Calibration of flow differential pressure coefficients on pump-turbine by thermodynamic method

differential pressure in the flow when the runner is rotating based on the basic theory of Bernoulli Eq. (Guo and Liu 1994). Zheng et al. performed a site test on prototype turbine by differential-pressure method using the coefficients derived from model test and found that the calculated flow rate differed from the predicted value (Zheng et al. 1997). Gao and Yan conducted several measures to ensure the


Introduction
The flow rate of pump-turbine is a main parameter for prototype turbines, and large flow measurement is an international technical challenge (Wang et al. 2022).The differential pressure-flow relationship of the flow passage is not affected by changes in head and opening, and can be used for long time with a one-time calibration, which has natural advantages (Shen and Zhou 2018).Hence, the flow rate of pump-turbine can be worked out basing on the flow differential pressure measured by the differential pressure gauge in the flow passage and the coefficients determined on model test, which is called the differential-pressure method (Gans et al. 2023).Guo and Liu discussed the formation reason of the differential pressure in the flow when the runner is rotating based on the basic theory of Bernoulli Eq. (Guo and Liu 1994).Zheng et al. performed a site test on prototype turbine by differential-pressure method using the coefficients derived from model test and found that the calculated flow rate differed from the predicted value (Zheng et al. 1997).Gao and Yan conducted several measures to ensure the

Abstract
On-line monitoring of flow rate is rather important to evaluate the water consumption of the pump-turbine and the overall efficiency of the pumped storage station.Flow rate can be worked out based on the flow differential pressure measured in the flow passage and coefficients determined on model test.But the coefficients must be calibrated to obtain the reliable flow rate.As thermodynamic method is a common way to measure the discharge of prototype pump-turbine, site test is carried out on a vertical pump-turbine with 600 m rated head to calibrated the coefficients in this paper.Measuring devices are designed to satisfy both the turbine and pump operating conditions, and test results of flow rate at both turbine and pump operating conditions are presented and show that the calibrated coefficients are different to the predicted values.The calibrated results indicate the importance of site calibration on prototype turbine to ensure the monitoring validity on real-time flow rate.
measurement accuracy of differential pressure (Gao and Yan 2001).However, due to the limited quality of manufacture or installation, the flow passage of prototype turbine is not geometrically similar to model turbine.The coefficients of flow measurement must be calibrated when differential-pressure method is applied on prototype turbine.
Zheng introduced a way to work out the coefficients of differential-pressure method according to the dimensions of flow passage, but the results were theoretical values (Zheng 1996).Xu calibrated the coefficients with the relative efficiency to the nominal maximum efficiency of prototype turbine based on an assumption that the maximum efficiency of prototype turbine is same with the predicted value determined on model test (Xu 2011).To accurately measure the flow rate through the hydraulic unit at any head, the flow differential pressure coefficients of the flow passage must be calibrated by another high-precision method.The internationally recognized methods for flow rate and efficiency test of large and medium-sized hydroelectric power plant units include flow meter method, water hammer method, indicator method, tracking method, ultrasonic method, and thermodynamic method for directly measuring unit efficiency (Ilinca and Ilinca 2017).Thermodynamic method is a common way to measure the turbine efficiency with rated head above 100 m, especially for the pump-turbines.Beside the efficiency, the discharge through the turbine also can be worked out indirectly by thermodynamic method (Teymoori 2013).For pumped storage power stations, on-line monitoring of flow rate is rather important evaluate the overall efficiency and the benefit of the station.Furthermore, as the hydraulic unit operates on turbine or pump mode, the pumped storage power station plays an important role to maintain the system stability, determine the load distribution, and compensate the intermittent renewables in the grid.Site test is carried out on a vertical pump-turbine with 600 m net head by thermodynamic method.With the measured discharge both at turbine and pump operation conditions, flow differential pressure coefficients are determined and calibrated.

Differential pressure method
When water flow in the passage of hydraulic machine, there is pressure difference at different locations of passage inner wall due to pressure impact.For rotating hydraulic machine, such as turbine, the higher pressure occurs on the outer circle of the passage inner wall, and the lower pressure on the inner circle (Javadian and Gaurav 2020).So, turbine discharge can be calculated by the differential pressure of the flow passage as following: Where, Q, turbine discharge; Δh, differential pressure of spiral case or draft tube; K, n, flow differential coefficients (W-K coefficients).
For pump-turbine, there are two operation conditions, turbine and pump modes.In turbine mode, the differential pressure is presented as the pressure difference between the outer circle and the inner circle of the spiral case.In pump mode, the differential pressure is presented as the pressure difference between the inlet (cone or bend) and outlet of draft tube (Stanilov et al. 2023).
In model test of a pump turbine, the flow differential pressure coefficients are determined by the measured differential pressure and discharge of model turbine as following (Kerschberger and Gehrer 2010): The discharge of prototype of turbine is determined by: Where, Q p , discharge of prototype turbine; Δh p , differential pressure of prototype turbine; D p , diameter of prototype turbine; D m , diameter of model turbine; K p , n, flow differential coefficients (W-K coefficients) of prototype turbine.Therefore, the flow differential coefficients of prototype turbine can be worked out by the coefficients determined in model test.However, due to the limited quality of manufacture and installation of turbine, the actual coefficients of prototype turbine are usually different with predicted values.With another absolute measurement method, the actual coefficients of prototype turbine may be calibrated (Zhang et al. 2016).During site test, the differential pressure Δh of the spiral case and draft tube shall be recorded in parallel with other method, such as thermodynamic method.The absolute flow discharge of turbine Q with various active power outputs can be measured by the thermodynamic method.The Winter-Kennedy coefficients K and n can be calculated by the equation, which shall be compared with the W-K coefficients determined on model test.

