*3.2. Effect of Air Friction Inside ACST*

We now consider Torpa HPP with each of the turbine units rated at 75 MW as a single entity, for simplification, with 150 MW with input *u*<sup>v</sup> 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*D,a modeled using Darcy's friction factor *f*<sup>D</sup> inside the ACST of Torpa HPP is considered using Equation (12) for the case of water level *h* ≤ *H*t, and using Equation (19) for the case of water level *h* > *H*t. The input to the turbine with valve signal *u*v for the simulation purpose is given by

$$\mu\_{\rm V} = \begin{cases} 0.5 & 0 < t \le 500 \,\text{s} \\ 0.95 & 500 \,\text{s} < t \le 1500 \,\text{s} \end{cases}$$

where the hydro-turbine is loaded from half-load to nominal load at time *t* = 500 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*c, and the inlet turbine pressure *p*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*c for the ACST modeled with and without the air friction consideration is in the order of 10−<sup>5</sup> bar = 1 Pa, even for the turbine loaded from half load to the nominal operation. This is because of the fact that fluid frictional force *F*<sup>f</sup> depends on Darcy's friction factor *f*D, and *<sup>f</sup>*<sup>D</sup> depends on Reynolds' number *<sup>N</sup>*Re = *<sup>ρ</sup>*|*v*|*<sup>D</sup> <sup>μ</sup>* where *μ* is the dynamic viscosity of the fluid. At STP, *<sup>μ</sup>*air = 1.81 · <sup>10</sup>−<sup>5</sup> Pa · <sup>s</sup> and *<sup>μ</sup>*water = 8.90 · <sup>10</sup>−<sup>4</sup> Pa · <sup>s</sup> which can be approximated as *μ*water ≈ 100 *μ*air.

**Figure 6.** ACST model with and without frictional force due to the air inside ACST for Torpa HPP, (**a**) turbine valve signal *u*v, (**b**) water level *h* inside ACST, (**c**) air cushion pressure *p*c, and (**d**) turbine inlet pressure *p*tr.

#### *3.3. Operations of ACST in Load Acceptance and Rejection*

Load acceptance and rejection are created by changing the turbine valve signal *u*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 s. At *t* = 500 s, a different load acceptance condition is created by changing the turbine valve signal *u*v, and the hydraulic behavior of the ACST is observed for the next 1500 s. The turbine valve signal *u*v is generated as

$$\mu\_{\rm V} = \begin{cases} 0 & 0 < t \le 500 \,\text{s} \\ \mu\_{\rm va} & 500 \,\text{s} < t \le 2000 \,\text{s} \end{cases}$$

where *u*va ∈ {0.25, 0.5, 0.75, 1.0} 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 s. At *t* = 500 s, a different load rejection condition is created by changing the turbine valve signal *u*v, and the hydraulic behavior of the ACST is observed for the next 1500 s. The turbine valve signal *u*v is generated as

$$\mu\_{\rm V} = \begin{cases} 1.0 & 0 < t \le 500 \,\text{s} \\ \mu\_{\rm vr} & 500 \,\text{s} < t \le 2000 \,\text{s} \end{cases}$$

where *u*vr ∈ {0.75, 0.5, 0.25, 0.0} 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*v, the air pressure *p*c, the turbine inlet pressure *p*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*v, *p*c, *p*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 s, shows that from the no load operation to TLA, the difference in the air pressure *p*c inside the ACST is around 4 bar. Similarly, Figure 7e shows that the difference in turbine inlet pressure *p*tr is around 3 bar, and Figure 7e shows that the difference in the water level *h* inside the ACST is around 1 m. In addition, Figure 7c shows that the difference in *p*c from no load operation to 25% load acceptance, 50% load acceptance and 75% load acceptance are around 1 bar, 2 bar and 3 bar, respectively. Similarly, results can be obtained for *p*tr (Figure 7e) and *h* (Figure 7g). For *p*c, *p*tr and *h* oscillation dies out as the time progresses for *t* > 500 s.

Figure 7b shows the turbine valve signal generated for load rejections of 25%, 50%, 75%, and 100%. Figure 7d, at *t* = 500 s, shows that from full load operation to TLR, the difference in *p*<sup>c</sup> is around 4 bar as similar in the case of TLA. Similarly, the difference is around 3 bar in the case of *p*tr, as shown in Figure 7f. The difference in *h* from full load operation to TLR is also 1 m, as in the case of TLA. Similarly, from Figure 7d, the difference in *p*c from full load operation to load rejections of 25%, 50% and 75% are around 1 bar, 2 bar and 3 bar, respectively. Similar results can be obtained for *p*tr (Figure 7f) and *h* (Figure 7h). For *p*c, *p*tr and *h*, oscillation dies out for *t* > 500 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 m for both TLA and TLR. Similarly, the difference in the air pressure is around 4 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.

**Figure 7.** Hydraulic performance of the ACST for Torpa HPP for the different percentage change in the load acceptances and the load rejections, (**a**) turbine valve signal *u*v as an input to the load acceptances, (**b**) turbine valve signal *u*v as an input to the load rejections, (**c**) air pressure *p*c for the load acceptances, (**d**) air pressure *p*c for the load rejections, (**e**) turbine inlet pressure *p*tr for the load acceptances, (**f**) turbine inlet pressure *p*tr for the load rejections, (**g**) water level inside the ACST *h* for the load acceptances, and (**h**) water level inside the ACST for the load rejections.

#### **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 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].

**Author Contributions:** Conceptualization, M.P., K.V., R.S. and B.L.; methodology, M.P., R.S. and B.L.; software, M.P. and D.W.; validation, M.P., K.V. and B.L.; formal analysis, M.P.; investigation, M.P.; resources, M.P., D.W. and K.V.; writing—original draft preparation, M.P.; writing—review and editing, M.P.; visualization, M.P. and B.L.; supervision, D.W., K.V. and B.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** Help and discussions with Liubomyr Vytvytsky, ABB Oslo, regarding model tuning is gratefully acknowledged.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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