Mechanistic Model of an Air Cushion Surge Tank for Hydro Power Plants

Abstract

Due to the increasing use of renewable energy sources, and to counter the effects of fossil fuels, renewable dispatchable hydro power can be used for balancing load and generation from intermittent sources (solar and wind). During higher percentage change in load acceptance or rejection in the intermittent grid, the operations of surge tanks are crucial in terms of water mass oscillation and water hammer pressure, and to avoid wear and tear in actuators and other equipment, such as hydro turbines. Surge tanks are broadly classified as open types, with access to open air, and closed types, with a closed volume of pressurized air. Closed surge tanks are considered to have a more flexible operation in terms of suppressing water mass oscillation and water hammer pressure. In this paper, a mechanistic model of an air cushion surge tank (ACST) for hydro power plants is developed based on the ordinary differential equations (ODEs) for mass and momentum balances. The developed mechanistic model of the ACST is a feature extension to an existing open-source hydro power library—OpenHPL. The developed model is validated with experimental data from the Torpa hydro power plant (HPP) in Norway. Results show that the air friction inside the ACST is negligible as compared to the water friction. The results also indicate that a hydro power plant with an ACST is a potential candidate as a flexible hydro power in an interconnected power system grid supplied with intermittent energy sources. Conclusions are drawn based on the simulation results from hydraulic performance of the ACST.

1. Introduction

1.1. Background

Electricity generation from renewable energy is increasing because of oil insecurity, climatic concern, the nuclear power debate, and carbon emission prices. In a growing trend of renewable energy, today’s power systems are a combination of intermittent and dispatchable renewable sources in a common interconnected grid. Intermittent sources include sources like solar power plants and wind power plants, whose variability can be balanced using a dispatchable renewable source like a hydro power plant, as discussed in [1,2]. In an interconnected power grid with both intermittent and dispatchable sources, a sudden loss in generation from the intermittent sources, for example, shadowing a large number of solar panels as in the case of solar power plants, a shutdown of the wind generators for unacceptable wind velocity as in the case of wind power plants, hydro power plants must be able to operate with a higher percentage of load acceptance to cope with the loss in generation, and to protect the power grid from a blackout. Similarly, when there is a sudden increase in production from the intermittent generation, hydro power plants must be able to operate with a higher percentage of load rejection to cope with grid instability and blackout. This indicates the need for flexible operation of dispatchable hydro power plants. In [3,4], the concept of flexible hydro power is coined for the interconnected power grid. Similarly, in [5] cascaded hydro power plants are considered as one of the candidates for flexible hydro power plants. In relation to the concept of flexible hydro power, hydro power plants with open surge tanks are relatively less able to tackle a higher percentage of load acceptance and rejection. However, power plants with ACST are more likely to tackle a higher percentage of load acceptance and rejection as ACST can be placed very near to the turbine. Hydraulic behavior of the open surge tanks studied in [6] outlines their operational limits in terms of their design heights and water hammer effects. As the percentage of load acceptance and rejection increases in the case of the open surge tanks, water mass oscillation inside the surge tanks may exceed the maximum allowed height and the operational limit of the power plant equipment due to an excessive water hammer effect. Similarly, in [7,8] the benefits of ACST with respect to open surge tanks are given.

In this regard, it is of interest to study the hydraulic behavior of an ACST (closed surge tank) with respect to open surge tanks. A simple mechanistic model of an ACST was developed and studied previously in [9] as a feature extension to an open-source hydro power library—OpenHPL. OpenHPL is based on an equation-based language—Modelica. OpenHPL is under development at the University of South-Eastern Norway. This paper primarily focuses on the model improvements from [9], validation of the improved model with experimental data from [10], and hydraulic behavior of an ACST in relation to flexible hydro power plants.

1.2. Previous Work and Contributions

The model of hydraulic transients inside the surge tank is a well-established theory using Newton’s second law [11,12]. The use of hydraulic resistances in the inlet of the surge tank helps to reduce water hammer effects. Different types of surge tanks designed with respect to the hydraulic resistances are presented in [13]. The time evolution equations for developing a mechanistic model of the surge tank are given in [14]. The hydraulic resistance at the inlet of different kinds of surge tanks can be studied from [14,15]. Closed surge tanks or ACST are important in terms of suppressing water mass oscillation due to the cushioning of air during hydraulic transients [16]. A hydraulic scale model of an ACST was studied in [10] based on 1D mass and momentum balances. In [17], a simulation study was carried out considering 1D mass and momentum equations for both water and air inside the ACST. In the paper, it is shown that the mass and momentum balances for air inside the ACST can be further simplified with an ideal gas relation. Other studies include the gas seepage theory for air loss through the ACST chamber in [18], a monitoring method for the hydraulic behavior of the ACST in [19], stability analysis of the ACST in [20], etc. The model developed in most of the previous work assumes an adiabatic process for the cushioning of air inside the ACST. The polytropic constant for air $\gamma$ is considered around $1.4$ for almost all the models of the ACST. However, previous work lacks modeling of the ACST with a possible consideration of friction due to air flow inside the ACST during its operation. The following research contributions are provided in this paper:

• a mechanistic model of an ACST, and
• a comparison between the ACST models with and without air friction.

