*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

**Citation:** Pandey, M.; Winkler, D.; Vereide, K.; Sharma, R.; Lie, B. Mechanistic Model of an Air Cushion Surge Tank for Hydro Power Plants. *Energies* **2022**, *15*, 2824. https:// doi.org/10.3390/en15082824

Academic Editors: Adam Adamkowski and Anton Bergant

Received: 17 March 2022 Accepted: 11 April 2022 Published: 13 April 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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 *γ* 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:


#### *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 *V*˙ flows towards the air chamber through the access tunnel with length *L*<sup>t</sup> and diameter *D*t. The intake-penstock manifold pressure at the bottom of the tank is represented by *p*m, and the air pressure at the air chamber due to the cushioning of the air is represented by *p*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*<sup>f</sup> is the fluid friction against *<sup>V</sup>*˙ , *<sup>F</sup>*<sup>g</sup> is the force due to gravity in the downward direction, and *<sup>F</sup>V*˙ <sup>g</sup> is the projection of *F*g in the alignment of the flow.

**Figure 1.** ACST with an access tunnel and an air chamber. (**a**) general schematic of ACST and (**b**) flow diagram.

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

$$\frac{dm}{dt} = \dot{m} \tag{1}$$

$$\frac{d\mathcal{M}}{dt} = \dot{\mathcal{M}} + F \tag{2}$$

where *<sup>m</sup>*˙ and <sup>M</sup>˙ represent the mass flow rate and the momentum flow rate, respectively. Equations (1) and (2) are expressed with a series of algebraic equations as

$$
\dot{m} = \rho \dot{V} \tag{3}
$$

$$\mathcal{M} = m\upsilon\tag{4}$$

$$
\dot{\mathcal{M}} = \dot{m}v\tag{5}
$$

$$F = F\_{\rm P} - F\_{\rm g}^{\mathcal{V}} - F\_{\rm f} \tag{6}$$

where *ρ* 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*<sup>p</sup> is the pressure force, and *F*<sup>f</sup> 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

$$m = m\_{\rm w} + m\_{\rm a} \tag{7}$$

where *m*w and *m*a are the masses of the water and the air inside the surge tank, respectively. *m*<sup>a</sup> is constant inside the chamber and is determined based on the initial air cushion pressure *p*c0 which is considered to be a design parameter for the hydraulic performance

of the surge tank. If *h*c0 is the initial water level inside the surge tank for the initial air cushion pressure *p*c0, 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 *γ* of the air inside the ACST, considering standard temperature and pressure (STP), the relation *pV<sup>γ</sup>* = constant is assumed where *γ* 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*c0 and the initial volume *A <sup>L</sup>* <sup>−</sup> *<sup>h</sup>*c0 *<sup>L</sup> H* given by

$$m\_{\rm a} = \frac{p\_{\rm c0}A\left(L - h\_{\rm c0}\frac{L}{H}\right)M\_{\rm a}}{RT^{\circ}}\tag{8}$$

where *M*a is the molar mass of air, *R* is the universal gas constant and *T*◦ is the temperature taken at STP. Similarly, *A* is the area of the air chamber expressed as *A* = *π <sup>D</sup>*<sup>2</sup> 4 .

From Equation (2) formulating *pc*0*V<sup>γ</sup>* <sup>0</sup> = *pcVγ*, the air cushion pressure during the operation of the surge tank is given by

$$p\_{\mathbb{C}} = p\_{\mathbb{C}0} \left( \frac{L - h\_{\mathbb{C}0} \frac{L}{H}}{L - \ell} \right)^{\gamma} \tag{9}$$

where *p*<sup>c</sup> depends on the length inside the ACST.

During the operation of the surge tank, the mass of the water inside the surge tank *m*w varies according to the variation in *h*. Thus, the expression for *m*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* ≤ *H*<sup>t</sup> and second we consider (ii) *h* > *H*t. Furthermore, we also formulate expressions for *F*<sup>p</sup> and *F*<sup>f</sup> for both of the scenarios of the water level *h*.
