On the Implementation of Variable Speed in Pump-Turbine Units Providing Primary and Secondary Load-Frequency Control in Generating Mode

This paper analyses different control strategies for the speed control loop of a variable-speed pump-turbine unit equipped with a doubly fed induction generator, operating in generating mode in an isolated power system with high penetration of intermittent renewable energy. The control strategies are evaluated and compared to each other in terms of the amount of water discharged through the pump-turbine and of the wicket gates fatigue while providing primary and secondary load-frequency control. The influence of the penstock length and the initial operating point on the performance of each control strategy is studied in detail. For these purposes, several simulations have been performed with a suitable dynamic model of the pumped-storage hydropower plant and the power system. The results of the paper indicate that a proper control strategy would consist in updating the reference speed according to the power generation schedule and keeping it constant within each scheduling period (typically 1 h).


Introduction
The integration of intermittent energy sources in the grid is one of the most important challenges for hydropower plants nowadays [1].Excess wind and solar energy is a real fact, especially in isolated systems, such as small or medium size island electric power systems [2].The penetration of these intermittent sources is very high in order to avoid environmental and social costs of fossil fuel-based electricity generation [3,4].Pumped storage hydropower plants (PSHPs) are useful to hedge against the uncertainty in intermittent renewable energy (wind and solar) [5].PSHPs can contribute significantly to the load-frequency control in generating mode.In pumping mode, variable-speed or hydraulic short-circuit operation is required to provide load-frequency control [6,7].
Doubly fed induction generators (DFIG) are widely used for variable speed pumped-storage units [8,9].In [10] two control schemes are proposed for variable speed pumped-storage units operating in generating mode.In the load control scheme, the power injected into the grid is regulated by the rotor-side converter and the turbine governor is in charge of restoring the unit running speed to the reference value.In this way, the electric power delivered by the unit is free from water hammer effects since the variations in the turbine mechanical power are absorbed by the inertia of the rotational masses.In addition, the reference speed can be modified in a speed control loop in order to increase the hydraulic efficiency [11].
The dynamic response of a PSHP equipped with a DFIG and load control scheme has been analyzed in detail in [12][13][14].In [12,13] the speed reference is kept constant during the simulations, whereas in [14] only a sudden decrease of the active power set point is simulated.There are therefore several questions regarding the speed control loop which have not been answered yet.Variable speed pumped-storage units are expected to provide primary and secondary load-frequency control both in generating and pumping mode.As it is well known, variable speed allows increasing the efficiency in turbine mode.However, to authors' knowledge there is no previous work on how to update the reference speed of a hydropower unit when it provides load-frequency control, as most hydropower units do.How often the optimal speed (as referred to in [14]) should be updated in order to provide primary and secondary load-frequency control in an optimum way is the main question addressed in this paper.In order to ask this question, it is necessary to define a criterion to evaluate the performance of the pump-turbine unit.The criterion used in this paper for this purpose is based on the amount of water discharged through the pump-turbine, and on the fatigue of the wicket gates servo mechanism, while providing load-frequency control: the lower the water usage (and therefore the necessary energy consumption for pumping) and the wicket gates fatigue, the better the pump-turbine performance, provided that the response meets certain quality standards.
This paper studies and compares to each other different control strategies for the speed control loop of a PSHP equipped with a pump-turbine and a DFIG when it operates in generating mode.The PSHP is connected to a small isolated power system with thermal, wind and solar generation, and which provides secondary load-frequency control under the orders of an automatic generation control (AGC) system.
A dynamic model of the PSHP and the power system has been developed and several simulations have been carried out in order to evaluate and compare to each other the proposed control strategies.Moreover, the influence of the initial operating point (initial power and gross head) and the penstock length on the performance of each control strategy has been analyzed in detail.
The remainder of the paper is organized as follows: in Section 2, the main characteristics of the PSHP and the power system where the study is carried out are presented; in Section 3 the non-linear dynamic model of the considered power system is described; the control strategies formulated for the speed control loop are described in Section 4; the simulations carried out in this study are described in Section 5; the results of the simulations are presented in Section 6; finally, the main conclusions of the paper are summarized in Section 7.