1.3. Outline

Section 2 provides a mechanistic model of an ACST based on mass and momentum balances. In Section 3, model fitting and simulation results are outlined through a case study of the ACST used in Torpa Hydro Power Plant (HPP). Section 4 provides conclusions and future work.

2. Mechanistic Model of ACST

A general schematic and a flow diagram of an ACST is shown in Figure 1. The free water surface inside the surge tank is filled with pressurized air. Figure 1a shows the general schematic of an ACST where the water with volumetric flow rate $\dot{V}$ flows towards the air chamber through the access tunnel with length $L_{\mathrm{t}}$ and diameter $D_{\mathrm{t}}$. The intake-penstock manifold pressure at the bottom of the tank is represented by $p_{\mathrm{m}}$, and the air pressure at the air chamber due to the cushioning of the air is represented by $p_{\mathrm{c}}$. The diameter of the air chamber is D. H is the total height of the surge tank and L is the total vertical slant length of the surge tank. In the figure, h represents the water level inside the tank during the operation of the ACST, and the dotted line in Figure 1a indicates that h is a variable quantity. Figure 1b shows a flow diagram inside the surge tank where $F_{\mathrm{f}}$ is the fluid friction against $\dot{V}$, $F_{\mathrm{g}}$ is the force due to gravity in the downward direction, and $F_{\mathrm{g}}^{\dot{V}}$ is the projection of $F_{\mathrm{g}}$ in the alignment of the flow.

Models developed in OpenHPL are based on a semi-explicit DAE formulation with a differential equation for the mass and the momentum balances as described in [21] and given by

$$\begin{array}{cc}\hfill \frac{dm}{dt}& =\dot{m}\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \frac{d\mathcal{M}}{dt}& =\dot{\mathcal{M}}+F\hfill \end{array}$$

(2)

where $\dot{m}$ and $\dot{\mathcal{M}}$ represent the mass flow rate and the momentum flow rate, respectively.

Equations (1) and (2) are expressed with a series of algebraic equations as

$$\begin{array}{c}\dot{m}=\rho \dot{V}\end{array}$$

(3)

$$\begin{array}{c}\mathcal{M}=mv\end{array}$$

(4)

$$\begin{array}{c}\dot{\mathcal{M}}=\dot{m}v\end{array}$$

(5)

$$\begin{array}{c}F=F_{\mathrm{p}}-F_{\mathrm{g}}^{\dot{V}}-F_{\mathrm{f}}\end{array}$$

(6)

where $\rho$ is the density of the water, m is the mass of air and water inside the ACST, v is the average velocity of the flow, V is the volume of the ACST, F is the total force acting in the surge tank, $F_{\mathrm{p}}$ is the pressure force, and $F_{\mathrm{f}}$ is the fluid frictional force. The expressions for all the variables are given in the sequel. A general idea regarding mathematical formulations of these variables is taken from [9].

The total mass inside the surge tank is expressed as

$$m=m_{\mathrm{w}}+m_{\mathrm{a}}$$

(7)

where $m_{\mathrm{w}}$ and $m_{\mathrm{a}}$ are the masses of the water and the air inside the surge tank, respectively. $m_{\mathrm{a}}$ is constant inside the chamber and is determined based on the initial air cushion pressure $p_{\mathrm{c}0}$ which is considered to be a design parameter for the hydraulic performance of the surge tank. If $h_{\mathrm{c}0}$ is the initial water level inside the surge tank for the initial air cushion pressure $p_{\mathrm{c}0}$, then the expression for the mass of the air inside the surge tank is found from an adiabatic compression and rarefaction of the air inside the surge tank during operation. It is found that for an ACST with a larger diameter, the heat transfer between air and water, air to the walls of the ACST, etc., can be neglected, and an adiabatic process of compression and rarefaction of the air inside the ACST can be assumed [16]. For an adiabatic process with pressure p, volume V, and $\gamma$ of the air inside the ACST, considering standard temperature and pressure (STP), the relation $p{V}^{\gamma }=\mathrm{constant}$ is assumed where $\gamma$ is the ratio of specific heats at constant pressure and at constant volume. The mass of the air is then calculated formulating an ideal gas relation with the initial air pressure $p_{\mathrm{c}0}$ and the initial volume $A\left(L-h_{\mathrm{c}0}\frac{L}{H}\right)$ given by

$$m_{\mathrm{a}}=\frac{p_{\mathrm{c}0}A\left(L-h_{\mathrm{c}0}\frac{L}{H}\right)M_{\mathrm{a}}}{R{T}^{\circ }}$$

