Hydraulic Oscillation and Instability of A Hydraulic System with Two Di ﬀ erent Pump-Turbines in Turbine Operation

: Hydraulic oscillation mainly reveals the undesirable pressure ﬂuctuations which can cause catastrophic failure of any hydraulic system. The behavior of a hydraulic system equipped with two di ﬀ erent pump-turbines was investigated through hydraulic oscillation analysis to demonstrate severe consequences induced in turbine operation, including S-shaped characteristics. The impedance of a pump-turbine has an essential role in the determination of the instability of the hydraulic system. The conventional way to determine the instability solely using the slope of a characteristic curve was improved, including the e ﬀ ect of guide vane opening in pump-turbine impedance, which consequently modiﬁed the instability expression. With this pump-turbine impedance, hydraulic oscillation analysis, including free oscillation analysis and frequency response analysis, was carried out. The free oscillation analysis entails the computation of complex natural frequencies and corresponding mode shapes of the system. These computations provided necessary information about the vulnerable position of vital hydraulic components and the scenario for self-excited oscillation. Further, the analysis illustrates the signiﬁcant role of guide vane opening to prevent the system from becoming unstable. Lastly, frequency response analysis was performed for the system with an oscillating guide vane to obtain the frequency response spectrum, which revealed that the resonating frequencies are consistent with natural frequencies, and it supported free oscillation results.


Introduction
As of now, the pumped storage power plant [1] is the most flexible and reliable load balancing scheme which stores energy as potential energy of water. It also displays a fast response to sudden load change and thereby keeps the frequency and voltage stable on the electric grid. These features advocate the utilization of a pump-turbine and pumped storage plant as the optimum solution for energy storage. The most widely used pump-turbine [2] is the reversible pump-turbine, which is the advanced contrivance of the Francis turbine and Centrifugal pump. The reversible pump-turbine has a comparatively wide range of operating heads and a large installed capacity. During operation, the pumping head surpasses the generation head due to losses along the waterways. As a result, the pump-turbine is basically designed as a pump, not a turbine, specifically prioritizing the high value of the pumping head. The main alterations implemented on the Francis turbine in order to achieve the pumping head are elongated blades, fewer blades, and blades bending backward. These suitable alterations have economic importance, and this is a comparatively better method of energy storage than other schemes, but it leads to off-design operation of the pump-turbine during turbine operation. Low superposition, which is detrimental for components of the system. Thus, the hydraulic oscillation analysis is necessary and should be implemented as a preliminary study for every pumped storage system in order to assure the safe and reliable operation. This research is a holistic investigation based on hydraulic oscillation analysis, which provides insight into acute perturbation in the system with two different pump-turbines working in turbine operation. The effect of the guide vane was taken as a significant factor; therefore, it was included in the pump-turbine impedance to obtain the instability expression and frequency response spectrum during the analysis.

S-shaped Characteristics of Pump-turbine
The compromise for the pumping head in design causes some anomalies in the pump-turbine and results in off-design working in turbine operation. Despite several aforementioned challenges during off-design operation of a pump-turbine, this article's sole concern is turbine operation incorporating S-shaped characteristics, which compel the system to become unstable.

Flow Characteristics
The flow characteristics of a pump-turbine [40] are achieved with a unit discharge (Q11) vs unit speed (n11) diagram. It consists of four quadrants in which unit speed and discharge are plotted along the X and Y axis of the cartesian coordinate system, respectively. The plot includes guide vane opening contours and a runaway curve. Each quadrant represents the different operating conditions of the pump-turbine. They are turbine operation, pump brake operation (dissipation), pump operation, and reverse pump operation, represented by the first (I), second (II), third (III), and fourth (IV) quadrant of the plot, respectively, as shown in Figure 1.

Demarcation of S-shaped Region
The S-shaped characteristics [8] start with a steep positive slope and the entire S-shaped region (SSR) in turbine operation holds this property. The portion of characteristic curves before the Sshaped region in the first quadrant has a negative slope for respective guide vane openings and this region is defined as the Regular Generation Region (RGR). The S-shaped characteristics [35] contribute to the uncontrollable transient, which is in fact undesirable for a smooth and effective operation. Also, the measure of S-shaped characteristics is negligible for small guide vane openings. The flow characteristics of the pump-turbine are accountable for the hydraulic properties at a constant rotation speed. The slope of the characteristic curve (dQ11/dn11) that defines the S-shaped characteristics is not an inclusive representation of the pump-turbine to determine the system stability. It fails to explain the contribution of guide vane opening in the system stability. Indeed, the guide vane opening variation brings change in the guide vane passage area, which has a significant role in system stability. Therefore, a certain artifice was considered to impart the influence of the guide vane in the analysis, and modified instability criterion of the system, accordingly. This artifice is thoroughly explained in the following sections.

Demarcation of S-Shaped Region
The S-shaped characteristics [8] start with a steep positive slope and the entire S-shaped region (SSR) in turbine operation holds this property. The portion of characteristic curves before the S-shaped region in the first quadrant has a negative slope for respective guide vane openings and this region is defined as the Regular Generation Region (RGR). The S-shaped characteristics [35] contribute to the uncontrollable transient, which is in fact undesirable for a smooth and effective operation. Also, the measure of S-shaped characteristics is negligible for small guide vane openings. The flow characteristics of the pump-turbine are accountable for the hydraulic properties at a constant rotation speed. The slope of the characteristic curve (dQ 11 /dn 11 ) that defines the S-shaped characteristics is not an inclusive representation of the pump-turbine to determine the system stability. It fails to explain the contribution of guide vane opening in the system stability. Indeed, the guide vane opening variation brings change in the guide vane passage area, which has a significant role in system stability. Therefore, a certain artifice was considered to impart the influence of the guide vane in the analysis, and modified instability criterion of the system, accordingly. This artifice is thoroughly explained in the following sections.