Description of the Pumped Storage Hydropower Plant and the Power System
The PSHP modelled in this paper is equipped with a pump-turbine unit coupled to a DFIG and the rated power of the unit is 14.97 MW.Two solutions with different penstock length (1350-m and 300-m) have been studied in order to gain insight on the influence of the conduit water inertia on the dynamic response of the PSHP.There is no surge tank in the scheme and the tail race tunnel is not considered due to its short length.The power system to which the PSHP is connected comprises six fuel-oil generating units, five diesel-oil generating units, seven gas-oil generating units and two combined cycle gas units.The total installed thermal generation power in July 2012 was 852 MW [7]; in addition, there were 35 MW and 24 MW of wind and photovoltaic generation at that time.

Dynamic Model
Figure 1 shows the block diagram of the simulation model developed in Matlab-Simulink.Each block of the diagram is briefly described below.

Penstock
Hydraulic transients in close conduits are properly described by a couple of partial differential equations, usually referred to as mass and momentum conservation equations [15].In this paper, a lumped parameters model has been used to transform the above-mentioned system of partial differential equations in a system of ordinary differential equations [13,16].For the long penstock configuration, the penstock has been modelled through Equations ( 1) and ( 2), considering ten Γ-shaped consecutive elements (n ge = 10), Figure 2 [7,17].
In Equations ( 1) and ( 2), nge is the number of elements, hi (p.u.) is the head at the end of the i-th  element, qi (p.u.) is the flow in the i-th  element, Tw (s) is the water starting time, L (m) is the penstock length, g (m/s 2 ) is the gravity acceleration, F (m 2 ) is the penstock cross section, Qb (m 3 /s) is base flow, Hb (m) is the base head, Te (s) is the water elastic time, ap (m/s) is the penstock wave speed and r (p.u.) is twice the head loss in the penstock with rated flow.
For the short penstock configuration, the penstock has been modelled through Equations ( 3) and (4), considering a single Π-shaped element, Figure 3.The element has one series branch and two shunt branches.The total head losses are included in the series branch of Equation (3), whereas the elasticity effects are included in the shunt branches of Equation (4).Expression in Equation ( 4) includes a correction coefficient β to match the first peak of the frequency response of the continuous model [18].1.
In Equations ( 1) and (2), nge is the number of elements, hi (p.u.) is the head at the end of the i-th  element, qi (p.u.) is the flow in the i-th  element, Tw (s) is the water starting time, L (m) is the penstock length, g (m/s 2 ) is the gravity acceleration, F (m 2 ) is the penstock cross section, Qb (m 3 /s) is base flow, Hb (m) is the base head, Te (s) is the water elastic time, ap (m/s) is the penstock wave speed and r (p.u.) is twice the head loss in the penstock with rated flow.
For the short penstock configuration, the penstock has been modelled through Equations ( 3) and (4), considering a single Π-shaped element, Figure 3.The element has one series branch and two shunt branches.The total head losses are included in the series branch of Equation (3), whereas the elasticity effects are included in the shunt branches of Equation ( 4).Expression in Equation ( 4) includes a correction coefficient β to match the first peak of the frequency response of the continuous model [18].
T e 2 pq i ´qi`1 q and T e " In Equations ( 1) and (2), n ge is the number of elements, h i (p.u.) is the head at the end of the i-th Γ element, q i (p.u.) is the flow in the i-th Γ element, T w (s) is the water starting time, L (m) is the penstock length, g (m/s 2 ) is the gravity acceleration, F (m 2 ) is the penstock cross section, Q b (m 3 /s) is base flow, H b (m) is the base head, T e (s) is the water elastic time, a p (m/s) is the penstock wave speed and r (p.u.) is twice the head loss in the penstock with rated flow.
For the short penstock configuration, the penstock has been modelled through Equations ( 3) and ( 4), considering a single Π-shaped element, Figure 3.The element has one series branch and two shunt branches.The total head losses are included in the series branch of Equation (3), whereas Energies 2015, 8, 13559-13575 the elasticity effects are included in the shunt branches of Equation (4).Expression in Equation ( 4) includes a correction coefficient β to match the first peak of the frequency response of the continuous model [18].

