Hydraulic Scale Modeling of Mass Oscillations in a Pumped Storage Plant with Multiple Surge Tanks

The evaluation and validation of hydraulic scale modeling of PSPs with multiple surge tanks was conducted through a case study of an existing HPP currently investigated for upgrading to PSPs. The Roskrepp hydropower plant in South Norway has a 50 MW Francis turbine with a $70{\mathrm{m}}^3/\mathrm{s}$ nominal discharge and a 83-m nominal head and was commissioned in 1978. The unlined tunnel system was constructed with drill-and-blast tunneling and had a total length of 3.8 km and a cross section of $38{\mathrm{m}}^2$. Two surge tanks were located along the headrace tunnel: a brook intake with inflow (Surge tank 1) and a two-chamber surge tank (Surge tank 2). Two surge tanks were constructed along the tailrace tunnel: a surge tank with an expansion chamber (Surge tank 3) and an unplugged adit (Surge tank 4), which could alternatively be filled or partially filled, depending on the water level in the lower reservoir. The system schematic is presented in Fig. 1. The water level in the upper and lower reservoirs varies between 890 and 929 and 820 and 837 meters above sea level (masl), respectively. The overflow weir in Surge tanks 2 and 3 were located at 940 and 840 masl, respectively. The outlets of Surge tanks 1 and 4 were at 946.5 and 829 masl, respectively.

Field measurements during power plant operation were collected in September 2017. Fig. 2 shows the power output and turbine inlet pressure. During the measurements, the power plant performed the following operations: start-up to full load; load decrease and increase of 20%, 40%, and 60%; and emergency shutdown (ESD) from full load. The field data collected were the pressure upstream and downstream of the turbine, produced power, rotational speed, guide vane position, and water levels in upstream and downstream reservoirs. The upstream pressure tap was located in the wall of the turbine inlet pipe at elevation 823 masl, between the main inlet valve and the spiral case, and the downstream pressure tap was located downstream of the turbine runner, on the draft tube wall at elevation 820.7 masl. The field measurements were used to tune and validate the numerical and hydraulic scale models.

Numerical and hydraulic scale modeling depend on having accurate boundary conditions and geometry. However, for older HPPs with unlined tunneling, the available information may be limited to the design drawings without as-built documentation. These present a good indication of the system, but in the case of drill-and-blast tunnels, they should be controlled with field measurements such as three-dimensional (3D) scanning. Both construction drawings and a 3D scan were collected in this work. The 3D scanning using a Leica Scan Station P20 (Leica) with a point density of 25 mm was carried out for the headrace tunnel, brook intake, headrace surge tank, powerhouse, tailrace surge tank (partly), access tunnels, and cable tunnel. Sections that were not included are the tailrace tunnel and the unplugged adit since these were not dewatered at the time of scanning.

1D numerical models were established using the LVTrans freeware (Svingen 2014), utilizing the Method of Characteristics (MOC) to solve the partial differential equations for continuity and momentum for elastic pipe flow shown in Eqs. (1) and (2)

Two numerical models were developed: one for the 1:1 prototype scale (PSNM) and one for the 1:70 hydraulic model scale (MSNM). The main elements included in the models are Surge tank 1, included as a long inclined tunnel with a controllable inflow; Surge tank 2, included with a variable cross section for the shaft, and upper and lower chambers; Surge tank 3, included as a simple surge tank simplifying the upper chamber; and Surge tank 4, included as an inclined tunnel connected to a stable water level identical to the lower reservoir. In the PSNM, the turbine was described as a standard Francis turbine modeled according to Nielsen (1990). The cross section of the tunnel system was simplified from a D-shape tunnel cross section in the prototype to a circular cross section in the numerical models, maintaining the cross-section area. The geometry used in the models was based on the 3D scan and the available construction drawings. The implementation of friction and local losses in the models are presented in the next subsection. A standard proportional integral derivative controller (PID), with the parameters $P=1$, $T_i=7$, and $T_d=0$, was used to get a similar behavior between the PSNM and the existing prototype. In the MSNM, the turbine was replaced with a valve element since the turbine did not scale as the tunnel system hydraulics.