Mathematical Model
The fundamental flow equations [29], which explain the transient flow phenomenon in closed pipelines, are hyperbolic and partial differential equations. The governing equations of the unsteady flow are: Momentum Equation: ∂H ∂x Continuity Equation: where, H is the instantaneous head, Q is the instantaneous discharge, t is the time, x is the distance along the pipe length, A is the cross-section area of the pipe, d is the diameter of the pipe, f is the friction factor, a is the velocity of the wave, and g is the acceleration due to gravity. These equations are transformed into a linearized form and expressed as also, H = H + h , Q = Q + q , h = H(x)e st , q = Q(x)e st , L = 1 gA , C = gA a 2 and R = f Q gdA 2 , where, H and Q are the average head and discharge, respectively; h and q are the oscillatory head and discharge, respectively; s is the complex frequency; L, C, and R are the hydraulic inductance, capacitance, and resistance, respectively; and H(x) and Q(x) are the oscillation head and discharge, respectively, in the frequency domain.
The solution of the above Equations (3) and (4) gives From Equations (5) and (6) for a definite length of pipe l, a matrix system which comprises the state vectors and transfer matrix is given as where, subscripts U and D represent upstream and downstream at a section, respectively. γ = Cs(R + sL) and Z C = γ Cs are the propagation constant and characteristic impedance, respectively. Another essential term is hydraulic impedance, defined as the ratio of head and discharge at the particular section along the pipeline in the frequency domain. It is useful in establishing the transfer matrix relationship for hydraulic oscillation analysis. Equation (7) is the general matrix equation for a simple pipe system.

Transfer Matrix
The column vectors that describe head and discharge, two quantities of interest, at any section of the pipe are called state vectors. The transfer matrix is a square matrix which relates two state vectors in the matrix equation. Equation (7) contains two state vectors, one on the left-hand side and another on the right-hand side of the equation, whereas the square matrix on the right-hand side represents a transfer matrix. There are two types of transfer matrix.

Field Matrix
A field matrix establishes the connection between two state vectors at two adjoining sections of a pipe. The field matrix is expressed as

Point Matrix
The point matrices come into play when the system at any section includes discontinuity, such as pipe series, a junction, an orifice, a valve, parallel pipes, turbines, etc. These point matrices relate the left and right state vectors of discontinuity, which lies in the pipe system. Various boundary conditions appear in pipes of the pumped storage system, which have specific point matrices.

Pipeline in Series
A pipeline system usually has many pipes connected, which are of different diameters. This situation obeys the following relation: where, k = integer number (1, 2, 3, . . . . . . . . . .) and subscript k of U and D designates the pipe serial number. Neglecting the minor losses, the point matrix is expressed as

Valve
The general equations for valve located between two pipes k and k + 1 is are , where, τ is the dimensionless valve position; C d A G is the coefficient of the discharge times area of valve opening; and τ' is a small perturbation of the guide vane position with amplitude, T V , and angular frequency, ω, Water 2019, 11, 692 7 of 26 for time, t. "Re" represents the real part of the complex value within parentheses and subscript "0" stands for the initial steady condition. Equations (12) and (13) are represented in the matrix form as Equation (14) represents the matrix expression for the valve boundary condition. The additional on the right-hand side defines the motion of the valve, and for constant opening, this term is equal to zero. Finally, the point matrix for a valve with constant opening is expressed as Also, the impedance of valve with constant opening is expressed as

Throttled Surge Tank
All surge tanks [41][42][43] are accepted as a lumped mass system where mass oscillation and head loss at the inlet provide the necessary impedance to govern the surge control. For a throttled surge tank [44] installed in a system, the base of the surge tank is considered as upstream (U) and the position of the water level as downstream (D). Then, the water oscillation inside the surge tank is governed by the following equations: where, Q s , A c , b, and β are the discharge, cross-sectional area, height of the water column, and head loss coefficient of the throttled surge tank, respectively. After applying oscillatory parameters and simplification, Equations (17) and (18) reduce to Now, Equations (19) and (20) give From Equation (21), the impedance of throttled surge tank is Therefore, the point matrix of the throttled surge tank is For turbine operation [34] of the pump-turbine based on the flow characteristic curve, Q 11 vs. n 11 , the equation defined below is valid: where, , and D 1 and n are the diameter and rotation speed of the pump-turbine, respectively.
After applying oscillatory parameters and simplification, Equation (24) reduces to From Equation (25), the impedance of the pump-turbine is There is a limitation in this impedance formula, which is that it lacks a guide vane opening effect, as mentioned in the above section. The artifice provided was used to analyze a pump-turbine as the combination of a cascaded valve, which imparts a guide vane effect, and a turbine represented by flow characteristics. The impedance of the valve with constant opening described in Section 3.2.2.2 gives the impedance of the guide vane. It provides a constant value at the operating points with the same guide vane opening, but varies correspondingly with respective change in the opening. The artifice transmits the influence of guide vane opening and the flow characteristic curve to compute the total impedance of the pump-turbine. The total impedance of the pump-turbine in turbine operation is where, P T represents the point matrix for the pump-turbine in turbine operation.