(8)

where $M_{\mathrm{a}}$ is the molar mass of air, R is the universal gas constant and $T^{\circ }$ is the temperature taken at STP. Similarly, A is the area of the air chamber expressed as $A=\pi \frac{D^2}{4}.$

From Equation (2) formulating $p_{c0}{V}_0^{\gamma }=p_c{V}^{\gamma }$, the air cushion pressure during the operation of the surge tank is given by

$$p_{\mathrm{c}}=p_{\mathrm{c}0}{\left(\frac{L-h_{\mathrm{c}0}\frac{L}{H}}{L-\ell }\right)}^{\gamma }$$

(9)

where $p_{\mathrm{c}}$ depends on the length ℓ inside the ACST.

During the operation of the surge tank, the mass of the water inside the surge tank $m_{\mathrm{w}}$ varies according to the variation in h. Thus, the expression for $m_{\mathrm{w}}$ is formulated considering two different scenarios inside the surge tank based on the variation of the water level h. First we consider (i) $h\le H_{\mathrm{t}}$ and second we consider (ii) $h>H_{\mathrm{t}}$. Furthermore, we also formulate expressions for $F_{\mathrm{p}}$ and $F_{\mathrm{f}}$ for both of the scenarios of the water level h.

2.1. Case $h\le H_{\mathrm{t}}$

When the water level is up to the tip of the access tunnel or below the tip of the access tunnel, $m_{\mathrm{w}}$ is given by $m_{\mathrm{w}}=\rho A_{\mathrm{t}}\ell$ where ℓ is the slant height for h as shown in Figure 1a. $m_{\mathrm{w}}$ is further expressed as

$$m_{\mathrm{w}}=\rho A_{\mathrm{t}}h\frac{L}{H}.$$

(10)

The pressure force $F_{\mathrm{p}}$ is formulated based on the pressure difference at the manifold and the air pressure with an expression

$$F_{\mathrm{p}}=\left(p_{\mathrm{m}}-p_{\mathrm{c}}\right)A_{\mathrm{t}}.$$

(11)

The frictional force $F_{\mathrm{f}}$ is expressed as

$$\begin{array}{cc}\hfill F_{\mathrm{f}}& =F_{\mathrm{D},\mathrm{w}}+F_{\mathrm{D},\mathrm{a}}\hfill \end{array}$$

(12)

where $F_{\mathrm{D},\mathrm{w}}$ is the frictional force formulated for water flow inside the surge tank based on Darcy’s friction factor for water, $f_{\mathrm{D},\mathrm{w}}$. Similarly, $F_{\mathrm{D},\mathrm{a}}$ is the frictional force formulated for air flow inside the surge tank based on Darcy’s friction factor for air, $f_{\mathrm{D},\mathrm{a}}$. Both $f_{\mathrm{D},\mathrm{w}}$ and $f_{\mathrm{D},\mathrm{a}}$ are calculated as in [9]. The general expression for Darcy’s friction factor $f_{\mathrm{D}}$ is based on Reynolds’ number $N_{\mathrm{Re}}=\frac{\rho \mid v\mid D}{\mu }$ and expressed as

$$f_D=\left\{\begin{array}{cc}\frac{64}{N_{\mathrm{Re}}}\hfill & N_{\mathrm{Re}}<2100\hfill \\ a{N}_{\mathrm{Re}}^3+b{N}_{\mathrm{Re}}^2+c{N}_{\mathrm{Re}}+d\hfill & 2100\le N_{\mathrm{Re}}\le 2300\hfill \\ \frac{1}{{\left(2{log}_{10}\left(\frac{\epsilon }{3.7D}+\frac{5.7}{N_{\mathrm{Re}}^{0.9}}\right)\right)}^2}\hfill & N_{\mathrm{Re}}>2300\hfill \end{array}\right.$$

where $\mu$ is the dynamic viscosity of the fluid, $\epsilon$ is the pipe roughness height. For the region $2100\le N_{\mathrm{Re}}\le 2300$, $f_{\mathrm{D}}$ is calculated from a cubic interpolation, with the coefficients $a,b,c,$ and d, differentiable at the boundaries. The final expression for $F_{\mathrm{f}}$ is calculated as in [9] given as

$$F_{\mathrm{f}}=\frac{1}{2}\rho v\mid v\mid \left(A_{\mathrm{w},\mathrm{w}}\frac{f_{\mathrm{D},\mathrm{w}}}{4}+A_{\mathrm{w},\mathrm{a}}\frac{f_{\mathrm{D},\mathrm{a}}}{4}\right)$$

