AbstractAs power systems include more intermittent renewable energy sources, energy storage solutions are needed to support them. Pumped hydro is a reliable alternative for long-term energy storage. A solution for bringing more pumped hydro into the power system is to upgrade existing hydropower plants into pumped hydro. To investigate this possibility, a case study with a complex tunnel system and multiple surge tanks along both the headrace and the tailrace tunnels is selected. A hydraulic scale model and validation methods were developed. The challenges in developing and validating complex models involving multiple surge tanks, throttles, head loss, and limited prototype information are detailed.IntroductionIncreased renewable energy share in the energy systems results in a need to balance supply and demand, which necessitates developing energy storage solutions. Batteries can only cover the short-term energy storage demand (Kusko and Dedad 2007). Today, the dominating technology for long-term electrical storage is pumped hydro (Hunt et al. 2020; Richter et al. 2020).The motivation of this work is to develop new solutions to reduce costs associated with pumped storage plants (PSPs) development. A promising solution is the reconstruction of existing hydropower plants (HPPs) into PSPs (Lia et al. 2016; Peran and Suarez 2019). Reconstruction of HPPs into PSPs is especially interesting in Norway because the country currently holds over 80  TWh of storage capacity in existing hydropower reservoirs and over 30 GW of existing HPP capacity. The country currently only holds 10 PSPs with a total installed capacity of 1,000 MW in pumping (Pitorac et al. 2020). By reconstructing existing HPPs, the existing reservoirs and tunnel system can be utilized, which will significantly lower both construction costs and environmental impacts. On the other hand, such upgrade comes with limitations owing to existing infrastructure or more complex hydraulic transients or both. Especially, surge tanks could represent a challenge in terms of limitations to the operation of the power plant (Anderson 1984). Surge tanks along the headrace and tailrace tunnels may need upgrades because of changes of the hydraulic grade line and hydraulic transients, which can lead to overpressure and flooding, or underpressure and air entrainment. Many existing HPPs have long tunnels with complex surge tanks or brook intakes (BI) that function as additional surge tanks (ST). The hydraulics of such HPPs is complex, and upgrading them into PSPs should be investigated carefully.The paper is limited to the evaluation and validation of hydraulic scale modeling as a method for investigating mass oscillations in hydropower plants and pumped storage plants with multiple surge tanks. Hydraulic scale modeling has been previously used as a method for mass oscillations assessment for hydropower plants with a single surge tank in Richter et al. (2012) and Svee (1970). In the present work, the method is evaluated and validated for multiple surge tanks using field measurements and a one-dimensional (1D) numerical model.Methods and MaterialsCase StudyThe 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  m3/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  m2. 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 MeasurementsField 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.Three-Dimensional ScanningNumerical 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.Comparison of 3D Scan with Construction DrawingsThis section compares the 3D scan and the construction drawings for the Roskrepp HPP. An example of the 3D scan is shown in Fig. 3. For older hydropower plants, the limited availability of documentation of unlined tunnels could pose a challenge for the accurate modeling of the prototype. The presented comparison demonstrates that the deviations between drawings and as-built can be significant. Their impact on the results of the numerical and hydraulic scale models (also called physical models) is discussed in the “Discussion” section.Table 1 compares the 3D scan and the construction drawings. The most significant differences involve the Surge tank 1 tunnel slope and cross section, the cross section of the shaft in Surge tank 2, and the cross section of Surge tank 3. The differences in slope and cross section of Surge tank 1 have a significant influence on the water table cross section, which is almost double the one calculated based on the construction drawings. The cross-section difference of the shaft in Surge tank 2 is smaller than the construction drawings design, which could result in a more unstable system than initially designed.Table 1. Comparison of parameters in construction drawings versus 3D scanTable 1. Comparison of parameters in construction drawings versus 3D scanSectionDimensionConstruction drawings3D scanUnitTunnel lengthIntake to brook intake2,2012,118mBrook intake to surge tank970967mSurge tank to penstock341347mPenstock5151mTunnel cross-sectional areaIntake to brook intake3840m2Brook intake to surge tank3839m2Surge tank to penstock3839m2Penstock12.612.6m2Brook intake (Surge tank 1)Length of the tunnel504643mArea (in length axis)812m2Water table area (horizontally)69112m2Slope1/8.41/8.7—Headrace surge tank (Surge