3
 element, qi (p.u.) is the flow in the i-th  element, Tw (s) is the water starting time, L (m) is the penstock length, g (m/s 2 ) is the gravity acceleration, F (m 2 ) is the penstock cross section, Qb (m 3 /s) is base flow, Hb (m) is the base head, Te (s) is the water elastic time, ap (m/s) is the penstock wave speed and r (p.u.) is twice the head loss in the penstock with rated flow.
For the short penstock configuration, the penstock has been modelled through Equations ( 3) and ( 4), considering a single Π-shaped element, Figure 3.The element has one series branch and two shunt branches.The total head losses are included in the series branch of Equation ( 3), whereas the elasticity effects are included in the shunt branches of Equation ( 4).Expression in Equation ( 4) includes a correction coefficient β to match the first peak of the frequency response of the continuous model [18].
In Equations ( 3) and ( 4), h c is the gross head, h is the net head at the pump-turbine inlet, q t is the flow in the penstock, and q is the flow through the pump-turbine.

Pump-Turbine
The performance curves of the pump-turbine unit in generating mode have been approximated by those used in [17] and are shown in Figure 4.In this case the hill chart, expressed as a function of the unit speed n 11 and flow q 11 in Equation ( 5), is experimental, where z is the wicket gates position.
In the absence of experimental data, the selection of the pump-turbine model can be crucial to accurately predict its behavior with variable speed [19].
Energies 2015, 88, page-page In Equations ( 3) and ( 4), hc is the gross head, h is the net head at the pump-turbine inlet, qt is the flow in the penstock, and q is the flow through the pump-turbine.

Pump-Turbine
The performance curves of the pump-turbine unit in generating mode have been approximated by those used in [17] and are shown in Figure 4.In this case the hill chart, expressed as a function of the unit speed n11 and flow q11 in Equation ( 5), is experimental, where z is the wicket gates position.
In the absence of experimental data, the selection of the pump-turbine model can be crucial to accurately predict its behavior with variable speed [19].The "pump-turbine" block calculates the flow discharged through the pump-turbine runner q, as well as the shaft mechanical power pt, by interpolating in the above-mentioned performance curves.Turbine dynamics has been neglected.

Optimal Speed
The "optimal speed" block determines the optimal running speed of the unit for a given mechanical power pt and gross head hc, by interpolating iteratively in the pump-turbine performance curves.The optimal running speed is defined as the one which allows generating the given The "pump-turbine" block calculates the flow discharged through the pump-turbine runner q, as well as the shaft mechanical power p t , by interpolating in the above-mentioned performance curves.Turbine dynamics has been neglected.

Optimal Speed
The "optimal speed" block determines the optimal running speed of the unit for a given mechanical power p t and gross head h c , by interpolating iteratively in the pump-turbine performance curves.The optimal running speed is defined as the one which allows generating the given mechanical power with the highest efficiency, or rather, with the lowest water discharge.The block searches for the optimal speed within the pump-turbine hill chart, which have been previously parameterized.

Pump-Turbine-Doubly Fed Induction Generator Inertial Model
The "pump-turbine-DFIG inertial model" block calculates the unit running speed n from the unbalance between the electrical power delivered to the grid p g and the shaft mechanical power p t in Equation ( 6): T m (s) is the mechanical starting time of the unit.

Speed Governor
The control diagram proposed in [10] for the speed governor of a hydro unit with DFIG is used in the present work.The "optimal speed" block sends the reference speed n ref to the "speed governor" block and a conventional PI controller in Equation ( 7) acts on the wicket gates opening z in order to follow the optimal speed signal: The controller gains are expressed in terms of the parameters used to model the classical mechanical hydraulic governor, δ is transient speed drop, and T r (s) is dashpot time constant.Two different pairs of controller gains have been used depending on the penstock length.

