system composed of an upstream reservoir, a long pressurized pipe, and a downstream valve was selected as the system plant. Numerical simulations of the valve closure process and frequency responses of different oscillation patterns were conducted.

The remainder of this paper is organized as follows. First, the necessary background knowledge of the time- and frequency-domain modeling techniques of hydraulic systems is briefly introduced in Section 2. Then, Section 3 discusses the methodology of different formulations of the ECM for hydraulic systems. Subsequently, the case study and result analysis are presented in Section 4. Finally, several conclusions are condensed in Section 5.

BACKGROUND KNOWLEDGE

In this section, the basic theories of frequency-domain and time-domain modeling schemes for the hydraulic systems are presented.

Transfer matrix method

Assuming there is no vertical displacement of the pipe, the one-dimensional Saint–Venant equations of a pressurized pipe^5,22 are given in the following equation: [Image Omitted. See PDF]where, the subscript i denotes the pipe number. Hydraulic phenomena are characterized by a high wave speed a and a relatively much lower flow velocity v, thus the convective terms $${v}_{i}\,\cdot \,\partial /\partial x$$ related to the transport characteristic can be neglected with respect to the propagative terms $$\partial /\partial t$$ in Equation (1).

Suppose that $${q}_{i}={q}_{i}(x){e}^{st}$$, $${h}_{i}={h}_{i}(x){e}^{st}$$, where the complex frequency $$s=\sigma +{\rm{j}}\omega $$. The discharge and piezometric head at the location x along the i^th pipe are stated as the following equation:⁵ [Image Omitted. See PDF]where, the complex wave number $${\mu }_{i}=\sqrt{{s}^{2}/{a}_{i}^{2}+{\lambda }_{i}| {Q}_{i}| s/(2{D}_{i}{A}_{i}{a}_{i}^{2})}$$, and the characteristic impedance $${Z}_{ci}=\frac{{a}_{i}}{g{A}_{i}}\sqrt{1+\frac{{\lambda }_{i}| {Q}_{i}| }{2{D}_{i}{A}_{i}s}}$$.

Suppose that $${q}_{iu}={q}_{i}(0)$$, $${h}_{iu}={h}_{i}(0)$$, $${q}_{id}={q}_{i}({L}_{i})$$, $${h}_{id}={h}_{i}({L}_{i})$$, the transfer matrix of a pressurized pipe can be written as the following equation:⁵ [Image Omitted. See PDF]

Hydraulic impedance method

The complex characteristic impedance $${Z}_{ci}$$ characterizes the water hammer wave propagating in the hydraulic system. It defines the ratio between the piezometric head phasor and the discharge phasor at a given location x of the pipe for progressive and retrograde waves in reflectionless systems.¹⁰ In a system with reflections, it is necessary to evaluate the specific hydraulic impedance $${Z}_{ai}$$ which is a function of the location x and of the known boundary conditions, as stated in the following equation: [Image Omitted. See PDF]

Therefore, $${Z}_{ai}(L)$$ for the location x = L in the ith pipe can be expressed as the following equation: [Image Omitted. See PDF]

The specific impedance for any location x can be calculated using the known boundary at one end of a pipe. Specifically, for a pressurized pipe with infinite length and uniform sectional area, it is totally anechoic, and therefore the specific impedance at either end is identical to the characteristic impedance, that is, $${Z}_{ai}(0)={Z}_{ai}(\infty )={Z}_{ci}$$.

MOC

The aforementioned analytical solutions are all derived for continuous systems, but only can be used for the linearized systems restricted to small perturbations. It is difficult for the analytical solutions to investigate the characteristics of large dimensional systems and thus time-domain numerical methods are needed. However, the discretization in the mathematical modeling of nonlinear systems inevitably introduces approximation errors that have to be considered.

Equation (1) can be transformed into the ordinary differential equations² expressed in Equation (6a) and Equation (6b) with the characteristic method by defining the characteristic lines $$dx/dt={Q}_{i}/{A}_{i}\pm {a}_{i}$$. Since the wave velocity is much faster than the flow velocity $${v}_{i}={Q}_{i}/{A}_{i}$$, the relative small term $$\frac{1}{g{A}_{i}}\frac{\partial {Q}_{i}}{\partial t}$$ is omitted. [Image Omitted. See PDF] [Image Omitted. See PDF]

The basic theory of tempo-spatial discretization in the MOC is illustrated in Figure 1. Suppose that the pressurized pipe is divided into n segments, and the length of each segment is $$\Delta x$$. Then, the sampling interval can be obtained as $$\Delta T=\Delta x/a$$ and the head and discharge at the ith node at each sample are obtained according to the relevant state values at the (i − 1)th node and the (i + 1)th node of the last time sample.