tank 2)Surge tank throttleYesNo—Length, lower chamber7573mLength, shaft5650mLength, upper chamber85116mArea, connection tunnel2027m2Area, lower surge chamber3438m2Area, shaft6053m2Area, upper surge chamber3427m2Tailrace surge tank (Surge tank 3)Area, surge tank shaft90110m2Numerical Models1D 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) (1) (2) g∂H∂x+∂v∂t+fv|v|2D=0Two 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, Ti=7, and Td=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  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  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 1,200  m/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.Tuning of Head Loss with Field MeasurementsHydropower tunnel systems with multiple surge tanks have a similar behavior to pipe network systems, where resistance along tunnel sections determine the direction and amount of flow on each stretch. Thus, local losses and friction losses can be reduced to a total head loss for each relevant stretch, representing the total resistance of the section.The prototype had unknown tunnel roughness and local losses that needed to be determined. It was not possible to know the exact distribution of friction losses and local losses only based on turbine inlet and outlet pressure measurements, together with reservoirs water levels. Therefore, the friction losses were based on empirical formulas for unlined tunnels (Rønn and Skog 1997) and the local losses along the headrace and tailrace were tuned to match the steady state in field measurements. The friction loss (hf) in the applied numerical simulation software was modeled with the Darcy-Weisbach equation (Moody 1944), and the friction factor f for the unlined tunnel sections was set to 0.05. The local losses included the inlet, several T-junctions, cross-section changes, and the outlet. In the numerical model, the local losses were cumulated and included in one point on each relevant tunnel and pipe branch, as shown in Fig. 4. The local losses were modeled with the local loss equation from Idelchik (2008). The applied friction and local loss models and factors [Eqs. (3) and (4)] are discussed in the “Discussion” section (3) (4) where f = friction factor; ξ = local loss factor; L = tunnel length; D = tunnel cross-section diameter; v = water velocity; and g = gravitational acceleration.Field measurements of transient power plant operation were used to tune the local loss parameters in each surge tank. For the tuning of the transient operations, the mass oscillation periods and amplitudes were used. In the case of multiple surge tanks, several periods and amplitudes were identified. From an Fast Fourier Transform (FFT) analysis of the field measurements, the period of the mass oscillations was determined. The amplitude of the mass oscillations was determined from time series measurements. Because the period was independent of the hydraulic losses and was primarily a function of the geometrical parameters of the system, no adjustments were needed. The amplitude for oscillations in the headrace tunnel was adjusted by tuning the distribution of the local loss for the various tunnel sections (Fig. 4). This was done stepwise, as follows: 1.Start the tuning with Surge tank 2 and remove any other surge tanks. Run the transient simulations while adjusting the local loss in surge tank (hs4) until the amplitude matches. At this moment, hs1+hs2+hs4 is adjusted; thus, hs4 is known.2.Remove Surge tank 2 from the model and reconnect Surge tank 1. Apply the same tuning as in Step 1 to determine hs1+hs3.3.Isolate the reservoir and apply the same method to determine hs2+hs3.4.With hs1+hs2, hs1+hs3, and hs2+hs3 known, the losses are redistributed between hs1, hs2, and hs3.5.The entire tunnel system is reconnected, and the model is verified and further validated using a different set of field measurements.The same method was applied for tuning the tailrace tunnel.Hydraulic Scale ModelDimensional AnalysisTo determine the hydraulic scale model parameters, a dimensional analysis using the Buckingham π theorem based on Buckingham (1914) was conducted and is shown in Table 2. Based on the results of the dimensional analysis, mass oscillations were scaled according to the Euler scaling law (relation of inertia and pressure forces), following the scaling equations in Table 3.Table 2. The π-terms for the hydraulics systemTable 2. The π-terms for the hydraulics systemNotationExpressionNameπ1vgDFroude numberπ2HDHead factorπ3LDLength factorπ4sTunnel slopeπ5fFriction factorπ6QD2vDischarge factorπ7vcMach numberπ8vDρμReynolds numberπ9votDKeulegan-Carpenter numberπ10pρv2Euler numberTable 3. Euler scaling factorsTable 3. Euler scaling factorsParameterScaling equationLengthLrTimeLrMassLr3DischargeLr5/2For water, where density was approximately constant, Euler scaling gave the same results as Froude scaling (inertia to gravity forces). The water hammer was scaled with the Mach number (relation between speed of sound to water velocity) and was not scaled correctly when applying the Euler scaling law. During ESD, the turbine closing was 7 s in the prototype, but only 3 s in the model; even though this influenced the water hammer, it had only a negligible effect on mass oscillations.Model