(13)

where $\mid v\mid$ preserves the fluid frictional force against both directions of flow; flow induced from the access tunnel towards the air chamber, and vice-versa. $A_{\mathrm{w},\mathrm{w}}$ is the wetted area due to water flow inside the surge tank given by

$$A_{\mathrm{w},\mathrm{w}}=\pi D_{\mathrm{t}}\ell$$

(14)

and $A_{\mathrm{w},\mathrm{a}}$ is the wetted area due to the air during adiabatic compression and rarefaction inside the surge tank, and expressed as

$$A_{\mathrm{w},\mathrm{a}}=\pi \left[D\left(L-L_{\mathrm{t}}\right)+D_{\mathrm{t}}\left(L_{\mathrm{t}}-\ell \right)\right].$$

(15)

2.2. Case $h>H_{\mathrm{t}}$

When the water level inside the surge tank is above the access tunnel expression for $m_{\mathrm{w}}$ is formulated by summing the mass of water inside the access tunnel and the mass of water inside the air chamber, and is expressed as

$$m_{\mathrm{w}}=\rho \left[A_{\mathrm{t}}{L}_{\mathrm{t}}+A\left(\ell -L_{\mathrm{t}}\right)\right].$$

(16)

For $\ell >L_{\mathrm{t}}$ we consider Figure 2 for finding the total pressure force $F_{\mathrm{p}}$ in the direction of the flow. The calculation of the fluid frictional force is given in Figure 3. From Figure 2, the pressure force $F_{\mathrm{p}}$ is calculated based on the junction pressure $p_{\mathrm{j}}$ between the junction of the access tunnel and the air chamber. $p_{\mathrm{j}}$ is expressed as the sum of the air pressure $p_{\mathrm{c}}$ and the hydrostatic pressure due to the difference in liquid-level $h-H_{\mathrm{t}}$. The junction pressure is then expressed as

$$p_{\mathrm{j}}=p_{\mathrm{c}}+\rho g\left(\ell -L_{\mathrm{t}}\right)\frac{H}{L}$$

(17)

which relates in the final expression for $F_{\mathrm{f}}$ as

$$F_{\mathrm{p}}=\left(p_{\mathrm{m}}-p_{\mathrm{j}}\right)A_{\mathrm{t}}+\left(p_{\mathrm{j}}-p_{\mathrm{c}}\right)A.$$

(18)

From Figure 2, the overall fluid frictional force $F_{\mathrm{f}}$ is calculated with an expression given as

$$F_{\mathrm{f}}=F_{\mathrm{D},\mathrm{w}}+F_{\varphi }+F_{\mathrm{D},\mathrm{a}}$$

(19)

where $F_{\mathrm{D},\mathrm{w}}+F_{\mathrm{D},\mathrm{a}}$ is given as

$$F_{\mathrm{D},\mathrm{w}}+F_{\mathrm{D},\mathrm{a}}=\frac{1}{2}\rho v\mid v\mid \left(A_{\mathrm{w},\mathrm{w}}\frac{f_{\mathrm{D},\mathrm{w}}}{4}+A_{\mathrm{w},\mathrm{a}}\frac{f_{\mathrm{D},\mathrm{a}}}{4}\right)$$

where $A_{\mathrm{w},\mathrm{w}}=\pi \left[D_{\mathrm{t}}{L}_{\mathrm{t}}+D\left(\ell -L_{\mathrm{t}}\right)\right]$ and $A_{\mathrm{w},\mathrm{a}}=\pi D\left(L-\ell \right)$; the calculations were similarly performed as in Equations (14) and (15).

In Equation (19), $F_{\varphi }$ is the fluid frictional force due to water flow from the access tunnel towards the air chamber, and vice-versa. $F_{\varphi }$ can be expressed in terms of the pressure drop (alternatively can be expressed in terms of the head loss). When the water is flowing from the access tunnel towards the air chamber, we consider the pressure drop due to the square expansion type of fitting as shown in Figure 3a, and when the water is flowing from the air chamber towards the access tunnel, we consider the pressure drop due to the square reduction type of fitting as shown in Figure 3b. Thus, $F_{\varphi }$ is calculated based on the generalized friction factors ${\varphi }_{\mathrm{se}}$ for the square expansion type of fitting and ${\varphi }_{\mathrm{sr}}$ for the square reduction type of fitting. Additionally, for both types of flows as shown in Figure 3, we assume an average cross-sectional area

$$\overline{A}=\frac{A+A_{\mathrm{t}}}{2}.$$

If $\Delta p_{\varphi }$ is the pressure drop due to the fittings, there exists a relationship between $\Delta p_{\varphi }$, the average kinetic energy of the fluid per volume $K^{‴}=\frac{1}{2}\rho v\mid v\mid$ and the friction factor $\varphi =\left\{{\varphi }_{\mathrm{se}},{\varphi }_{\mathrm{sr}}\right\}$. The relationship between $\Delta p_{\varphi }$, $K^{‴}$, and $\varphi$ is given by