Figure 1. Basic theory representation of the MOC. MOC, method of characteristics.

With the MOC, the recursive equations that reflect the dynamic behavior of the piezometric head and discharge at the i^th node along a pressurized pipe can be transformed to the following equation: [Image Omitted. See PDF]where, $$i=2,\ldots ,n$$, the parameters $${C}_{Pi}={H}_{i-1\_1}+B\cdot {Q}_{i-1\_1}$$, $${B}_{Pi}=B+r\cdot | {Q}_{i-1\_1}| $$, $${C}_{Mi}={H}_{i+1\_1}-B\,\cdot {Q}_{i+1\_1}$$, $${B}_{Mi}=B+r\cdot | {Q}_{i+1\_1}| $$, and the constants $$B=a/gA$$, $$r=\lambda \Delta x/2gD{A}^{2}$$. The inner nodes of the pipeline in the system plant at every sample can be obtained with Equation (7) iteratively.

The CFL numerical stability criteria linking the spatial step dx and the time step dt through the water hammer wave speed a of the pipe ensures the causality of the system because the information in the system cannot transit faster than the wave speed, that is, $$dt\le dx/a$$. In the case of the MOC, the critical condition of CFL criteria, that is, $$dt=dx/a$$, is proved to achieve satisfactory performance in modeling precision.^5,12,21 If $$dt\lt dx/a$$, the additional interpolation should be applied to the MOC algorithm, otherwise the numerical precision would deteriorate significantly although the computation still keeps convergent.⁵

METHODOLOGY OF THE ECM

The telegraphist's equation^16–19 which is used to describe the electromagnetic wave propagation characteristic in conductors can be stated as the following equation: [Image Omitted. See PDF]

The equivalent RLC circuit derived from the telegraphist's equation in Equation (8) is illustrated in Figure 2, where $${U}_{i+1/2}$$ represents the voltage at the center of this conductor section.

Figure 2. The RLC circuit of the transmission line per unit length

ECM of the pressurized pipe

Taking the pressurized head h and discharge Q as state variables, the Saint-Venant equation in Equation (1) is identical to the telegraphist's equation in the following:¹⁹ [Image Omitted. See PDF]where, the equivalent RLC parameters of the piping system are expressed as the following equation: [Image Omitted. See PDF]

The equivalent resistance R_ei, the equivalent inductance L_ei and the equivalent capacitance C_ei are related to the heat losses through the pipe, the inertia effect of the water, and the storage effect due to the pressure increase, respectively. The RLC components are necessary to fully describe the hydraulic characteristics of a pressurized pipe and the ignorance of any component will cause a negative effect on the modeling accuracy. If the equivalent resistance R_ei is ignored, the pipe is assumed to be frictionless. The assumption of a frictionless pipe is often used for the rough calculation of steady-state flow and the analytical resolution of the frequency responses of the system in applied hydrodynamics. If the equivalent inductance is ignored. The important water inertia that results in the water hammer effect will no longer exist, which is unacceptable. If the equivalent capacitance is ignored, it means that the elasticity of the water and the pipe wall won't be considered. In fact, the rigid water hammer theory based on this simplification can be applied to the modeling of short pressurized pipes under small perturbation conditions.

The equivalent circuit of a long pressurized pipe containing n segments is displayed in Figure 3. Since ECM belongs to a discrete numerical model, the hyperbolic partial differential equation set can be solved numerically and the values of system state variables are obtained only for given locations x and given times t, according to the predefined time step and spatial step. Besides this, the compact time step and spatial step will lead to higher computational costs.

Figure 3. The equivalent circuit of a long pressurized pipe19

Time-domain ECM

The hyperbolic partial differential equations in Equation (9) for the ith pipe segment can be transformed as an ordinary differential equation matrix in the following equation: [Image Omitted. See PDF]where, $${R}_{i}={R}_{ei}\cdot dx$$, $${L}_{i}={L}_{ei}\cdot dx$$, $${C}_{i}={C}_{ei}\cdot dx$$.