Construction and MaterialsThe hydraulic scale model was a 1:70 scale of Roskrepp HPP with a 55 m long tunnel with a 0.1 m inner diameter built with 3 mm thick AISI304 stainless steel pipes. The surge tanks were geometrically similar and were transparent to visually control the water level. Some simplifications of the tunnel system were necessary; thus, the tunnel diameter was constant throughout the entire model, meaning that the variable unlined cross section, niches, sand traps, and five shorter sections with concrete lining were not modeled. However, their effect on the head loss was included for each stretch. The turbine was modeled using a globe valve with electric actuator for flow control and a fast-acting butterfly valve with pneumatic actuator for the ESD tests. In addition, knife gate valves at the entrance in each surge tank were included to allow to disconnect them as needed. To release air from the system after water filling, automatic air release valves were placed in locations of potential air pockets.The friction in the steel pipes was not equal to the unlined rock surface in the prototype. To account for the total friction losses, butterfly valves were installed in each tunnel section and adjusted to account for the sum of the friction and local losses. This method was previously demonstrated and validated by Vereide et al. (2015).The instrumentation included six 0.3 bar pressure sensors (PS), two electromagnetic flow meters (FM), and two ultrasonic level sensors (US) in the reservoirs, as shown in Fig. 5. The pressure sensors had an uncertainty of ±0.04% full-scale accuracy, which resulted in a maximum error of ±1.2  mm in model scale. This was equivalent to 0.08 m in prototype scale; thus, it was not significant. The ultrasonic sensors had an uncertainty of ±1% and a reproducibility ±0.15%. The flow meters had an uncertainty of ±0.4%, which resulted in an error up to 0.007  l/s in model scale.The parameters that could be controlled in the model were the water level in both reservoirs, the turbine valve (globe valve with parabolic plug with a pneumatic actuator), the fast-acting valve (butterfly valve with a pneumatic actuator), the valves for disconnecting the surge tanks (knife gate valves with a pneumatic actuators), local loss valves (butterfly valves with manual handles) along each relevant tunnel section, and local loss valves at the bottom of the surge tanks (butterfly valves with manual handles). The control of the model, as well as the data acquisition, was done using an input/output (I/O) cabinet with National Instruments (NI) modules incorporating a total of 16 analog input, 16 digital input, 8 analogue output, and 8 relay output channels. The analog modules had a programmable range between 0 and 20 mA. The digital modules had a voltage range of 0–30 VDC. The system control- and data-acquisition software was programmed by the main author in LabView version 2018. The modules did not have antialiasing filters, but distortion of the original signal was avoided through oversampling with the sampling frequency of 10 Hz, far higher than the 0.12 Hz Nyquist rate.Tuning of Head Loss in the Hydraulic Scale ModelThe hydraulic scale model tuning was done using the turbine inlet and outlet pressure from the field measurements and the local loss parameters determined with the numerical models. For the steady-state adjustment of the head losses, the results from the numerical models were used to adjust the butterfly valves on each tunnel section. For the transient tuning, the local loss valves for each of the surge tanks were adjusted separately by disconnecting the other surge tanks according to the same method described for the numerical model. After the tuning of each surge tank, all surge tanks were reconnected to the system, and a final verification and validation of the full system was performed.ResultsThis section presents the results for assessment of the hydraulic scale modeling accuracy. Two versions of the hydraulic scale model were tested, and it has been shown why and how the different models were implemented. In the following sections, the modeling process is presented in chronological order; then, detailed descriptions for each individual model result are provided.The work started with an initial numerical model based on construction drawings and tuned using field measurements (Pitorac et al. 2018). Based on the results, an initial version of the hydraulic scale model was built with similar simplifications as the initial numerical model (Pitorac et al. 2020). However, the tuning of the model was unsuccessful. To determine the problems of this model, a new numerical model was built, called prototype scale numerical model or PSNM, based on the 3D scan of the prototype. This model was successfully tuned. Comparison between the initial numerical model and the PSNM showed unphysical parameters in the former. An additional numerical model of the hydraulic model, called model scale numerical model or MSNM, was established to investigate any possible scaling issues. Both the PSNM and MSNM showed similar results. The modifications and boundary conditions from the latter numerical models were thus implemented in the final hydraulic scale model (HSM), which was validated