The boundary conditions of the two numerical models were defined by the upper reservoir water level, lower reservoir water level, water inflow into Surge tank 1, and the power demand or the guide vane opening. In the PSNM, simulations were run with a timestep of $dt=0.1\mathrm{s}$ for about 500 s, accounting for 5 s of computational time. In the MSNM, the simulations were run with a timestep of $dt=0.01\mathrm{s}$ for about 60 s. The timestep of the PSNM was selected to respect the Nyquist criterion, considering the water hammer as the highest relevant frequency in the system, i.e., the selected sampling frequency was six times higher than the water hammer frequency. A mesh-independence test with smaller timestep had been run to verify that the results had indeed converged. For the MSNM, the influence of the water hammer was disregarded, because its effects were not relevant for the study of mass oscillations, thus a higher timestep could be implemented. In the applied simulation software, LVTrans, the space resolution $dx$ was automatically calculated from the MOC, as $dx=cdt$, where $c$, the speed of sound, was assumed to be $\mathrm{1,200}\mathrm{m}/\mathrm{s}$ for unlined rock tunnels. Minor adjustments of the speed of sound in the individual elements were done automatically by the software to obtain integer values of $dx$.

Before constructing the hydraulic scale model, an initial numerical model was established based on the construction drawings to verify if numerical and hydraulic scale modeling were possible. Surge tank 1 was simplified to a vertical shaft instead of the actual inclined tunnel. Fig. 6 shows a comparison of the tuned numerical model with field measurements. These results were obtained by adjusting the different parameters in the numerical model. Based on the results, the construction of the hydraulic scale model was initiated. Detailed descriptions of the model and simulations are presented in Pitorac et al. (2018).

The first version of the hydraulic scale model was built based on the same geometrical dimensions as the initial numerical model. The same simplification for Surge tank 1 was implemented in the physical model as well. Fig. 7 compares the hydraulic scale model with the field measurements. The results from the hydraulic scale model are presented as a moving average to filter out signal noise and water hammer. Detailed descriptions of the model and experiments are presented in Pitorac et al. (2020). As the results did not match the field measurements, further investigations were needed to identify the source of the disparity.

To find the error in the hydraulic scale modeling, a revised numerical model of the prototype was utilized (PSNM). The identified problems included simplification of Surge tank 1 and errors in the local loss factors. The distribution of the local losses was found to be very sensitive in the case of an HPP with multiple surge tanks. Based on these findings, Surge tank 1 was redesigned, and the local loss factors were determined using the method previously presented. The revised parameters used in the numerical model are shown in Fig. 8. An additional verification was performed using an MSNM. The results are presented in Fig. 9 demonstrating the improvements.

Using the revised numerical models, the physical model was reconstructed. The following modifications were implemented: local loss valves were included at the bottom of the surge tanks, the local losses were tuned, and Surge tank 1 (the brook intake) was rebuilt into an inclined tunnel. Fig. 10 presents the results from the revised physical model.

This section compares the results from the revised numerical models and the revised hydraulic scale model. The main hydraulic values from the field measurements before and after the ESD in the prototype are included in Table 4. These parameters were used for the steady-state tuning of the numerical and physical models.

Table 5 presents a summary of the results from the final hydraulic scale model, scaled up to the prototype scale, and the final full-scale numerical model (PSNM) compared with the field measurements. The details of the results are presented subsequently. The difference in head loss on the headrace in the physical model came from model construction reasons, and it was accounted for in the measurements. The damping factor was calculated as the division between the amplitude of the first oscillation period and the amplitude of the second oscillation period. The accuracy is determined as shown in Eq. (5)where $P_m$ = model parameter; and $P_p$ = prototype parameter that the accuracy is calculated for.

Tuning with prototype data is a common method used in numerical and hydraulic scale modeling of hydropower plants. However, tuning of the transient behavior for systems with multiple surge tanks is particularly challenging owing to superposed mass oscillations. Tuning is also challenging without detailed information about the geometry and local loss factors in specific sections from the prototype. When some of the parameters are not available from as-built documentation, the combination of a numerical model and a hydraulic scale model can allow for a mutual tuning and validation process. In the present work, it was found that applying only a single numerical or hydraulic scale model separately could result in a wrong representation of the actual system. This can be observed from the tuning of the initial numerical model presented in the “Results” section. For this case, the model was tuned and found accurate based on a comparison with the field measurements (Fig. 6), but it was finally revealed to have several errors. In complex systems in which boundary conditions are difficult to obtain, combining numerical with hydraulic scale modeling reduces uncertainty and the risk of such errors.

For systems with multiple surge tanks resulting in multiple T-junctions, the local loss factors in T-junctions are especially sensitive, and a stepwise tuning process to obtain the correct values is proposed. This is necessary to ensure that water is flowing in the correct direction in each T-junction during transient flow. Also, tuning with field measurements of transient flow is necessary, because the steady-state flow does not reveal the significant effect of the tridirectional T-junction local losses.