Parallel Pipe System
The overall transfer matrix to establish the expression between upstream main conduit k and downstream main conduit k + 1 separated with any number of parallel pipes in between is given as The overall transfer matrix for each branch is given as where, subscript j represents the branch number in the parallel pipe system. From Equation (30), this can be expressed as Taking summation on both sides of Equation (31), we get where, v 12 and µ = e 1 u 12 . Similarly, taking summation on both sides of Equation (32), we get where, Q U k+1 = Q D j , ξ = v 22 v 12 and ψ = v 22 µ − e 2 . Therefore, the point matrix of parallel pipe system is given by P PP = y 11 y 12 y 21 y 22 (35) where, y 11 = ς η , y 12 = 1 η , y 21 = ξς η − η and y 22 = ξ η . Also, the elements of the overall column matrix for the parallel pipe system are c 1 = µ η and c 2 = ξµ η − ψ. These column matrices e 1 e 2 and c 1 c 2 are computed when the forcing function is applied at any branch, and they are equal to zero otherwise. Also, the determinant of the point and field matrices must be equal to one, which is the essential property of these matrices.

Hydraulic Oscillation Analysis
Hydraulic oscillation analysis reveals the oscillatory state vectors (head and discharge) induced periodically and illustrates their impacts on a hydraulic system in the frequency domain. This analysis also determines the status of the entire system regarding its stability. Hydraulic oscillation is a general characteristic and response phenomenon of the system when the system is stimulated inherently or by external excitation. On this basis, hydraulic oscillation analysis comprises free oscillation and frequency response analysis.

Free Oscillation Analysis
Free oscillation analysis is the investigation of various oscillation modes of a system in the absence of an external forcing function. In fact, the system perturbed initially, where disturbance is removed leaving the system oscillating freely on its own, is free oscillation. Practically, we can discover such an event for each small fraction of residual time interval during both load acceptance and load rejection of the pump-turbine in the turbine operation. The hydraulic system goes through free oscillation for a few seconds during its activity in between the change of operating points in turbine operation. Essentially, the natural frequency and mode shape computation describe the characteristics of free oscillation. The natural frequencies are independent of the initial amplitude and application point of disturbance, and instead depend on the intrinsic property of system components. The system undergoes different modes of oscillation driven by the natural frequencies of a hydraulic system. The complex natural frequency is the pivotal parameter necessary to determine and study the peculiarity of a hydraulic system in free oscillation. It comprises real and imaginary parts which represent the damping factor (σ) and natural angular frequency (ω). The complex natural frequency for a certain order k is expressed as In each natural angular frequency, the system oscillates in different ways, represented by corresponding mode shapes. The evaluation of modes leads to the speculation of undesirable oscillation in different components of a system. A hydraulic system has infinite modes of oscillation, which results in infinite natural frequencies. The first natural frequency is called the fundamental frequency, and others are called higher harmonics. The damping factor determines the stability of a complete system oscillating freely and depends on the impedance of various components of the system. Therefore, the damping factor is the property of the entire system under oscillation at a particular frequency, not only that of an individual component.

Frequency Response Oscillation
The oscillation driven by a known forcing function is defined as frequency response oscillation. In this oscillation, damping factor σ k = 0 and the complex frequency contains only an imaginary part: The absence of a damping factor explains that the system is completely under the influence of a forcing function, which propels oscillation in the system. The entire system oscillates with the frequency of the known forcing function. Severe flow oscillation develops when resonance evolves in the hydraulic system. If more than one forcing function is acting on a system, the principle of superposition is applied to obtain the resultant impact.

System Description
A typical pumped storage plant was considered with an upper reservoir; upstream throttled surge tank; parallel system with two branches, with each consisting of a pump-turbine; downstream surge tank; and lower reservoir, as shown in Figure 2. These components were connected through various pipes of different lengths and diameters. The hydraulic system was built and compiled in Visual Fortran to simulate multiple operating points of two pump-turbines. In each natural angular frequency, the system oscillates in different ways, represented by corresponding mode shapes. The evaluation of modes leads to the speculation of undesirable oscillation in different components of a system. A hydraulic system has infinite modes of oscillation, which results in infinite natural frequencies. The first natural frequency is called the fundamental frequency, and others are called higher harmonics. The damping factor determines the stability of a complete system oscillating freely and depends on the impedance of various components of the system. Therefore, the damping factor is the property of the entire system under oscillation at a particular frequency, not only that of an individual component.

Frequency Response Oscillation
The oscillation driven by a known forcing function is defined as frequency response oscillation. In this oscillation, damping factor 0 k   and the complex frequency contains only an imaginary part: The absence of a damping factor explains that the system is completely under the influence of a forcing function, which propels oscillation in the system. The entire system oscillates with the frequency of the known forcing function. Severe flow oscillation develops when resonance evolves in the hydraulic system. If more than one forcing function is acting on a system, the principle of superposition is applied to obtain the resultant impact.

