Numerical Study of Constant Pressure Systems with Variable Speed Electric Pumps

This work focuses on the modeling and simulation of constant pressure systems based on variable speed pumps, with the aim of studying and evaluating their performance from a multidisciplinary approach. Using the physical models of the Simscape library, from MATLAB/Simulink R2019b, two study cases are assembled consisting of: piping system, a hydropneumatic tank, centrifugal pumps with an induction motor, variable speed drives, and a control system. Case one is comprised of one pump at a fixed speed and another at variable speed, and case two with both pumps at variable speed. For the parameterization of the models, data from manufacturers and process requirements are used. The different stages of the control system are integrated and configured; these are constant V/f control, slip compensation, space vector modulation (SVM,) and pressure controller. The dynamic response of the system, power saving, transient current at startup, and harmonic distortion are evaluated. The results showed that both cases kept the pressure constant in the face of variable flow demand and smoothed out the current during startup. Case two saved more energy (between 28 and 49%) but generated more harmonic distortion. In addition, both cases have better performance compared with traditional fixed-speed technologies.


Introduction
At an industrial, commercial, and residential level, applications with pumps, compressors, fans, and others driven by electric motors represent 53% of total energy consumption, and there are new technologies that have the potential to reduce this electricity demand by between 20 to 30% [1]. In pumping applications with variable flow demand, the implementation of variable-frequency drives adapted to electric motors should be evaluated because of the potential energy savings that can be achieved and improvements in the overall performance of the system [2][3][4][5]. In these applications, the scalar control strategy with a constant V/f ratio is the most feasible, mainly because of its simplicity and low-cost implementation [6][7][8]. There are several studies where scalar control with a constant V/f ratio was simulated for the speed control of an induction motor; for example, in an open loop with voltage and slip compensation to improve the performance in low frequencies [9], and adding a PID controller for automatic regulation in a closed loop to eliminate the speed position error [8,10,11]. The performances of the control system in the open and Using the MATALAB/Simulink R2019b Simscape library, the blocks are reviewed, selected, and integrated to form two case studies, as described in Section 2.3. The models are parameterized using data from manufacturers and process requirements. Then all the stages of the control system are adjusted to meet the process requirements; these are: PI slip compensator, V/f scalar control, space vector modulation (SVM), and PI pressure controller. The simulation is run to solve the mathematical models, data and trend graphs are obtained for the subsequent analysis and evaluation. Finally, a comprehensive analysis of the hydraulic and electrical aspects of each case study is carried out, evaluating the dynamic response of the control system, power consumption and savings, starting transient current, and harmonic distortion. Based on these results, the case studies are compared to determine which one has the best performance. The novelty of this methodology lies in the use of Simscape physical models and the multidisciplinary analysis of the results. In other studies, linearized and approximate models are used, and the analysis is partial depending on the discipline (purely hydraulic or electric).

Capacity Regulation Methods in Centrifugal Pumps
Typical methods to regulate the capacity of a pumping system are on-off control, control valve, and speed variation. On-off control is justified for applications where there are tanks or reservoirs and where such precise control is not required. When demand flow adjustment is required, a control valve or variable speed drive should be used [21]. Flow regulation with a control valve can be seen in the laboratory work developed by [22], where a PID algorithm was implemented on a control valve placed in the discharge of a centrifugal pump operating at a fixed speed. The results showed a good performance of Using the MATALAB/Simulink R2019b Simscape library, the blocks are reviewed, selected, and integrated to form two case studies, as described in Section 2.3. The models are parameterized using data from manufacturers and process requirements. Then all the stages of the control system are adjusted to meet the process requirements; these are: PI slip compensator, V/f scalar control, space vector modulation (SVM), and PI pressure controller. The simulation is run to solve the mathematical models, data and trend graphs are obtained for the subsequent analysis and evaluation. Finally, a comprehensive analysis of the hydraulic and electrical aspects of each case study is carried out, evaluating the dynamic response of the control system, power consumption and savings, starting transient current, and harmonic distortion. Based on these results, the case studies are compared to determine which one has the best performance. The novelty of this methodology lies in the use of Simscape physical models and the multidisciplinary analysis of the results. In other studies, linearized and approximate models are used, and the analysis is partial depending on the discipline (purely hydraulic or electric).