Power Control
The changes in the electrical power required to the DFIG ∆p ref in Equation ( 8), are calculated from the variations in the power set-point signal sent by the AGC system psp hp (p.u.), and the system frequency deviation F g (rd/s), in the "power control" block.The variations in the pump-turbine power set-point due to the contribution of the PSHP to the primary load-frequency control p R1 (p.u.), has been modelled in a similar way to [20] in Equation (9).Two different control actions are considered: a droop-based control action p R1E (p.u.) in Equation (10), and an inertia emulation control action p R1I (p.u.) in Equation ( 11), [21,22]: p R1E " K¨F g ω 0 (10) In expression of Equation (10), ω 0 (rd/s) is the base system frequency.In expression of Equation ( 11) T d (s) is the derivative gain and T fnf (s) is the time constant of the Power control block.

Doubly Fed Induction Generator Electrical Model
The electromagnetic torque is assumed to be controlled by the rotor-side inverter [23,24] in an instantaneous manner, comparing to the dynamics studied in the paper.In the "DFIG electrical model" block the electromagnetic torque, c em (p.u.) and the rotor power, p r (p.u.) are calculated from the running speed, n and the required electric power, (p 0 + ∆p ref ) (Figure 5).If none of these values exceeds its corresponding upper limit, the electric power output matches exactly the required electric power; otherwise, the electric power output is conveniently limited.The power losses in the generator are neglected.

Power System
A single node inertial model has been used in this paper to represent the isolated power system.This assumption is accepted for the time scale of the dynamics analyzed in this work [25,26].The changes in the system frequency are calculated from the unbalance between total generation and demand in Equation ( 12), in the "power system" block: Wind and solar generators are supposed to be connected to the system through frequency converters and do not contribute to system inertia.The contribution of the thermal generation to the primary load-frequency control is represented by R1 (p.u.) in Equation ( 13) [27].The contribution of the thermal generation to the secondary load-frequency control R2 (p.u.) is modelled through the transfer function proposed in [28], Figure 6.Although modern wind and solar generators could contribute to the load-frequency regulation, this contribution entails a cost: some energy will be lost [18].Therefore, in this paper no contribution to frequency regulation from the wind and solar farms is assumed.

Power System
A single node inertial model has been used in this paper to represent the isolated power system.This assumption is accepted for the time scale of the dynamics analyzed in this work [25,26].The changes in the system frequency are calculated from the unbalance between total generation and demand in Equation ( 12), in the "power system" block: Wind and solar generators are supposed to be connected to the system through frequency converters and do not contribute to system inertia.The contribution of the thermal generation to the primary load-frequency control is represented by R1 (p.u.) in Equation ( 13) [27].The contribution of the thermal generation to the secondary load-frequency control R2 (p.u.) is modelled through the transfer function proposed in [28], Figure 6.Although modern wind and solar generators could contribute to the load-frequency regulation, this contribution entails a cost: some energy will be lost [18].Therefore, in this paper no contribution to frequency regulation from the wind and solar farms is assumed.
Energies 2015, 8, 13559-13575 6 primary load-frequency control is represented by R1 (p.u.) in Equation ( 13) [27].The contribution of the thermal generation to the secondary load-frequency control R2 (p.u.) is modelled through the transfer function proposed in [28], Figure 6.Although modern wind and solar generators could contribute to the load-frequency regulation, this contribution entails a cost: some energy will be lost [18].Therefore, in this paper no contribution to frequency regulation from the wind and solar farms is assumed.In expressions of Equations ( 12) and ( 13), ∆p d (p.u.) is the variation in the load demand at rated frequency, ∆p g (p.u.) is the variation in the electric power delivered to grid by the PSHP, M (s) is the equivalent mechanical inertia of the power system, R (p.u.) is the equivalent permanent droop of the synchronized thermal units, D net (p.u./p.u.) the power system damping constant and T net (s) is the time constant of the primary load-frequency control provided by the synchronized thermal unit.