System Description
A typical pumped storage plant was considered with an upper reservoir; upstream throttled surge tank; parallel system with two branches, with each consisting of a pump-turbine; downstream surge tank; and lower reservoir, as shown in Figure 2. These components were connected through various pipes of different lengths and diameters. The hydraulic system was built and compiled in Visual Fortran to simulate multiple operating points of two pump-turbines. The above hydraulic system is a commonly used layout of a pumped storage plant for which the measurement and characteristics of hydraulic components are shown in Tables A1, A2 and A3 of Appendix A. The changes in diameter, length, and roughness coefficient of pipes in series, which are usually encountered in real projects, are taken into account in the above hydraulic system. The surge tanks equipped in the system were identical. The longer branch consisting of pump-turbine I was considered as a part of the main conduit. Besides, pump-turbine I and pump-turbine II were of equal dimensions, but dissimilar characteristic curves which is shown in Figures A1 and A2 of Appendix A. This sort of situation is encountered when pump-turbines installed in a pumped storage plant are manufactured by different companies with their captive guidelines and design criteria. Despite the same measurements, two pump-turbines are not considered identical because of differences in the S (1)(2)(3)(4)(5) S (  The above hydraulic system is a commonly used layout of a pumped storage plant for which the measurement and characteristics of hydraulic components are shown in Tables A1-A3 of Appendix A. The changes in diameter, length, and roughness coefficient of pipes in series, which are usually encountered in real projects, are taken into account in the above hydraulic system. The surge tanks equipped in the system were identical. The longer branch consisting of pump-turbine I was considered as a part of the main conduit. Besides, pump-turbine I and pump-turbine II were of equal dimensions, but dissimilar characteristic curves which is shown in Figures A1 and A2 of Appendix A. This sort of situation is encountered when pump-turbines installed in a pumped storage plant are manufactured by different companies with their captive guidelines and design criteria. Despite the same measurements, two pump-turbines are not considered identical because of differences in the characteristic curves which determine the vital parameter, hydraulic impedance. Free oscillation and frequency response analyses were carried out to study the peculiar features of the system when the pump-turbines were working constantly at a rated speed in turbine operation.

Natural Frequency and Mode Shape Computation
The overall transfer matrix at the downstream end of the last pipe, m, of the hydraulic system is given as where, u 11 u 12 u 21 u 22 is the overall transfer matrix of the system at the end of the last pipe, m: For a hydraulic system equipped with an upper and lower reservoir at each end, we have The solution of Equation (42) gives the complex natural frequency (s k = σ k + iω k ) of the entire hydraulic system. The mode shapes were plotted for the first five natural frequencies, which are based on corresponding complex natural frequencies.

Frequency Response Computation
The system was under the influence of external disturbance at pump-turbine I, which was the oscillating guide vanes at different excitation frequencies. In the study, the oscillating guide vane was considered equivalent to the oscillating valve, and pump-turbines were considered to run continuously at a rated speed. The overall matrix system at the downstream reservoir is given by where, C 1 C 2 is the overall column matrix of the system, which includes the forcing function at any branch of the parallel system: From Equations (43) and (44), we obtain From Equations (45) and (46), we get Equation (47) is the required expression for complex discharge at the lower reservoir when the forcing function is acting on any branch of the parallel system.

Operation Scenarios, Operating Points, and Natural Frequencies
The above hydraulic system was simulated based on the mathematical model for numerous operating points of pump-turbines extracted from their characteristic curves with corresponding guide vane openings. Numerous points of RGR and SSR were retrieved from characteristic curves of each pump-turbine. These points were accessed for the simulation to obtain hydraulic impedances, damping factors, natural angular frequencies, and mode shapes. Among many operating points, some of them are listed below in Table 1. These chosen data rendered the overall behavior of the system at specific conditions and also represented the tentative nature of those operating points that are not included in Table 1. The operating points for pump-turbines I and II were extracted such that the parallel pipe system criteria for steady state conditions were met, i.e., discharge was conserved and head loss across each branch was equal. Some selected operating points of two pump-turbines which produced various operation scenarios in the system are summarized in Tables 1 and 2.
Tables 1 and 2 entail various features of the hydraulic system when pump-turbines were performed at different operating points in the absence of external excitation. In addition, these tables provide the necessary information about the influence of operating points on the hydraulic system subjected to free oscillation. Considering the nature of the tangent slope at operating points, the first quadrant was divided into two regions for convenience in analysis: Regular generation region (RGR) and S-shaped region (SSR). The negative and positive values of the slope distinguished the two regions, respectively. According to the investigation, the first five natural angular frequencies (ω) of the entire system remain almost the same, with negligible differences after the decimal point. This difference was ignored [33], taking an average of all data obtained that gives natural angular frequencies of ω 1 = 1.309 rad/s, ω 2 = 2.541 rad/s, ω 3 = 2.890 rad/s, ω 4 = 3.625 rad/s, and ω 5 = 4.332 rad/s for the first five modes, regardless of the change in operating points of pump-turbines. However, the damping factor, σ, varies with the operating points of pump-turbines, which subsequently demonstrates that the damping factor depends on the impedance of pump-turbines.