$$\Delta p_{\varphi }=\varphi K^{‴}.$$

The pressure drop $\Delta p_{\varphi }$ is related to $F_{\varphi }$ through the average cross-sectional area $\overline{A}$ and given as

$$F_{\varphi }\approx \Delta p_{\varphi }\overline{A}$$

which can be further expressed as

$$F_{\varphi }\approx \frac{1}{2}\rho v\mid v\mid \overline{A}\varphi ,\varphi =\left\{{\varphi }_{\mathrm{se}},{\varphi }_{\mathrm{sr}}\right\}.$$

The final expression for overall fluid frictional force $F_{\mathrm{f}}$ is then given as

$$F_{\mathrm{f}}\approx \frac{1}{2}\rho v\mid v\mid \left(A_{\mathrm{w},\mathrm{w}}\frac{f_{\mathrm{D},\mathrm{w}}}{4}+A_{\mathrm{w},\mathrm{a}}\frac{f_{\mathrm{D},\mathrm{a}}}{4}+\overline{A}\varphi \right)\varphi =\left\{{\varphi }_{\mathrm{se}},{\varphi }_{\mathrm{sr}}\right\}.$$

(20)

This completes the expressions for variables m, $F_{\mathrm{p}}$ and $F_{\mathrm{f}}$ for the two scenarios of the liquid level inside the surge tank, viz., $h\le H_{\mathrm{t}}$ and $h>H_{\mathrm{t}}$. To further complete the information of variables in Equation (6), the expression for $F_{\mathrm{g}}^{\dot{V}}$ is calculated as

$$F_{\mathrm{g}}^{\dot{V}}=mg\frac{H}{L},$$

(21)

as shown in the flow diagram of Figure 1a. Finally, the mechanistic model of the ACST needs an expression for the average velocity v expressed as

$$v=\frac{\dot{V}}{\overline{A}}.$$

(22)

Equations (1)–(6), in addition to other associated algebraic relations from Equations (7)–(22), represent a semi-explicit DAEs formulation for the ACST, and can be modeled in a equation-based modeling language like Modelica. The developed mechanistic model of the ACST is implemented in OpenHPL as a feature extension, and the case study was carried out for Torpa HPP.

3. Case Study

Figure 4a shows the layout diagram of Torpa HPP. Similarly, Figure 4b shows the simulation model of Torpa HPP created in OpenHPL. In Figure 4b, the reservoir model, the intake tunnel model, the penstock model, and the discharge model are developed as in [21]. A detailed model of the penstock considering water compressibility and pipe elasticity can be formulated from [22]. However, we consider the penstock model as a simple pipe model. Similarly, the Francis turbine mechanistic model for the case study is modeled as in [23]. The mechanistic model for the tailrace is taken as an exact mirror replica of the reservoir model.

The dimensions of the ACST shown in Figure 4a are found based on the piezometric diagram for Torpa HPP from [10]. The model developed in Section 2 is based on a cylindrical access tunnel and a cylindrical air chamber. Thus, the hydraulic diameters for the access tunnel $D_{\mathrm{t}}$ and the air chamber D are evaluated based on the volume of air inside the chamber using the operating conditions.

Table 1. Parameters and operating conditions of the ACST for Torpa HPP.

Quantity	Symbol	Value
Hydraulic diameter of the throat	$D_{\mathrm{t}}$	$15\mathrm{m}$
Hydraulic diameter of the chamber	D	$24\mathrm{m}$
Length of the throat	$L_{\mathrm{t}}$	$29\mathrm{m}$
Total height	H	$50\mathrm{m}$
Total length	L	$58\mathrm{m}$
Pipe roughness height	$\epsilon$	$0.9\mathrm{mm}$
Total volume	−	$17·{10}^3{\mathrm{m}}^3$
Operating temperature	$T^{\circ }$	$293\mathrm{K}$
Adiabatic exponent for air at STP	$\gamma$	$1.4$
Molar mass of air at STP	$M_{\mathrm{a}}$	$29·{10}^{-3}\mathrm{kg}{\mathrm{mol}}^{-1}$
Universal gas constant	R	$8.314{\mathrm{JK}}^{-1}{\mathrm{mol}}^{-1}$
Initial pressure of air	$p_{\mathrm{c}}\left(0\right)=p_{\mathrm{c}0}$	$41·{10}^5\mathrm{Pa}$
Initial water level	$h\left(0\right)=h_{\mathrm{c}0}$	$27\mathrm{m}$
Initial volume of air	$V_0$	$13·{10}^3{\mathrm{m}}^3$

shows the parameters and the operating conditions of the ACST for Torpa HPP.