Taking n piezometric heads $${h}_{1+1/2}$$, …, $${h}_{n+1/2}$$ along the pipe and n + 1 discharges $${Q}_{1}$$, …, $${Q}_{n+1}$$ at both ends of n pipe segments as the system states, the mathematical expression of the overall system can be expressed in the following equation: [Image Omitted. See PDF]where, [Image Omitted. See PDF] [Image Omitted. See PDF] [Image Omitted. See PDF] [Image Omitted. See PDF]

Frequency-domain ECM

Apart from the time-domain differential equations for numerical simulation, the ECM can also be utilized to investigate the frequency features of the discrete-time model including the transfer matrix and the hydraulic impedance.

The discrete-time transfer matrix of the equivalent circuit of the ith pipe segment is stated in Equation (13) according to the Kirchhoff's law, [Image Omitted. See PDF]

If a pipe is modeled by n T-shape equivalent circuit elements, the equivalent impedance is calculated recursively from one end of the pipe according to the circuit theory, the equivalent impedance of the ith loop can be obtained with the following equation: [Image Omitted. See PDF]

In frequency domain analysis, it is difficult for the traditional continuous hydraulic impedance model to treat complex piping systems with branches and loops. However, for the frequency-domain ECM, the well-developed circuit theory such as the series and parallel operations can be used to calculate the hydraulic impedance of piping systems with complicated connection structures.

For frictionless systems, the system impedance can be calculated at one end for a given range of frequencies and identify which satisfies the known boundary conditions. For dissipative systems, the problem becomes tougher and the complex frequencies that satisfy all the boundary conditions need to be searched. This usually leads to an iterative optimization that searches for the minimum of the specific objective function. For instance, when the system is frictionless, the waterfall diagram of the system impedance with respect to different frequencies at different locations is depicted in Figure 4.

Figure 4. The magnitude of the impedance with different frequencies at different locations

Boundary treatments Reservoir

In hydraulic transient analysis, the water level of the reservoir is often assumed to stay constant during the whole dynamic process regardless of the possible variation of discharges. Therefore, the reservoir boundary can be viewed as an open end, that is, the impedance $${Z}_{r}=0$$.

Valve

When the water passes through a valve, the piezometric head at the outlet decreases compared with that of the inlet due to the head losses. The head losses can be stated as the following equation: [Image Omitted. See PDF]where,  the valve hydraulic resistance $${R}_{v}={K}_{v}| {Q}_{i}| /(2g{A}_{v}^{2})$$.

For a given valve stroke position, the valve impedance is expressed as the following equation:, [Image Omitted. See PDF]

CASE STUDY AND RESULT ANALYSIS Introduction of the system plant

Focused on investigating the time- and frequency-domain characteristics of ECM, a typical hydraulic system composed of an upstream reservoir, a long pressurized pipe, and a downstream valve was chosen as the system plant in this study. In this system, the discharge that flows through the pipe was controlled by the downstream valve. The schematic of the reservoir-pipe-valve system is illustrated in Figure 5 and the system parameters are given in Table 1.

Figure 5. Schematic of the reservoir-pipe-valve system

Table 1 Parameters setting of the system plant

Parameter	L (m)	a (m/s)	D (m)	A (m2)	λ
Value	600	1200	0.8	0.502	0.00

Comparison of the time-domain simulation of MOC and ECM

MOC has been widely used in academic study and engineering practice for the prediction of hydraulic transient behaviors.^3,7 It is capable of characterizing the time evolutions of the pressurized heads and discharges at different nodes along the pipeline. In this study, the simulation results of ECM were compared with those of MOC to evaluate its effectiveness in numerical simulation. Assume that the pressurized head at the upstream reservoir was set to 60 m water column and remained constant during the whole process, and the initial steady discharge that passes through the valve was set to 0.5 m³/s, and the downstream valve closed at a constant speed within 2.1 s. The pipe was divided into 100 segments in the model. Time evolutions of the piezometric head at the downstream valve inlet and the discharge at the upstream reservoir boundary are illustrated in Figure 6A,B, respectively. The two evolution curves of the piezometric head and discharge of the two numerical methods nearly coincide with each other, which indicates that the ECM can achieve satisfactory modeling precision compared with the state-of-the-art methods in the hydraulic transient simulation if the parameters strictly yield to the critical CFL criteria (i.e., dx = a·dt).