Mode Shapes
Mode shapes are the intrinsic properties of a hydraulic system since it is evaluated in the absence of any external disturbance imposed on the system. The mode shape of the system was calculated with the initial assumed value of Q U = 0.001 m 3 /s at the upper reservoir. Mode shapes of OS 1 were plotted for the first five natural frequencies, which is shown in Figure 3. Figure 3 describes the oscillation behavior at natural frequencies when no forcing agent was acting on the system. Also, when the forcing function is anonymous in any system, mode shapes are applied to investigate certain components contributing detrimental oscillation.  Figure 3 describes the oscillation behavior at natural frequencies when no forcing agent was acting on the system. Also, when the forcing function is anonymous in any system, mode shapes are applied to investigate certain components contributing detrimental oscillation.
These mode shapes are also used to determine the inept position of hydraulic components in the system. The position of pressure nodes and anti-nodes decides where the particular hydraulic components should not be located; otherwise, special attention and additional protection measures are required. The nodes and antinodes of pressure or discharge [30] describe the minimum and maximum values of perturbation, respectively. Nodes are important to determine the effective location of the surge tank, pump-turbine, and valve. If the pressure node falls at the base of surge tank in any mode of oscillation, then pressure waves are minimally prevented from being transmitted to the upstream area of the surge tank. This results in a dysfunctional and ineffective mechanism of the surge tank for that particular mode, even if all other design criteria for the surge tank are satisfied. These mode shapes are also used to determine the inept position of hydraulic components in the system. The position of pressure nodes and anti-nodes decides where the particular hydraulic components should not be located; otherwise, special attention and additional protection measures are required. The nodes and antinodes of pressure or discharge [30] describe the minimum and maximum values of perturbation, respectively. Nodes are important to determine the effective location of the surge tank, pump-turbine, and valve. If the pressure node falls at the base of surge tank in any mode of oscillation, then pressure waves are minimally prevented from being transmitted to the upstream area of the surge tank. This results in a dysfunctional and ineffective mechanism of the surge tank for that particular mode, even if all other design criteria for the surge tank are satisfied. Meanwhile, the formation of the pressure node at the pump-turbine in any mode affects the transmitting of hydraulic energy to mechanical energy and reduces its efficiency in that particular mode. In the case of a valve, the formation of the discharge node at its place makes the valve maneuver inoperative.
On the contrary, the position of anti-nodes is crucial for the design of pipes because pipes encounter harmful pressure oscillation at anti-nodes that can burst or collapse the pipe. Therefore, at the location of the surge tank and pump-turbine, pressure nodes are avoided as far as possible. Similarly, it is preferred that the valve installed location escapes discharge node, while the fatigue problems should be considered for the pipes at antinodes.
Figure 3e,f provide the variation of absolute impedance (|Z|) along the pipeline for various natural frequencies. The place of large impedance in the system indicates that a subtle fluctuation in the head or discharge at a particular frequency may render excess oscillation and make the vital component at that location vulnerable. It is concluded that for both pump-turbines, mode 2 and 5 were undesirable from the above graph of impedance, as shown in Figure 3. All this analysis imparts the significance of natural frequencies and mode shapes of the hydraulic system for both pump-turbines operating in RGR.

Impedance of Pump-Turbines
The influence of pump-turbines operating in the S-shaped region is ambiguous from mode shapes. The displacements of the mode shape diagram are dependent on the free variable value provided at the upper reservoir, which indicates that the amplitude of mode shape is arbitrary. Mode shape is not a head or discharge at a particular natural frequency; rather, it is merely a shape. Although the amplitude values of mode shape are nearly insignificant, the distribution is useful. The distribution value of parameters depends on the natural frequency and its rank; whether it is first, second, or third natural frequency and so on. The mode shape depends on the natural angular frequency, which does not change for the same mode, despite the change in operating points. This results in similar mode shapes of the system for RGR and SSR operating points, which is not useful.
The determination of instability is an integral part of the analysis of the hydraulic system. The system becomes unstable when the damping factor obtained is greater than zero and stable when it is negative. The physical meaning of the negative sign is that the resulting oscillation dampens with an increase in time, and the positive sign explains that the oscillation further grows as time proceeds. The impedance of the pump-turbine has a substantial effect on the stability of the hydraulic system. From Table 1 for OS 1 and OS 2 , it can be concluded that both pump-turbines with positive impedance do not bring any instability in the system. Nevertheless, the conditions with negative impedance should be tackled with a special technique. There were two pump-turbines placed at each branch in a parallel pipe system and the whole hydraulic system was analyzed for two categories, i.e., both pump-turbines operating in the same region and each pump-turbine operating at different regions.

Both Pump-Turbines Operating in Same Region
Two possible cases were discovered for this category: the first was both pump-turbines operating in RGR, and the other was both pump-turbines operating in SSR. The graph of the damping factor vs. impedance was plotted independently for each pump-turbine, which is shown in Figure 4. From Figure 4, it is seen that two pump-turbines displayed slight differences in the nature of polyline and point distribution for the same mode. There is a prominent role of the dissimilar flow characteristic curve of pump-turbines for the difference in nature. It is also evident that for a particular impedance of any pump-turbine, the magnitude of the damping factor is relatively higher for lower modes. The magnitude of the damping factor determines the rate at which oscillation grows or dampens exponentially. The relatively higher positive value of the damping factor in lower modes further supports the selection of the first five modes for instability analysis in this paper.
The slope at the operating point determines whether or not the pump-turbine is in the S-Shaped region. If the slope is negative, the pump-turbine operates in RGR; otherwise, with a positive slope, the pump-turbine operates in SSR. The damping factor of the system is always negative for both pump-turbines operating in RGR. The pump-turbines with negative impedance, operating in SSR, induce a positive damping factor in the system. However, for OS5 and OS6 in Tables 1 and 2, the guide vane opening of pump-turbines was small and both operated in the S-shaped region with a positive impedance, resulting in a negative damping factor. This shows the significant influence of guide vane opening on the system stability and advocates that the effect of S-shaped characteristics on the system is undermined with an increase in head loss at the guide vane by reducing the opening area. Table 2 shows that for mode 2 and 5, although pump-turbines were operated in the S-shaped region, the damping factor was negative for all extracted operating points. It suggests that the system was stable, regardless of operating points and the region in which they resided. The damping factors of mode 2 and 5 are not plotted in Figure 4 because the system was stable and magnitudes were negligible compared to other modes. Therefore, instability of the system was determined with the remaining modes. Figure 4, and Tables 1 and 2 suggest that the self-excited oscillation was induced in mode 1, 3, and 4 when both pump-turbines exhibited negative impedance in the same operation scenario.