against results from both the prototype, the PSNM, and the MSNM.Initial Numerical ModelBefore 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).Initial Hydraulic Scale ModelThe 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.Revised Numerical ModelTo 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.Revised Hydraulic Scale ModelUsing 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.Comparison of the MethodsThis 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 4. Initial and final emergency shut-down test resultsTable 4. Initial and final emergency shut-down test resultsParameterInitialEndInitialEndInitialEndPower (MW)500500——Upper reservoir water level (masl)925925925925925925Lower reservoir water level (masl)833833833833822822Turbine inlet pressure (mWC)95.75103.595.75103.596.2103.3Turbine outlet pressure (mWC)811.311.811.317.411.5Headrace head loss (m)7.807.807.10Tailrace head loss (m)1.100.601.70Table 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) (5) where Pm = model parameter; and Pp = prototype parameter that the accuracy is calculated for.Table 5. Results comparison for the prototype, numerical model, and physical modelTable 5. Results comparison for the prototype, numerical model, and physical modelParameterPrototypeNumerical modelHydraulic scale modelMaximum amplitude level (masl)933.8934934Period (s)217224219Damping factor2.23.02.6Accuracy of maximum amplitude (%)—99.699.6Accuracy of period (%)—96.899.1Accuracy of damping (%)—6182DiscussionComparison of Model Results and Field MeasurementsTuning 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.Hydraulic Scale Modeling TechniquesIn 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 hl=hf+hs=kQ2, 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).Secondary Results: Comparison of 3D Scan and Construction DrawingsThe 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.ConclusionsBased on the results, it is concluded that both numerical modeling and hydraulic scale modeling of PSPs with multiple surge tanks are possible. For complex systems with limited available documentation, such as the presented case study, a combination was found necessary to obtain reliable results. More specifically, this was critical to obtain accurate tuning owing to limited documentation of the geometry and the unknown distribution of head losses in the system.Hydraulic scale modeling achieved a 94%, 99%, and 82% accuracy of modeling the amplitude, period, and damping of the mass oscillations. The numerical model achieved a 94%, 99%, and 61% accuracy for amplitude, period, and damping, respectively. The main challenge in both the numerical modeling and hydraulic scale modeling is to correctly model the dampening of mass oscillations over time. This error is expected to be the result of transient friction and transient local loss. However, for design and construction of PSPs, it is the maximum amplitudes that are of importance, and both numerical and hydraulic scale models are able to model this with high accuracy.For older HPPs, there can be large inaccuracies between construction drawings and real tunnel dimensions, and documentation of the actual tunnel dimensions may be limited. Up to 50% inaccuracy was found in this case study. A 3D scan of an unlined tunnel system of the presented case study proved beneficial for identifying differences including tunnel diameter inaccuracies, roughness differences, invert erosion, and potential air pocket locations.Data Availability Statement•Some or all data, models, or code generated or used during the study are available in a repository online in accordance with funder data retention policies (field measurements of the pressure head at turbine inlet and outlet, power output, guide vane position, for all operating points;•Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request (available data: numerical models, full information regarding the hydraulic model, raw data for the results).•Some or all data, models, or code generated or used during the study are proprietary or confidential in nature and may only be provided with restrictions (tunnel scan).AcknowledgmentsThis research was financed by the Norwegian Research Center for Hydropower Technology (HydroCen). The tunnel scanning was financed by Sira-Kvina Hydropower Company. The field measurements were obtained with help and support from Sira-Kvina Hydropower Company. The construction of the hydraulic scale model was done with the help of the technicians from the Hydraulic Laboratory at Norwegian University of Science and Technology, M-Tech Workshop, and Danielsen AS.NotationThe following symbols are used in this paper:c=speed of sound (m/s);D=tunnel diameter (m);dt=time step (s);f=friction factor;g=gravitational acceleration (m/s2);H=head (m);hf=friction head loss (m);hs=local head loss (m);k=loss parameter (s2/m5);L=tunnel length (m);M=Manning number (m1/3/s);P=coefficient for proportional term of PID;Pm=model parameter;Pp=prototype parameter;Q=discharge (m3/s);Td=derivative time of PID;Ti=integration time of PID;t=time (s);v=water velocity (m/s);x=distance (m); andξ=local loss factor.References Hunt, J. D., E. Byers, Y. Wada, S. Parkinson, D. E. Gernaat, S. Langan, D. 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