For the model created in Figure 4b, it is of interest to:

• validate the model with the experimental data from [10],
• simulate the model considering air friction inside the ACST, and
• study the hydraulic behavior of the ACST at different load acceptances and rejections.

3.1. Simulation Versus Real Measurements

Figure 5 shows the simulated versus real measurement for Torpa HPP. As shown in Figure 4b, $u_{\mathrm{v}1}$ and $u_{\mathrm{v}2}$ are the turbine valve signals for the turbine unit-1 and the turbine unit-2, respectively, for controlling the volumetric discharge through the turbines. The input turbine valve signal for unit-1 is given by

$$\begin{array}{cc}\hfill u_{\mathrm{v}1}& =\left\{\begin{array}{cc}0.68\hfill & 0<t\le 500\mathrm{s}\hfill \\ \frac{0.68}{50}\left(t-550\right)+0.98\hfill & 500\mathrm{s}<t\le 550\mathrm{s}\hfill \\ 0.98\hfill & 550\mathrm{s}<t\le 1200\mathrm{s}\hfill \end{array}\right.,\hfill \end{array}$$

and the input turbine valve signal for unit-2 is given by,

$$u_{\mathrm{v}2}=\left\{\begin{array}{cc}0.55\hfill & 0<t\le 500\mathrm{s}\hfill \\ \frac{0.55}{50}\left(t-550\right)+0.93\hfill & 500\mathrm{s}<t\le 550\mathrm{s}\hfill \\ 0.93\hfill & 550\mathrm{s}<t\le 1200\mathrm{s}.\hfill \end{array}\right.$$

For inputs $u_{\mathrm{v}1}$ and $u_{\mathrm{v}2}$, the mechanical power outputs for the turbine unit-1 (Figure 5c) and the turbine unit-2 (Figure 5d), the turbines inlet pressure $p_{\mathrm{tr}}$ (Figure 5e), and the air pressure inside the surge tank $p_{\mathrm{c}}$ (Figure 5f) are recorded for $1200\mathrm{s}$ with the measurement samples taken at each second. The air pressure $p_{\mathrm{c}}$ is measured using the pressure sensor PARO scientific 8DP000-S with an error of less than 0.01% of full scale of 6 Mpa, the turbine inlet pressure $p_{\mathrm{tr}}$ is measured using the pressure sensor PARO scientific DIQ 73K with an error of less than 0.04% of full scale of 20 Mpa, and the measurements for the mechanical power outputs are provided by the plant owner from Torpa HPP. The information about Torpa HPP and its experimental procedures are taken from [24]. Figure 5 shows that the simulation corresponds well with the real measurements in the case of power productions from the turbines (Figure 5c,d). In the case of the turbine inlet pressure $p_{\mathrm{tr}}$ (Figure 5e) there is an steady-state error of 0.6 bar for $0<t\le 500\mathrm{s}$. We believe that the steady-state error in $p_{\mathrm{tr}}$ for $0<t\le 500\mathrm{s}$ can be eradicated by the inclusion of detailed geometrical dimensions for the headrace tunnel. In this paper, the headrace tunnel is considerd with a simple slanted pipe geometry as shown in Figure 4a. Similar steady-state error can be seen in the case of the height of water level inside the ACST h (Figure 5g) with negligible error of $0.05\mathrm{m}$. In the case of air pressure inside the ACST $p_{\mathrm{c}}$, the simulation and the measurement data are in good agreement. The measurement sampling rate in the case of water level h, air pressure $p_{\mathrm{c}}$, and turbine power outputs are slower and oscillatory because the data are only recorded after a minimum change in the measured value, which may be the reason for the steady-state errors and phase difference between the simulation and measurements shown in Figure 5c,d,f,g. In addition, in Figure 5f,g for $800\mathrm{s}<t\le 1200\mathrm{s}$, the simulated values have poorly damped oscillation while the measurement quickly reaches a steady value. The simulated and the experimental dynamics of the variables ($p_{\mathrm{c}}$ and h) are not captured well because of the slower and oscillatory sampling rate of the sensors. The simulation and the real measurements are matched by manual tuning of pipe roughness height of the headrace tunnel ($\epsilon \approx 0.4\mathrm{mm}$), hydraulic diameter of the access tunnel $D_{\mathrm{t}}\approx 15\mathrm{m}$, and hydraulic diameter of the air chamber $D\approx 24\mathrm{m}$.

3.2. Effect of Air Friction Inside ACST

We now consider Torpa HPP with each of the turbine units rated at $75\mathrm{MW}$ as a single entity, for simplification, with $150\mathrm{MW}$ with input $u_{\mathrm{v}}$ as the turbine valve signal. This simplification is made for studying the hydraulic behavior of the ACST in terms of the air friction inside the ACST, and the operation of Torpa HPP with respect to load acceptance and rejection (Section 3.3). Only simulated results will be presented in the sequel.