Pump-turbines Operating in Different Regions
Pump-turbine I operating in SSR while pump-turbine II in RGR, and pump-turbine I operating in RGR while pump-turbine II operating in SSR were the two possible conditions. The necessary operation scenarios were produced by extracting numerous operating points from the flow characteristic curves. The graphs shown in Figure 5 demonstrate damping factor vs impedance for pump-turbines in this category. From Figure 4, it is seen that two pump-turbines displayed slight differences in the nature of polyline and point distribution for the same mode. There is a prominent role of the dissimilar flow characteristic curve of pump-turbines for the difference in nature. It is also evident that for a particular impedance of any pump-turbine, the magnitude of the damping factor is relatively higher for lower modes. The magnitude of the damping factor determines the rate at which oscillation grows or dampens exponentially. The relatively higher positive value of the damping factor in lower modes further supports the selection of the first five modes for instability analysis in this paper.
The slope at the operating point determines whether or not the pump-turbine is in the S-Shaped region. If the slope is negative, the pump-turbine operates in RGR; otherwise, with a positive slope, the pump-turbine operates in SSR. The damping factor of the system is always negative for both pump-turbines operating in RGR. The pump-turbines with negative impedance, operating in SSR, induce a positive damping factor in the system. However, for OS 5 and OS 6 in Tables 1 and 2, the guide vane opening of pump-turbines was small and both operated in the S-shaped region with a positive impedance, resulting in a negative damping factor. This shows the significant influence of guide vane opening on the system stability and advocates that the effect of S-shaped characteristics on the system is undermined with an increase in head loss at the guide vane by reducing the opening area. Table 2 shows that for mode 2 and 5, although pump-turbines were operated in the S-shaped region, the damping factor was negative for all extracted operating points. It suggests that the system was stable, regardless of operating points and the region in which they resided. The damping factors of mode 2 and 5 are not plotted in Figure 4 because the system was stable and magnitudes were negligible compared to other modes. Therefore, instability of the system was determined with the remaining modes. Figure 4, and Tables 1 and 2 suggest that the self-excited oscillation was induced in mode 1, 3, and 4 when both pump-turbines exhibited negative impedance in the same operation scenario.

Pump-Turbines Operating in Different Regions
Pump-turbine I operating in SSR while pump-turbine II in RGR, and pump-turbine I operating in RGR while pump-turbine II operating in SSR were the two possible conditions. The necessary operation scenarios were produced by extracting numerous operating points from the flow characteristic curves. The graphs shown in Figure 5 demonstrate damping factor vs. impedance for pump-turbines in this category. It is seen in Figure 5, and Tables 1 and 2 that negative impedance produces a positive damping factor. Also, the magnitude of the damping factor is comparatively higher for lower modes. The results of OS8 and OS7 imply that when pump-turbine I got a negative impedance, mode 3 and 4 contributed to instability, while the rest of the modes were stable. Similarly, the results of OS10 and OS11 imply that when pump-turbine II got negative impedance, only mode 1 contributed to instability. Each pump-turbine produced instability at different modes of oscillation because these are two different pump-turbines as they have dissimilar characteristics curves. Also, it is evident from the results of OS9 and OS12 that guide vane opening affects the stability of the hydraulic system.

System Instability
The plot of the head and discharge perturbation at pump-turbines against time provides clear perception about hydraulic oscillation and instability. In OS3, both pump-turbines were operated in SSR and exhibited negative impedance, which generates a positive damping factor. The expression for head and discharge oscillation in the time domain is given by where, φ1 and φ2 are phase angles of H(x) and Q(x), respectively. The difference of φ1 and φ2 gives the phase angle between h' and q'. In Equations (48) and (49), only the real part of oscillatory head and discharge was taken because the real part defines the physical system behavior. The main interest of the analysis was the nature of oscillation with time, but not the exact value of the oscillation amplitude at pump-turbines. So, small arbitrary real values for head and discharge at pump-turbines were considered initially. With operating scenario OS3, head and discharge fluctuation values at pump-turbines for a short interval of time in the first five modes are presented in Table 3. It is seen in Figure 5, and Tables 1 and 2 that negative impedance produces a positive damping factor. Also, the magnitude of the damping factor is comparatively higher for lower modes. The results of OS 8 and OS 7 imply that when pump-turbine I got a negative impedance, mode 3 and 4 contributed to instability, while the rest of the modes were stable. Similarly, the results of OS 10 and OS 11 imply that when pump-turbine II got negative impedance, only mode 1 contributed to instability. Each pump-turbine produced instability at different modes of oscillation because these are two different pump-turbines as they have dissimilar characteristics curves. Also, it is evident from the results of OS 9 and OS 12 that guide vane opening affects the stability of the hydraulic system.