Automatic Generation Control
In the Spanish insular power systems, an AGC system is in charge of eliminating the frequency deviations after the performance of the primary load-frequency control.In this paper, the AGC system is modelled in a similar way to [7].The proportional coefficient in the control area K f (MW/Hz) is calculated using the recommendations of [29].The total regulation effort RR (p.u.), is proportional to the frequency deviation F g in Equation (14).The regulation effort is distributed among the synchronized units as a function of the participation factors K hp , K ps , (p.u.) in Equations ( 15) and ( 16) [30].In expressions of Equations ( 15) and ( 16), subscript hp refers to the PSHP and subscript ps refers to the power system; psp hp (p.u.) is the increment in the power set-point signal sent by the AGC, psp ps (p.u.) is the increment in the power set-point of the synchronized thermal units, T hp (s) is the time constant of the AGC system for the PSHP and T ps (s) is the time constant of the AGC system for the synchronized thermal units:

Model Numerical Values
In this section a summary of the main model magnitudes is presented.Table 1 contains the model base values.Table 2 lists the values of the model parameters.The parameters of the power system were properly calibrated in [7] from the system response during a load tripping event.
Energies 2015, 8, 13559-13575 Governor PI gains have been adjusted by using two different tuning criteria.On one hand, for the long penstock configuration, the criterion proposed in [18] has been used.This criterion considers the elasticity of both water and penstock, as it is required in long penstocks [31].On the other hand, for the short penstock configuration the tuning criterion proposed in [32] is used.This criterion is one of the most used ones for isolated power plants and usually yields moderate oscillations in the wicket gates position.As can be seen in Table 2, the head loss in the penstock, expressed in per unit values with respect to the base head, is assumed to be identical for both configurations.The possible differences that might exist between the two configurations are expected not to appreciably affect the parameters used to evaluate and compare the speed control strategies analyzed in the paper.

Control Strategies for the Speed Control Loop
As it was above-mentioned, different control strategies have been studied for the speed control loop of the pumped-storage unit.The studied control strategies are summarized below.It is important to mention that the option of bypassing the converter is not considered in the paper: Strategy A (n ref = n fix ).The reference speed n ref is kept constant during the whole simulation.Two different values have been adopted: the speed (n fix = 1.051) at which the unit provides the rated power (p g = p t = 1.0) with the best efficiency and an intermediate gross head (h c = 1.131); and the speed (n fix = 1.088) at which the unit provides an intermediate power (p g = p t = 0.7) with the best efficiency and the same gross head.The first speed corresponds to the design or rated conditions, whereas the second one corresponds to an intermediate value of power that maximizes the up and down power reserve and that will likely be the most frequent steady-state operating point.
Strategy B (n ref = n opt,fix ).The reference speed is optimum for the initial operating point (h c 0 and p 0 ) and is kept constant during the simulation.
Energies 2015, 8, 13559-13575 Strategy C (n ref = n opt,var ).The reference speed varies throughout the simulation as a function of the operating point (h c and p t ).The reference speed changes every T n seconds (hereinafter referred to as update period).

Simulations
The above-described strategies have been evaluated in terms of the amount of water discharged Vol, and the wicket gates fatigue, while providing load-frequency control, considering different initial operating points and penstock lengths, as well as different update periods for the strategy C. The wicket gates fatigue has been measured as the number of changes in the sign of the first derivative of the wicket gates position, zvar.Therefore, in each simulation zvar is an integer value but in some tables it is expressed as a decimal number resulting from the averaging of the multiple simulations.Table 3 summarizes the initial operating points considered in the simulations.The initial operating points have been superimposed on the pump-turbine hill chart in Figure 4.The dynamic response of the PSHP has been simulated with each strategy, initial operating point, and update periods for strategy C, in nine different scenarios of wind and solar power.
These scenarios have been built by combining the wind and solar photovoltaic power records presented in [33], conveniently scaled to the installed wind and photovoltaic power in the system under study.Said power records are shown in Figure 7, expressed in per unit values with respect to the maximum wind and solar power in the system analyzed in [33].The nine scenarios are characterized in Table 4, in terms of the wind and solar power output level.