Figure 6. Time evolutions of system states using different numerical methods with the same time step and space step: (A) the piezometric head at the downstream valve inlet and (B) the discharge at the upstream reservoir boundary

It is known that the CFL criterion $$dx\ge a\cdot dt$$ must be met in MOC to ensure the convergence of the numerical solution. If $$dx\gt a\cdot dt$$, the modeling precision deteriorates significantly although the time evolution curves of the system would not become divergent.⁵ The time evolutions of the system states of the simulation using MOC with the same number of pipe segments (n = 100) and different values of the time step dt are shown in Figure 7. When time step dt decreases, the underestimation of the fluctuation amplitudes of the system states gradually become evident. Although previous research showed that the additional interpolation in MOC can reduce modeling errors caused by the inappropriate dt settings to some extent, the complexity of the algorithm is increased which leads to heavier computational cost.

Figure 7. Time evolutions of system states using MOC with different time steps: (A) the piezometric head at the downstream valve inlet and (B) the discharge at the upstream reservoir boundary. MOC, method of characteristics.

The time evolutions of the head at the valve and the discharge at the reservoir outlet using ECM with different spatial steps are shown in Figure 8. With a different number of pipe sections n, the two curves of different parameter settings almost coincide with each other. The simulated fluctuations of the system state only attenuate very slightly when $$dx\gt a\cdot dt$$ as n decreases. The simulation results indicate that the ECM has much looser constraints on the CFL criteria than the MOC. Although the CFL criterion also works for ECM, the additional interpolation that is necessary for MOC is avoided in ECM to keep simulation precision when $$dx\gt a\cdot dt$$. This important characteristic enables ECM to coordinate the relationship between the time step and the spatial steps, especially in complex hydraulic systems containing cascading pipes, branches, or loops with different sectional areas.

Figure 8. Time evolutions of system states using ECM with different spatial steps: (A) the piezometric head at the downstream valve inlet and (B) the discharge at the upstream reservoir boundary. ECM, equivalent circuit model.

Frequency-domain analysis of the ECM Eigen analysis of the system plant

The mathematical equations of the reservoir-pipe-valve system in matrix form are given in Equation (17). For free oscillation analysis, the vector C on the right side of the equation must be 0 since no external excitation exists in this case. Therefore, the free oscillation analysis of the system with ECM corresponds to the problem of determining eigenvalues of the differential equation in the following equation: [Image Omitted. See PDF]where [Image Omitted. See PDF] [Image Omitted. See PDF]

Then, the eigenfrequencies and eigenvectors can be easily obtained through the eigenvalue analysis of the matrix $${{\bf{A}}}^{-1}{\bf{B}}$$.

When the downstream valve is fully open, the first 10 complex eigenfrequencies obtained from Equation (17) are displayed in Table 2, and the corresponding first 10 eigenmodes of the piezometric head at the downstream valve inlet are illustrated in Figure 9. In Figure 9, the eigenmodes of this condition are even harmonics (It is denoted that the base natural frequency $${f}_{0}=a/(4\cdot L)=0.5\text{Hz}$$). In addition, the complex frequencies that meet the downstream boundary condition $${Z}_{v}=0$$ obtained from the system impedance in Equation (14) derived from the discrete-time ECM are also given in Table 2. The corresponding decay coefficients and eigenfrequencies of the two approaches are in good agreement with each other.

Table 2 The decay coefficients and eigenfrequencies of the first 10 orders of the 2 ECM models under fully open valve condition

No.	Obtained from eigenvalue analysis of matrix A−1B		Obtained from the discrete system impedance Zdis of ECM
	Decay coeffici.	Eigen frequency	Decay coeffici.	Eigen frequency
1	−7.383e−15	1.000	−2.216e−8	1.000
2	3.996e−15	1.999	1.774e−9	2.000
3	−7.754e−16	2.998	1.546e−8	2.999
4	1.887e−15	3.997	−7.562e−9	3.996
5	−7.030e−15	4.994	−6.228e−9	4.991
6	−7.105e−15	5.991	8.766e−9	5.986
7	3.552e−15	6.985	−7.329e−9	6.971
8	1.287e−14	7.979	1.157e−8	7.975
9	4.801e−15	8.970	2.008e−8	8.953
10	−1.609e−15	9.958	2.237e−9	9.932

Abbreviation: ECM, equivalent circuit model.