System Instability
The plot of the head and discharge perturbation at pump-turbines against time provides clear perception about hydraulic oscillation and instability. In OS 3 , both pump-turbines were operated in SSR and exhibited negative impedance, which generates a positive damping factor. The expression for head and discharge oscillation in the time domain is given by where, ϕ 1 and ϕ 2 are phase angles of H(x) and Q(x), respectively. The difference of ϕ 1 and ϕ 2 gives the phase angle between h' and q'.
In Equations (48) and (49), only the real part of oscillatory head and discharge was taken because the real part defines the physical system behavior. The main interest of the analysis was the nature of oscillation with time, but not the exact value of the oscillation amplitude at pump-turbines. So, small arbitrary real values for head and discharge at pump-turbines were considered initially. With operating scenario OS 3 , head and discharge fluctuation values at pump-turbines for a short interval of time in the first five modes are presented in Table 3.   Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be   Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be   Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be   Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be   Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be   Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be   Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be   Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be   Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be   Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be   Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be   Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be  Eventually, from Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be  Eventually, from Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be  Eventually, from Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be  Eventually, from Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be  Eventually, from Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be  Eventually, from Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be  Eventually, from Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be  Eventually, from Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be  Table 3, it is concluded that the system gets self-excited when it oscillates with a positive damping factor. The amplitude growth of self-excited oscillation is observed to be relatively greater in lower modes. The amplitude of oscillation with a negative damping factor gradually decays with time. This type of self-excited oscillation was also analytically proposed by Zhou et al. [35] for the pumped storage plant when pump turbines operate in the S-shaped region.
The system holds a positive damping factor when either both or any one of the pump-turbines exhibits negative impedance that is possible only for pump-turbine operating in SSR. The self-excited oscillation was induced in mode 1, 3, and 4 when both pump-turbines exhibited negative impedance in the same operation scenario. Also, the self-excited oscillation was induced in mode 3 and 4 when only pump-turbine I exhibited negative impedance. Besides, self-excited oscillation was induced in mode 1 when only pump-turbine II exhibited negative impedance. Thus, the different pump-turbines installed at separate branches individually produce instability at different modes of oscillation. Between two different pump-turbines, pump-turbine II might be critical because pump-turbine II exhibited a positive damping factor in mode 1 which has a smaller angular frequency, lower energy state, and greater value of the damping factor compared to higher harmonics.
The establishment of a system instability criterion was considered with at least any one of pump-turbines operating in the S-shaped region because this was taken as the most probable adverse case for the hydraulic system. From the above analysis, it is concluded that the damping factor becomes positive in either one of the first five natural frequencies if any one of the pump-turbines operating in the S-shaped region produces negative impedance. Now, when any pump-turbine operates in SSR, the condition that produces instability is Z T < 0 (50) Equation (52) shows the adequate criterion for a particular pump-turbine operating in SSR to produce instability in the system. It means the negative impedance value obtained from the slope of the characteristic curve is canceled by the impedance due to guide vane opening, which occurs as the guide vane opening gets smaller. As a result, the system can have positive impedance followed by a negative damping factor, even though any one of the pump-turbines is operating in the S-shaped region and instability induced is nullified. It also advocates the provision of the throttle or valve [23] just before the pump-turbine, which renders additional positive impedance to overcome the negative impedance and countermeasures the effect of S-shaped instability in the system. The guide vane maneuver resembles the throttle within the pump-turbine itself. It also imparts stability to the system, thereby diminishing the effect prompted by S-shaped characteristics. In addition, the guide vane impedance makes the system immune to instability. It is the intrinsic property of the pump-turbine and cannot be discarded. Apart from this, it is clear that the larger guide vane angles have less guide vane impedance and this impedance becomes smaller as the guide vane opening increases. So, the second term of Equation (51) Equation (55) gives the condition for instability of a hydraulic system with a pump-turbine when guide vane opening is large, and corresponding impedance becomes insignificant.

Sensitivity Analysis
The results of the system taken for simulation might have some uncertainty because of model simplification and the input of various parameters. The sensitivity analysis is necessary to clarify the effects of some important parameters on hydraulic characteristics of the system and confirm the obtained results. Sensitivity analysis was carried out for OS 2 and OS 3 with reasonable change in the roughness coefficient and length of each pipe of the system. These considerations generated four different cases for the analysis. For SA R − , the roughness coefficient of all pipes in the system was reduced by 0.002, and for SA R + , it was increased by 0.002. For SA L − , the length of all pipes in the system was reduced by 5.00 m, and for SA L + , it was increased by 5.00 m. The total length of the main conduit was changed by ±110.00 m. These various cases which were established for analysis are presented in Table 4. In Table 4, SA 0 represents the output of OS 2 and OS 3 from Table 2 for comparison with the results of the above four cases. From Table 4, it is concluded that with a reasonable increment in the roughness coefficient and length of pipes, there is a slight decrement in damping factors for OS 2 and OS 3 . It indicates that the stability of the system has increased slightly because the increment in these parameters contributes to producing additional head loss, which is beneficial for the damping of the undesirable oscillation in the system. The sensitivity analysis advocates the reliability of the results concluded from free oscillation analysis.
In addition, the hydraulic system in Figure 2 is modified to the equivalent system consisting of eight different pipes. The various pipes in series (1-6), (6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16), (17,19,21), (18,20,22), (23,25,27,29,31), (24,26,28,30,32), (33,34), and (35-38) of the original system shown in Figure 2 were substituted by their respective single equivalent pipes [30]. The sensitivity analysis was carried out with the equivalent system for OS 2 and OS 3 . The change in the pipe roughness coefficient and length by 0.002 and 5.00 m, respectively, produced four different cases. For SE R − , the roughness coefficient of equivalent pipes was reduced by 0.002, and for SE R + , it was increased by 0.002. For SE L − , the length of each original pipe was reduced by 5.00 m and substituted by respective equivalent pipes, while for SA L + , it was increased by 5.00 m and substituted by respective equivalent pipes. The change in damping factors for OS 2 and OS 3 is shown in Table 5. SE 0 represents the results of the equivalent system for OS 2 and OS 3 without change in parameters. The results concluded from Table 5 confirm that the variation of damping factors with respect to pipe parameters is similar to the results of Table 4. It is found that there is a slight deviation in values of damping factors obtained from the original system and equivalent system. Therefore, the sensitivity analysis of the equivalent system also supports the reliability of the results from hydraulic oscillation analysis.