Simulation Results
The main results obtained in this study are discussed in the following subsections.The numerical values presented in the following tables and figures are the average values across the nine considered scenarios.

Update Period
Tables 5 and 6 show the results obtained for different update periods (strategy C) and compare these results with those obtained with strategies A and B. The results corresponding to the strategy A are used as a basis for the comparison.The results included in the tables are the average values across all considered initial operation points, (p 0 = 0.6, 0.7, 0.8 and h c 0 = 1.068, 1.131, 1.250).As can be seen in Table 5, for both long and short penstock configurations, there is an update period which minimizes Vol and another which minimizes zvar.The results shown in Tables 5 and 6 confirm that the strategy C allows reducing the amount of water discharged while providing load-frequency control, with respect to the strategies A and B. However, this benefit comes at the expense of an increase in the wicket gates fatigue, especially for the short penstock configuration.The strategy B yields a reduction in the amount of water discharged of a similar magnitude to that yielded by the strategy C, with a significantly lower wicket gates fatigue.
Figures 8 and 9 show the running speed and the reference speed with the three control strategies for the long and short penstock configurations, in scenario 1 and with a specific initial operating point.By comparing Figures 8 and 9 it is observed that the lower water inertia in the short penstock facilitates the governor to maintain the unit speed close to the reference speed albeit at the expense of a considerable increase in the wicket gates fatigue, as shown in Tables 5 and 6.
Figures 8 and 9 show the running speed and the reference speed with the three control strategies for the long and short penstock configurations, in scenario 1 and with a specific initial operating point.By comparing Figures 8 and 9 it is observed that the lower water inertia in the short penstock facilitates the governor to maintain the unit speed close to the reference speed albeit at the expense of a considerable increase in the wicket gates as shown in Tables 5 and 6.

Initial Power
Tables 7 and 8 show the results obtained with the three control strategies for different initial powers delivered to the system by the PSHP.The results corresponding to the strategy A are used as a basis for the comparison.The results included in the Tables are the average values across the three initial considered gross heads.The update periods which give the least Vol and the least zvar in Tables 5 and 6 have been used for the strategy C in this case.
On the one hand, the results shown in Tables 7 and 8 confirm those discussed in Section 6.1: the strategy B yields a reduction in the amount of water discharged similar to that yielded by the strategy C, with a considerable reduction in the wicket gates fatigue.On the other hand, the results shown in these tables demonstrate that the water savings yielded by the strategies B and C slightly decrease as

Initial Power
Tables 7 and 8 show the results obtained with the three control strategies for different initial powers delivered to the system by the PSHP.The results corresponding to the strategy A are used as a basis for the comparison.The results included in the Tables are the average values across the three initial considered gross heads.The update periods which give the least Vol and the least zvar in Tables 5 and 6 have been used for the strategy C in this case.
On the one hand, the results shown in Tables 7 and 8 confirm those discussed in Section 6.1: the strategy B yields a reduction in the amount of water discharged similar to that yielded by the strategy C, with a considerable reduction in the wicket gates fatigue.On the other hand, the results shown in these tables demonstrate that the water savings yielded by the strategies B and C slightly decrease as the initial power decreases; as can be seen in Figure 4, the efficiency barely changes with speed for the lowest initial power (p 0 = 0.6).