Thermodynamic method
The thermodynamic method results from the application of the principle of conservation of energy (first law of thermodynamics) to a transfer of energy between water and the runner through which it is flowing (Chai 2014).
In the actual operation, when the water passes through the turbine flow channel, it will produce a series of losses caused by friction, vortex, and flow separation and so on.All the losses will transfer to the thermal energy and make the water temperature difference between the turbine inlet and outlet section.The temperature difference is determined by the structure characteristics and work head of unit.Based on the temperature difference and water pressure, the discharge and efficiency of turbine can be worked out (Merry andThew 1984, Souri 2019).
In thermodynamic method, the measuring sections are defined in Fig. 1.The subscript 10 means high-pressure section of the pump-turbine, and subscript 11 means high-pressure measuring Sect.11 which locates at sampling vessel behind of main inlet valve of the pump-turbine.Subscript 20 means low-pressure section of the pump-turbine in the draft tube (Zhou 2019).
At high-pressure measuring section, it is difficult to directly measure Q, p and θ in the flow passage, so the water is discharged to a sampling vessel as the high-pressure measuring Sect.11.At low-pressure measuring Sect.20, the quantities of water are directly measured.According to the definition of hydraulic specific energy in Sect.2.3.6.2 of IEC60041, to establish the efficiency, the need to measure the discharge is thus eliminated by using the specific mechanical energy E m together with the specific hydraulic energy E h .
Where, η h , the hydraulic efficiency; η m , the turbine mechanical efficiency; P axial , the input power of generator; P m , the mechanical power output of turbine; P abs10 , P abs11 , water pressure in high pressure section; P abs20 , water pressure in low pressure section; v 10 , v 11 , water velocity in high pressure section; v 20 , water velocity in low pressure section; Z 10 , Z 11 , the level of high-pressure measuring section; Z 20 , the level of low-pressure measuring section; ρ, water density; g, local gravity; θ 1 , water temperature of high pressure section; θ 2 , water temperature of low pressure section; C P , the water specific heat capacity; a , the adiabatic coefficient; δE m , the corrective energy term of turbine mechanical energy; P g , generator output power; η g , the generator efficiency.P L_T , the turbine guide bearing loss; P L_T hrust , the loss of thrust bearing which attributed to the turbine in proportion; Q, the flow discharge of turbine; The flow discharge of turbine Q can be calculated with the turbine mechanical power and specific mechanical energy (Cao et al. 2021).

Measurement arrangement
Thermodynamic test is carried out on a vertical pump-turbine with 600 m rated net head by.Main turbine quantities are listed in Table 1.
As shown in Fig. 2, measuring pipes of flow differential pressure in the spiral case at turbine working condition locate at the inner wall of spiral case (WK1A, WK1B and WK1C).When the unit operates at turbine mode, two differential pressures shall be generated and recorded as the difference between WK1A & WK1C and the difference between WK1B and WK1C.At pump working condition, the high-pressure measuring pipes for flow differential pressure locate at the outlet section of draft tube (WK6A and WK6B), and low-pressure measuring pipes locate at draft tube cone (WK3A, WK4A  For pump-turbine, water in the penstock flows along two directions as the turbine and pump operating conditions.And the sampling probe must satisfy the two operating conditions.So in order to get the pressure and velocity and temperature parameters, such as P abs1,2 and v 1,2 and θ 1,2 , specific sampling vessels are designed and the specific sampling probes are installed on both sides of high-pressure sample vessels and low-pressure draft tube Sect.(Yang et al. 2022).As shown in Fig. 3, two opposite inner channels are arranged in the probe to sample the water from both directions.
The arrangement of measuring apparatus at high-pressure measuring section is shown in Fig. 3.The sampling vessel is isolated from the air by the heat isolation material, which can minimize the heat exchange between the sampling vessel and environment air.The high accuracy temperature sensor is installed on the bottom of sampling vessel, the pressure transducer and electric-magnetic flow meter are installed on the two sides, which are used to measure the inner pressure and flow discharge.We can regulate the valve after the electric-magnetic flow meter and change the flow discharge goes through the vessel, so as to regulate the inner water temperature.
On the low-pressure measuring section, the pressure and velocity and temperature parameters will be measured directly at draft tube temperature parameters will be measured directly at draft tube.The arrangement of temperature sensor at low-pressure measuring section is shown in Fig. 4. Metal measuring frame are located in the draft tube flow channel to support one thermometer on the center point, which can measure the average temperature of water flow leaded from four inlet pipes perpendicular to each other.The measuring method uses only one thermometer but can meet the requirements of IEC 60,041.To satisfy the two directions of water flow from turbine and pump operating conditions, two opposite non-return valves are located at the start of inlet pipes, so the water flowing in the pipes may be leaded to the temperature sensor.Finally, the related specific mechanical energy and specific hydraulic energy can be calculated by measuring the flow discharge, temperature and water pressure on the sides of high-pressure and low-pressure section.