Frequency Response Spectrum
The frequency response analysis was carried out for various frequencies of an oscillating guide vane as a forcing function for OS 3 and OS 4 , where both pump-turbines operate in SSR with a constant rated speed. The motion of the guide vane at pump-turbine I is given by The resultant discharge oscillation at the downstream reservoir due to the oscillating guide vane was obtained using Equation (47). The pressure oscillation at pump-turbines was obtained from the discharge oscillation at the downstream reservoir by applying the inverse of the transfer and point matrix to calculate the upstream state vectors from the downstream known section. For various values of angular frequency, the system response was plotted for pump-turbines operating in the S-shaped region, which is shown in Figure 6. It is apparent that when the forcing frequency, which produces oscillation, is equal to the natural frequency of the system, the amplitude peak appears. The values of the resonating angular frequencies are consistent with the natural angular frequencies of free oscillation analysis. This is the general characteristic of any hydraulic system and the result of frequency response analysis advocated the outcomes associated with free oscillation analysis. It is also suggested that oscillation of any component with the natural frequency of the hydraulic system should be avoided as far as possible; otherwise, the system suffers severe damage from hydraulic resonance. This frequency response spectrum renders the resultant feedback of the system when it is forced to oscillate with the known external excitation.

Conclusion
Hydraulic oscillation is a primary aspect of a pumped storage plant which should be properly acknowledged for effective and reliable operation. For a typical and commonly used layout of the pumped-storage plant, the features of the entire system, when two different pump-turbines work in turbine operation, were studied considering hydraulic oscillation analysis. All the analysis was carried out at a constant rated speed, and the effect of the guide vane mechanism, which has been overlooked in prior research, was taken to be equivalent to the valve mechanism. From the above analysis, the following conclusions are drawn: 1. Guide vane impedance was incorporated into the impedance of the pump turbine, and the instability criterion was modified for hydraulic oscillation analysis. The increase in guide vane impedance makes the system immune to instability. The sensitivity analysis was carried out, which supports the reliability of results from the model considered; 2. In free oscillation analysis, less efficient positions for various hydraulic components, such as the surge tank, valve, and pump-turbine, are analyzed based on the location of nodes in the mode shapes. Similarly, the pipe susceptibility at anti-nodes is clearly explained. In addition, when both pump turbines operated at RGR, mode 2 and mode 5 were comparatively vulnerable for pump turbines; 3. When two different pump-turbines with the same dimensions but dissimilar characteristic curves operate in the S-shaped region, individually, they trigger self-excited oscillation at different modes that is clear from the system instability analysis; 4. With the oscillating guide vane of a pump-turbine in the main conduit as external excitation, the frequency response analysis was conducted, which provided fundamental information about It is apparent that when the forcing frequency, which produces oscillation, is equal to the natural frequency of the system, the amplitude peak appears. The values of the resonating angular frequencies are consistent with the natural angular frequencies of free oscillation analysis. This is the general characteristic of any hydraulic system and the result of frequency response analysis advocated the outcomes associated with free oscillation analysis. It is also suggested that oscillation of any component with the natural frequency of the hydraulic system should be avoided as far as possible; otherwise, the system suffers severe damage from hydraulic resonance. This frequency response spectrum renders the resultant feedback of the system when it is forced to oscillate with the known external excitation.

Conclusions
Hydraulic oscillation is a primary aspect of a pumped storage plant which should be properly acknowledged for effective and reliable operation. For a typical and commonly used layout of the pumped-storage plant, the features of the entire system, when two different pump-turbines work in turbine operation, were studied considering hydraulic oscillation analysis. All the analysis was carried out at a constant rated speed, and the effect of the guide vane mechanism, which has been overlooked in prior research, was taken to be equivalent to the valve mechanism. From the above analysis, the following conclusions are drawn:

1.
Guide vane impedance was incorporated into the impedance of the pump turbine, and the instability criterion was modified for hydraulic oscillation analysis. The increase in guide vane impedance makes the system immune to instability. The sensitivity analysis was carried out, which supports the reliability of results from the model considered; 2.
In free oscillation analysis, less efficient positions for various hydraulic components, such as the surge tank, valve, and pump-turbine, are analyzed based on the location of nodes in the mode shapes. Similarly, the pipe susceptibility at anti-nodes is clearly explained. In addition, when both pump turbines operated at RGR, mode 2 and mode 5 were comparatively vulnerable for pump turbines;

3.
When two different pump-turbines with the same dimensions but dissimilar characteristic curves operate in the S-shaped region, individually, they trigger self-excited oscillation at different modes that is clear from the system instability analysis; 4.
With the oscillating guide vane of a pump-turbine in the main conduit as external excitation, the frequency response analysis was conducted, which provided fundamental information about resonating frequencies and resonance. The resonating frequencies obtained from the response spectrum coincide with natural frequencies and advocated results of free oscillation.
This entire analysis can be implemented as a preliminary investigation for the feasibility and design of the pumped storage plant. This research recommends future work on the influence of guide vanes on dynamic characteristics of pumped storage plants. n is the manning roughness coefficient and ξ turb is the local resistance coefficient of pipes for turbine operation.