An important aspect for obtaining correct modeling is the implementation of the head loss (friction loss and local losses) on each relevant tunnel stretch. The numerical simulations are done with the Darcy-Weisbach equation with a fixed friction factor. This yields some inaccuracy as the friction factor changes during transition through laminar flow during mass oscillations. However, this simplification is regarded by the authors to only have a minor effect for the study of mass oscillations, because the time in the laminar regime is limited and the velocity in this regime is low, hence resulting in limited friction losses. The hydraulic scale model, on the other hand, includes the real-world physics for transient flow, with the transition through laminar flow during mass oscillations having an influence on the head loss. A challenge in the scale model is that the transition between the turbulent and laminar regime occurs earlier, compared with a full-scale prototype, because the Reynolds number is not correctly scaled. This means that the flow takes a relatively longer time in the laminar regime in the physical scale model. Despite these known flaws in the methodology, the comparison of the results from the numerical models and hydraulic scale model shows that they do not have a significant influence on the results. The main error is the difference in dampening of the mass oscillations. However, for design and dimensioning of hydropower plants and pumped storage plants, it is the maximum amplitudes that is of highest importance. As can be seen from the results, the maximum amplitude is modeled with high accuracy in both the numerical and hydraulic scale models.

In hydraulic scale models, the friction loss can be implemented as local loss along each longitudinal section. This is considered correct for the purpose of the study, because both local losses and friction losses are functions of the discharge squared. In this work, it is assumed that the sum of friction losses and local losses on each tunnel stretch between surge tanks determine the hydraulic behavior of the system, and that this can, in turbulent flow, be expressed as $h_l=h_f+h_s=k{Q}^2$, where $k$ is a constant. According to this analytic description, it is indifferent if the head loss is modeled as friction loss or local loss. However, these analytic equations do not account for transient friction and transient local loss, which are regarded as the main error source in the hydraulic scale modeling. More research to determine both the transient dampening of mass oscillations in full-scale and model-scale hydropower plants is necessary.

The results show that a simplification of inclined shafts to an equivalent vertical shaft is not possible due to differences in inertia and head loss. This simplification significantly reduced the accuracy of the modeling compared with the ESD field measurements.

In the hydraulic scale model, the penstock was scaled according to the geometry in the prototype, but due to the very small scale, the relative head losses were larger in the hydraulic model than in the prototype. This resulted in a lower pressure head at the turbine inlet, than in the field measurements, which was corrected for in the presented results; however, this could be overcome by having a larger diameter of the penstock and adjusting the total head loss using a local loss valve instead to obtain the correct total head loss in the section. A simplified version of the pressure shaft is recommended by the authors to be applied in future studies of mass oscillations to avoid similar issues and to spare costs.

The impacts of atmospheric air pressure and air and water temperature were also considered in the design of the hydraulic scale model. These have been proven in previous studies to have a significant influence on the results when studying pressurized systems with closed surge tanks (Vereide et al. 2015). Even though the presented case study was a pressurized system, all the surge tanks were open surge tanks; thus, the variation in air pressure would have had an equivalent effect throughout the entire system and would not have influenced the results. The temperature of the water was also assumed to not have a significant influence. In this case study, the measurements were performed in different order, at different times of day; thus, parameters such as atmospheric pressure and air and water temperature varied, and it was observed that they did not have a significant influence on the results. This suggests that the assumption is acceptable.

Another aspect that was considered for the design of the model was the presence of air trapped inside the pipeline. Entrapped air may also have a significant influence on mass oscillations because they, in practice, can become equivalent to small air-cushion surge tanks (Vereide et al. 2015). There are two options for ensuring that air is not entrapped in the system: either giving a slope to the tunnel system or including air release valves in the relevant points, the latter being applied in the presented hydraulic model. Entrapped air will have significant influence on the overall results, especially if it is entrapped in a section where water hammer occurs, because water hammer can increase up to five times (Pitorac et al. 2016).

The analysis of the 3D scan shows a real cross section larger than in the construction drawings, presumably because the construction drawings give a minimum cross section that needs to be met by the contractor. The drill-and-blast tunnels are uneven by nature, and the final cross section normally becomes larger than in the drawings. This has a positive result for the head loss and loss of energy production. Another observation is that the slope of the tunnel can vary in construction drawings and has a significant influence on the effective surge tank water surface area. This was encountered in the case of Surge tank 1.

The 3D scan showed a location where an air pocket could be trapped in the tunnel system. The tunnel was constructed from two directions; the tunnel met in the middle, close to Surge tank 1. The slope was inclined upward from both sides, resulting in air being trapped at the crown during water filling of the system. This risk could be eliminated with a correct design of the entrance from the tunnel to Surge tank 1, but because of an unbeneficial construction of this transition in the prototype, air could be trapped. This air pocket is possibly eliminated during transient operations, and it is unknown if the air pocket is large enough to include an air cushion effect in the system that can influence both the hydraulic transients and the head losses.