Results and discussion
Measurements are taken with a load-adjusting test step by step at turbine working condition.Measured turbine efficiency by thermodynamic method on rated net head is shown in Fig. 5.The design values in Fig. 5 are read on the operating characteristics curves which derived from the model test of the prototype pump-turbine.As shown in Fig. 5, measured turbine efficiency by thermodynamic method is higher than design values derived from model test when turbine mechanical output power is higher than 200 MW.Measured relation between turbine discharge and flow differential pressure is shown in Fig. 6 (taking WK1A for example).As shown in Table 2, the Winter-Kennedy coefficients K and n calculated by the thermodynamic method results are different from that determined on model test.At turbine working condition, the coefficients K and n calculated by the thermodynamic method results are higher than that determined on model test for all measuring sections (WK1A and WK1B) of spiral case.The deviation of calibrated results of thermodynamics presents as positive deviation.It means that if coefficients derived from model test are taken to monitoring the real-time flow rate of turbine, a lower value would be obtained and then lead to a higher efficiency of turbine.
As shown in Table 3, at pump working condition the coefficients K and n calculated by the thermodynamic method results are lower than that determined on model test for all measuring sections (WK3A and WK4A and WK5A) of draft tube.The deviation of calibrated results of thermodynamics presents as negative deviation.It means that if coefficients derived from model test are taken to monitoring the real-time flow rate of pump, a higher value would be obtained and then lead to a higher efficiency of pump.
Test results of thermodynamics indicate the importance of calibration for the coefficients K and n when they are applied to the online flow monitoring, daily flow control or load adjustment.Due to the limitation of manufacture and installation, it is almost impossible to ensure the geometrical similarity of water passages between model and prototype.The coefficients K and n determined on model test must be calibrated when they are used before the pump-turbine goes into the commercial operation.With the calibrated coefficients of flow differential pressure, the prototype turbine could be monitored in real time conveniently.
The calibration uncertainty includes three parts which are the measuring uncertainty of thermodynamics, differential pressure and the fitting error for the calibrated coefficients.According to the uncertainty estimation of thermodynamics, the uncertainty of measured flow rate is 0.7%∼0.8%.The uncertainty of measurement of differential pressure is 0.2% taken the calibrated certificate of the sensor.Then the fitting error of least square method presents below 1%.The calibration uncertainty of flow coefficients by thermodynamics in this case is approximately 1%∼1.3%.

Conclusion
Thermodynamic test is carried out on a vertical pump-turbine with 600 m rated head.Measuring devices are designed to satisfy both the turbine and pump operating conditions.Test results of discharge and efficiency of both turbine and pump operating conditions are presented in this paper.With the measured discharge both at turbine and pump operation conditions, flow differential pressure coefficients are determined and calibrated.At turbine working condition, the coefficients K and n calculated by the thermodynamic method results are higher than that determined on model test.And at pump working condition, the coefficients K and n calculated by the thermodynamic method results are lower than that determined on model test.
Compared to previous calibration ways, for example, calculation according to dimensions of flow passage, calibration by the predicted flow rate or the maximum efficiency of prototype turbine, a high-precision method to determine the actual flow rate of prototype turbine may lead to a more reliable calibrated result.When the prototype turbine is designed, some specific measures are necessary to prepare for a site calibration by thermodynamics or other available methods.During manufacture and installation, the measuring pipes for online monitoring need to be carefully checked to ensure the geometrical similarity to model turbine in case that site calibration is inconvenient to conduct.

Fig. 1
Fig. 1 Definition of measuring sections at high and low-pressure section and WK5A).When the unit operates at pump mode, three differential pressures shall be generated and recorded as the difference between WK3A & WK6A/6B, WK4A & WK6A/6B and WK5A & WK6A/6B.All the measuring pipes are arranged geometricsimilarly to the flow differential measuring pipes on model test.

Fig. 2
Fig. 2 Location of the measuring pipes for flow differential pressure

Fig. 4
Fig. 4 Arrangement of measuring apparatus at low-pressure measuring section

Fig. 6
Fig. 6 Measured turbine discharge VS flow differential pressure of spiral case

Fig. 8
Fig. 8 Measured pump discharge VS flow differential pressure of draft tube

Table 1
Main quantities of the test pump-turbine

Table 2
Comparison to model test results (turbine working condition)

Table 3
Comparison to model test results (pump working condition)