Figure 9. The first 10 eigenmodes of the valve pressure with respect to the location × under the fully open valve condition

When the downstream valve is closed, the first 10 complex eigenvalues obtained from the matrix $${{\boldsymbol{A}}}^{-1}{\boldsymbol{B}}$$ and the discrete-time hydraulic impedance $${Z}_{v}$$ in Equation (14) are given in Table 3, and the corresponding first 10 eigen modes of the piezometric head at the downstream valve inlet are illustrated in Figure 10. The eigenmodes of this condition are odd harmonics. Similar to the case of the open valve condition, the corresponding decay coefficients and eigenfrequencies of the two approaches are also in good agreement. The analyses under the two typical conditions indicate that the ECM can solve both the time- and frequency-domain problems.

Table 3 The decay coefficients and eigenfrequencies of the first 10 orders of the 2 ECM models under fully open valve condition

No.	Obtained from eigenvalue analysis of matrix A−1B		Obtained from the discrete system impedance Zdis of ECM
	Decay coeffici.	Eigen frequency	Decay coeffici.	Eigen frequency
1	1.759e−8	0.500	2.268e−7	0.500
2	3.908e−8	1.499	−9.984e−8	1.499
3	2.647e−8	2.499	3.252e−7	2.497
4	6.089e−8	3.498	3.429e−6	3.495
5	4.238e−8	4.496	−1.306e−6	4.492
6	−5.671e−9	5.493	1.172e−7	5.483
7	2.429e−8	6.488	6.171e−8	6.471
8	−2.053e−8	7.482	8.992e−8	7.465
9	−6.545e−9	8.474	−2.973e−7	8.460
10	5.512e−9	9.464	1.136e−6	9.453

Abbreviation: ECM, equivalent circuit model.

Figure 10. The first 10 eigenmodes of the valve pressure with respect to the location × under the fully closed valve condition

Forced oscillation response analysis with the ECM

The eigenfrequencies of the system can be also obtained from a numerical simulation using the ECM and a frequency spectrum analysis of the time-domain responses. Suppose a white noise excitation is considered at the piezometric head of the upstream reservoir boundary. As the energy spectra of white noise is known to be distributed uniformly in the frequency domain,²³ it is an ideal excitation to fully excite the oscillation modes of the system of interest. To testify to the effectiveness of ECM in the forced oscillation scenarios, three typical valve conditions, i.e., the downstream valve stroke is fully open ($${Z}_{v}/{Z}_{c}\approx 0$$), the downstream valve stroke is fully closed ($${Z}_{v}/{Z}_{c}=\infty $$) and the downstream valve works at the critical anechoic condition ($${Z}_{v}/{Z}_{c}\approx 1$$), were chosen as the operating conditions in the case study.

Analyze with the frequency-domain ECM, the waterfall diagrams of pressure fluctuations at the downstream valve inlet under the above-predefined conditions are presented in Figure 11A–C, respectively. From Figure 11, the system's eigenmodes are even harmonics when the valve serves as an open end and odd harmonics when the valve serves as a dead end. In addition, nearly no frequency components can be excited in the critical anechoic condition. This is because the reservoir-pipe-valve coupling system under this condition can be viewed as a pipe with infinite length, thus there is no wave reflection being detected. The frequency amplitude spectra of the forced oscillation responses present good agreement with the analytical results of the eigenmodes obtained from the free oscillation analysis in Section 4.3.1.

Figure 11. Waterfall diagrams of pressure fluctuations at the downstream valve inlet under the three different valve conditions: (A) the valve is fully open, (B) the valve is fully closed, and (C) the valve operates at the critical anechoic condition.