The air friction force $F_{\mathrm{D},\mathrm{a}}$ modeled using Darcy’s friction factor $f_{\mathrm{D}}$ inside the ACST of Torpa HPP is considered using Equation (12) for the case of water level $h\le H_{\mathrm{t}}$, and using Equation (19) for the case of water level $h>H_{\mathrm{t}}$. The input to the turbine with valve signal $u_{\mathrm{v}}$ for the simulation purpose is given by

$$u_{\mathrm{v}}=\left\{\begin{array}{cc}0.5\hfill & 0<t\le 500\mathrm{s}\hfill \\ 0.95\hfill & 500\mathrm{s}<t\le 1500\mathrm{s}\hfill \end{array}\right.$$

where the hydro-turbine is loaded from half-load to nominal load at time $t=500\mathrm{s}$.

Figure 6 shows hydraulic behavior of the ACST for the turbine loading from 50% to 95%. Figure 6b–d show the water level h inside the ACST, the air cushion pressure $p_{\mathrm{c}}$, and the inlet turbine pressure $p_{\mathrm{tr}}$, respectively, for the ACST modeled with and without the air friction consideration. From Figure 6c, we see that the differences in air cushion pressure $p_{\mathrm{c}}$ for the ACST modeled with and without the air friction consideration is in the order of ${10}^{-5}\mathrm{bar}=1\mathrm{Pa}$, even for the turbine loaded from half load to the nominal operation. This is because of the fact that fluid frictional force $F_{\mathrm{f}}$ depends on Darcy’s friction factor $f_{\mathrm{D}}$, and $f_{\mathrm{D}}$ depends on Reynolds’ number $N_{\mathrm{Re}}=\frac{\rho \mid v\mid D}{\mu }$ where $\mu$ is the dynamic viscosity of the fluid. At STP, ${\mu }_{\mathrm{air}}=1.81·{10}^{-5}\mathrm{Pa}·\mathrm{s}$ and ${\mu }_{\mathrm{water}}=8.90·{10}^{-4}\mathrm{Pa}·\mathrm{s}$ which can be approximated as ${\mu }_{\mathrm{water}}\approx 100{\mu }_{\mathrm{air}}$.

3.3. Operations of ACST in Load Acceptance and Rejection

Load acceptance and rejection are created by changing the turbine valve signal $u_{\mathrm{v}}$ from one operating condition to another operating condition, and are described in the sequel.

3.3.1. Load Acceptances

We consider Torpa HPP running at no load condition for a time period of $500\mathrm{s}$. At $t=500\mathrm{s}$, a different load acceptance condition is created by changing the turbine valve signal $u_{\mathrm{v}}$, and the hydraulic behavior of the ACST is observed for the next $1500\mathrm{s}$. The turbine valve signal $u_{\mathrm{v}}$ is generated as

$$u_{\mathrm{v}}=\left\{\begin{array}{cc}0\hfill & 0<t\le 500\mathrm{s}\hfill \\ u_{\mathrm{va}}\hfill & 500\mathrm{s}<t\le 2000\mathrm{s}\hfill \end{array}\right.$$

where $u_{\mathrm{va}}\in \left\{0.25,0.5,0.75,1.0\right\}$ for load acceptances of $25\%$, $50\%$, $75\%$, and $100\%$, respectively. For a total load acceptance (TLA) the load acceptance is $100\%$.

3.3.2. Load Rejections

In contrast to the load acceptances, we now consider Torpa HPP running at full load condition for a time period of $500\mathrm{s}$. At $t=500\mathrm{s}$, a different load rejection condition is created by changing the turbine valve signal $u_{\mathrm{v}}$, and the hydraulic behavior of the ACST is observed for the next $1500\mathrm{s}$. The turbine valve signal $u_{\mathrm{v}}$ is generated as

$$u_{\mathrm{v}}=\left\{\begin{array}{cc}1.0\hfill & 0<t\le 500\mathrm{s}\hfill \\ u_{\mathrm{vr}}\hfill & 500\mathrm{s}<t\le 2000\mathrm{s}\hfill \end{array}\right.$$

where $u_{\mathrm{vr}}\in \left\{0.75,0.5,0.25,0.0\right\}$ for load rejections of $25\%$, $50\%$, $75\%$, and $100\%$, respectively. For a total load rejection (TLR), the load rejection is $100\%$.