Gross Head
The storage capacity of the upper reservoir is small and therefore both the water level and the gross head h c are expected to vary frequently.Tables 9 and 10 show the results obtained with the three control strategies for different initial gross heads.The results corresponding to the strategy A are used as a basis for the comparison.The results included in the Tables are the average values across the three initial considered powers (p 0 = 0.6, 0.7 and 0.8).The update periods which give the least Vol and the least zvar in Tables 5 and 6 have been used in this case.in the sign of the first derivative of the wicket gates position.Different T n (update periods) have been analyzed for strategy C. For these purposes, several simulations have been performed.The results of the simulations demonstrate that with respect to strategy A, strategies B and C allow reducing the amount of water discharged while providing load-frequency control.In general, strategy C yields higher water savings than strategy B, albeit at the expense of a significant increase in the wicket gates fatigue.Strategy B yields water savings of a similar magnitude to those yielded by strategy C, and a wicket gates fatigue lower in all cases than that of strategy C, and in most cases than that of strategy A.
The influence of the penstock length, and the initial operating point on the water usage and the wicket gates fatigue has also been analyzed.The results of the simulations show that neither the penstock length nor the initial operating point has a significant influence on the selection of the best control strategy.In general, the longer the penstock the lower the wicket gates fatigue with the three control strategies, what can be expected to a certain extent, due to the higher water inertia.The penstock length does have a considerable influence on the optimum update period for strategy C.
An interesting practical conclusion can be drawn from the results obtained in the paper.According to these results, the best way to implement variable speed in a pump-turbine unit that operates in generating mode and that provides load-frequency control, would be to update the reference speed according to the power generation schedule and keep it constant within each scheduling period (typically 1 h).
Finally, as a future line of work it would be interesting to carry out a similar study with other pump-turbines of different specific speeds.Future studies should also consider different PI tuning criteria as well as the possibility to find a trade-off solution between the hydraulic efficiency and the wicket gates fatigue by means of a multi-objective approach.Additionally, the benefits of variable speed operation from the point of view of the Transmission System Operator (TSO) should be also analyzed.

Figure 1 .
Figure 1.Block diagram of the dynamic model.All variables are expressed in p.u.values with respect to the base values included in Table1.

Figure 1 .
Figure 1.Block diagram of the dynamic model.All variables are expressed in p.u. values with respect to the base values included in Table1.

Figure 1 .
Figure 1.Block diagram of the dynamic model.All variables are expressed in p.u.values with respect to the base values included in Table1.

Figure 2 .
Figure 2. Scheme of the long penstock model.

Figure 2 .
Figure 2. Scheme of the long penstock model. dq

Figure 3 .
Figure 3. Scheme of the short penstock model.Figure 3. Scheme of the short penstock model.

Figure 3 .
Figure 3. Scheme of the short penstock model.Figure 3. Scheme of the short penstock model.

Figure 4 .
Figure 4. Pump-turbine performance curves in generating mode; considered initial operating points superimposed in the curves.

Figure 4 .
Figure 4. Pump-turbine performance curves in generating mode; considered initial operating points superimposed in the curves.

Figure 5 .
Figure 5. "DFIG electrical model" block.(T fp (s) is the converter time constant and n syn (p.u.) is the synchronous speed of the group).Doubly fed induction generators: DFIG.

Figure 6 .
Figure 6."Power system" block (Tsc (s) is the parameter of the transfer function used to model the secondary load frequency control of the synchronized thermal units).Figure 6. "Power system" block (T sc (s) is the parameter of the transfer function used to model the secondary load frequency control of the synchronized thermal units).

Figure 6 .
Figure 6."Power system" block (Tsc (s) is the parameter of the transfer function used to model the secondary load frequency control of the synchronized thermal units).Figure 6. "Power system" block (T sc (s) is the parameter of the transfer function used to model the secondary load frequency control of the synchronized thermal units).

Table 1 .
Model base values.

Table 3 .
Initial values for each operating point considered in this study.eff is the turbine efficiency.Superscripts 0 indicate initial conditions.

Table 4 .
Wind and solar power scenarios analyzed in the paper.H = high, M = medium, L = low.

Table 4 .
Wind and solar power scenarios analyzed in the paper.H = high, M = medium, L = low.

Table 5 .
Vol and zvar with the three control strategies and different update periods for the strategy C, n fix = 1.0508 p.u.

Table 6 .
Vol and zvar with the three control strategies and different update periods for the strategy C, n fix = 1.0882 p.u.

Table 7 .
Vol and zvar with the three control strategies, and different initial pumped storage hydropower plants (PSHP) powers p 0 , n fix = 1.0508 p.u.

Table 8 .
Vol and zvar with the three control strategies, and different initial PSHP powers p 0 , n fix = 1.0882 p.u.

Table 9 .
Vol and zvar with the three control strategies, and different initial gross heads h c 0 , n fix = 1.0508 p.u.