Influence of spatial discretization on the frequency responses of ECM

The amplitude of the discrete impedance can be applied to evaluate the precision of ECM in the frequency domain by taking the solutions of the continuous hyperbolic differential equation model as the reference. For the sake of reducing the error and enhancing the resolution of the discrete ECM model, the wave length of the phenomenon of interest is set to 10 times larger than the length of each pipe segment in discrete models, Therefore, the rated wavelength is fixed to $$\lambda /dx=10$$ in this study, and the maximal frequency is determined by the spatial discretization of the system. The rated amplitudes of system impedance $$|Z(dx)|/Zc$$ at the location $$x=dx$$ with respect to the frequency for the ECM with four different values of n are shown in Figure 12. The frequency response of each discrete model shows good agreement with the resolution of the continuous model within a certain range [0, $${f}_{\text{upperlim}}$$]. When $$n=20$$, $${f}_{\text{upperlim}}\approx 10\unicode{x0200A}\text{Hz}$$; when $$n=50$$, $${f}_{\text{upperlim}}\approx 20\unicode{x0200A}\text{Hz}$$; when $$n=100$$, $${f}_{\text{upperlim}}\approx 40\unicode{x0200A}\text{Hz}$$; when $$n=300$$, $${f}_{\text{upperlim}}$$ is close to the upper bound of the frequency of interest. The numerical results indicate that the spatial discretization in ECM introduces modeling errors in frequency responses of the system of interest. The model is more prone to characterize low-frequency responses of the system. Fine spatial discretization of the pipe can broaden the effective range of the frequency responses. Whereas it also induces the drawback of enlarging the dimension of the model, thus increasing the computational burden. It is therefore paramount to balance the modeling accuracy and computational cost in frequency analysis when modeling with ECM.

Figure 12. Frequency-related impedance comparison results of a pipe segment with a length of dx for the continuous model and ECM with different number of segments (A) n = 20, dx = 30; (B) n = 50, dx = 12; (C) n = 100, dx = 6; and (D) n = 300, dx = 2. ECM, equivalent circuit model.

CONCLUSION

Both time- and frequency-domain formulations of the ECM for hydraulic systems have been investigated in this study. Several conclusions and the future prospect are summarized below:

• (1)

  The effectiveness of the time-domain numerical simulation with ECM for hydraulic transient processes is investigated by taking the conventional MOC scheme as a reference. The tempo-spatial discretization restriction of ECM is much looser than that of MOC, which brings greater flexibility in determining the time step and the spatial step in system modeling.

• (2)

  The frequency-related responses to the free oscillation and the forced oscillation of the ECM are discussed by means of the eigenanalysis of the system matrix and the frequency spectrum analysis of the numerical simulation results, respectively. The eigenmodes of the system under different conditions obtained with ECM show good agreement with the theoretical analysis result.

• (3)

  The influences of the spatial discretization on the numerical simulation precision and the frequency response accuracy for ECM are revealed, respectively. This numerical scheme shows greater performance in the coordination of time and spatial discretization compared with MOC, whereas the refined spatial resolution contributes to the precise excitation responses to the higher frequency band.

• (4)

  The system plant is limited to the simple reservoir-pipe-valve system. However, as a powerful numerical tool for the time- and frequency-domain analysis of hydraulic systems. The applicability and superiority of the ECM in more complex engineering scenarios such as the hydropower station system, the pump station system, and the water delivery piping networks need further investigation in the future.

NOMENCLATURE

• A
• sectional area of the pipe (m²)
• a
• water hammer wave speed (m/s)
• L
• length of the pipe (m)
• n
• number of the segments in a pipe
• v
• flow velocity in the pipe (m/s)
• g
• gravitational acceleration (m/s²)
• D
• diameter of the pipe (m)
• dt
• time step (s)
• dx
• spatial Step (m)
• Re
• equivalent circuit resistance per unit length (s/m³)
• Le
• equivalent circuit inductance per unit length (m)
• Ce
• equivalent circuit capacitance per unit length (s²/m³)
• R
• equivalent circuit resistance with a length of dx (s/m²)
• L
• equivalent circuit resistance with a length of dx (m²)
• C
• equivalent circuit resistance with a length of dx (s²/m²)
• U
• voltage (V)
• I
• current (A)
• U_{i + 1/2}
• voltage at the center of the ith conductor section (V)
• h
• piezometric head (m)
• Q
• discharge (m³/s)
• h_{i +1/2}
• piezometric head at the center of the ith pipe segment (m)
• Z_equi
• equivalent discrete-time system impedance of the ith loop (s/m²)
• K_v
• valve head loss coefficient
• A_v
• sectional area that water flows through in the valve (m²)
• f_upperlim
• upper limit of the frequency that discrete ECM precisely modeled (Hz)

ACKNOWLEDGMENTS

This study was supported by theNational Natural Science Foundation of China (Grant No. 52009096) andFundamental Research Funds for the Central Universities (Grant No. 2042022kf1022).