Figure 7 shows hydraulic performance of the ACST during load acceptances and rejections for Torpa HPP. Figure 7a,c,e,g shows the turbine valve signal $u_{\mathrm{v}}$, the air pressure $p_{\mathrm{c}}$, the turbine inlet pressure $p_{\mathrm{tr}}$ and the water level inside ACST h, respectively, for the different percentage change in the load acceptances. Similarly, Figure 7b,d,f,h shows $u_{\mathrm{v}}$, $p_{\mathrm{c}}$, $p_{\mathrm{tr}}$ and h, respectively, for the different percentage change in the load rejections.

Figure 7a shows the turbine valve signal generated for load acceptances of $25\%$, $50\%$, $75\%$, and $100\%$. Figure 7c, at $t=500\mathrm{s}$, shows that from the no load operation to TLA, the difference in the air pressure $p_{\mathrm{c}}$ inside the ACST is around $4\mathrm{bar}$. Similarly, Figure 7e shows that the difference in turbine inlet pressure $p_{\mathrm{tr}}$ is around $3\mathrm{bar}$, and Figure 7e shows that the difference in the water level h inside the ACST is around $1\mathrm{m}$. In addition, Figure 7c shows that the difference in $p_{\mathrm{c}}$ from no load operation to $25\%$ load acceptance, $50\%$ load acceptance and $75\%$ load acceptance are around $1\mathrm{bar}$, $2\mathrm{bar}$ and $3\mathrm{bar}$, respectively. Similarly, results can be obtained for $p_{\mathrm{tr}}$ (Figure 7e) and h (Figure 7g). For $p_{\mathrm{c}}$, $p_{\mathrm{tr}}$ and h oscillation dies out as the time progresses for $t>500\mathrm{s}$.

Figure 7b shows the turbine valve signal generated for load rejections of $25\%$, $50\%$, $75\%$, and $100\%$. Figure 7d, at $t=500\mathrm{s}$, shows that from full load operation to TLR, the difference in $p_{\mathrm{c}}$ is around $4\mathrm{bar}$ as similar in the case of TLA. Similarly, the difference is around $3\mathrm{bar}$ in the case of $p_{\mathrm{tr}}$, as shown in Figure 7f. The difference in h from full load operation to TLR is also $1\mathrm{m}$, as in the case of TLA. Similarly, from Figure 7d, the difference in $p_{\mathrm{c}}$ from full load operation to load rejections of $25\%$, $50\%$ and $75\%$ are around $1\mathrm{bar}$, $2\mathrm{bar}$ and $3\mathrm{bar}$, respectively. Similar results can be obtained for $p_{\mathrm{tr}}$ (Figure 7f) and h (Figure 7h). For $p_{\mathrm{c}}$, $p_{\mathrm{tr}}$ and h, oscillation dies out for $t>500\mathrm{s}$, similar to the case of load acceptances. However, the oscillation dies out sooner in the case of TLA than TLR.

3.3.3. ACST as a Flexible Hydro Power

The results for Figure 7 show hydraulic behavior of the ACST in the case of load acceptance and rejection. The difference in the water level is around $1\mathrm{m}$ for both TLA and TLR. Similarly, the difference in the air pressure is around $4\mathrm{bar}$ for both TLA and TLR. Referring to the results on the hydraulic performance of the ACST from Section 3.3 and the study carried out for different types of open surge tanks in [6] clearly indicates that ACST has a robust performance on suppressing water mass oscillation and water hammer pressure during a higher percentage of load acceptances and rejections, unlike different types of open surge tanks. Since one of the prominent requirements of a flexible hydro power plant is to have a robust operation under various load acceptances and rejections, a hydro power plant operated with ACST makes it a potential candidate for participating in the concept of flexible hydro power.

4. Conclusions and Future Work

A mechanistic model of an ACST has been developed considering an access tunnel connected to an air chamber. The difference in diameters of the access tunnel and the air chamber has been taken into consideration. The model is further enhanced with the inclusion of Darcy’s friction force for air inside the ACST. Model fitting is done for the $150\mathrm{MW}$ Torpa HPP. The experimental data and the model simulation were matched by manual tuning of pipe roughness height of the headrace tunnel, and hydraulic diameters of the access tunnel and the air chamber of the ACST. Apart from the model fitting, simulation results show that the effect of air friction inside the ACST is negligible as compared to water friction. The simulation studies carried out for load acceptance and rejection show the robust hydraulic behaviors of the ACST in terms of suppressing water mass oscillation and water hammer pressure, which indicate that a hydro power plant with ACST makes it a potential candidate for flexible hydro power in case of an energy-mix (intermittent and dispatchable sources) interconnected power grid.

Future work includes the study of the hydraulic behavior of ACST in interconnected grids supplied with intermittent generation. In addition, the model for ACST can be improved using Lagrangian computational fluid dynamics. For the Lagrangian approach, the meshless discretization technique smoothed particle hydrodynamics (SPH) can be used to handle coupling between the free water surface and air inside the ACST [25,26].