Capacity Regulation Methods in Centrifugal Pumps
Typical methods to regulate the capacity of a pumping system are on-off control, control valve, and speed variation. On-off control is justified for applications where there are tanks or reservoirs and where such precise control is not required. When demand flow adjustment is required, a control valve or variable speed drive should be used [21]. Flow regulation with a control valve can be seen in the laboratory work developed by [22], where a PID algorithm was implemented on a control valve placed in the discharge of a centrifugal pump operating at a fixed speed. The results showed a good performance of the flow control system, but the study did not delve into the performance of the centrifugal pump. However, fixed speed operation with a control valve can waste energy, while variable Energies 2022, 15, 1918 4 of 22 speed operation can increase energy efficiency [23]. When a centrifugal pump operates at a variable speed, it is important to monitor its efficiency curve to determine its performance based on the flow it delivers. This curve defines the efficiency "from the cable to the water", that is to say, the total efficiency, which considers the mechanical losses in the pump as well as the electrical losses in the motor and the variable speed drive [24]. Figure 2a shows the characteristic curves of a variable speed centrifugal pump, which is the preferred choice for operating in a region where the required flow remains controlled at best possible efficiency. Figure 2a shows the characteristic curves of a variable speed centrifugal pump, which is the preferred choice for operating in a region where the required flow remains controlled at best possible efficiency.
An alternative to using a single pump in an application with variable demand is to use several smaller capacity pumps connected in parallel since the multiple variations in the demand of the system prevent a single pump from operating constantly, close to the best efficiency point (BEP). Operating a centrifugal pump too far from the BEP to either the right or left of the curve can put equipment at risk from adverse effects such as cavitation, vibration, impeller damage, recirculation in suction, or unloading, and reduced life of seals and bearings [21,24]. These damages can be minimized by maintaining the operating flow in the Preferred Operating Region (POR). The POR is the range of flow rates on both sides of the BEP, where the operational reliability and hydraulic efficiency of the pump are not substantially degraded, so operating the pump within the POR generally results in lower energy consumption, higher reliability, and less impact on the useful life of the equipment. Most centrifugal pumps have a POR between 70-120% of the BEP [25].  Figure 2b shows the performance curves of a multipump system at variable speeds for different speeds. Variation in speed allows greater flexibility in operation and provides better performance for the system. In the study developed by Ahmed et al. [26], two configurations of multiple pump systems are presented: the first with a single pump at a variable speed, and the second where all the pumps operate at variable speed. The diagrams of both configurations can be seen in Figure 3. An alternative to using a single pump in an application with variable demand is to use several smaller capacity pumps connected in parallel since the multiple variations in the demand of the system prevent a single pump from operating constantly, close to the best efficiency point (BEP). Operating a centrifugal pump too far from the BEP to either the right or left of the curve can put equipment at risk from adverse effects such as cavitation, vibration, impeller damage, recirculation in suction, or unloading, and reduced life of seals and bearings [21,24]. These damages can be minimized by maintaining the operating flow in the Preferred Operating Region (POR). The POR is the range of flow rates on both sides of the BEP, where the operational reliability and hydraulic efficiency of the pump are not substantially degraded, so operating the pump within the POR generally results in lower energy consumption, higher reliability, and less impact on the useful life of the equipment. Most centrifugal pumps have a POR between 70-120% of the BEP [25]. Figure 2b shows the performance curves of a multipump system at variable speeds for different speeds. Variation in speed allows greater flexibility in operation and provides better performance for the system. In the study developed by Ahmed et al. [26], two configurations of multiple pump systems are presented: the first with a single pump at a variable speed, and the second where all the pumps operate at variable speed. The diagrams of both configurations can be seen in Figure 3.

Electrical Power and Control System
There are different types of motors to drive pumps, but the induction motor is one of the most widely used because of the advantages it presents compared with other electric machines [11,14,27,28]. The accelerated advance of microprocessor systems and power electronics has made it possible to position variable speed drives as the main technology for motion control in applications driven by induction motors [10,29]. These types of equipment reduce the input energy to the motor when it is not operating at full load, which allows energy savings and operational improvements in systems that previously used motors without any type of speed control.
As concerns control techniques, there are two broad categories used by variable speed drives: scalar control and vector control. The most widely used scalar control technique is voltage-frequency (V/f) control, and vector control includes oriented field control and direct torque control [28][29][30]. Scalar control is less precise and presents a limited response in torque compared with other control techniques; it is simple to implement, it presents lower losses in switching, reduced computational processing, and a lower cost of implementation [4,11,12,14,30]. Some of the most common applications driven by the variable speed drive induction motor are pumps, compressors, fans, and small machines that operate at high speeds, but also, these types of applications require a strategy of simple control and moderate precision, so the V/f control is the most feasible in these cases [6][7][8].
Automatic process controllers can also help improve the performance of electromechanical systems. The implementation of better monitoring, control, and optimization strategies improves the energy performance directly through the reduction of waste (very frequently associated with undue oscillations in the process) and indirectly through better maintenance practices [31]. The most widely used automatic process controller is the PID controller, present in more than 95% of control loops, constituting an important tool for improving productivity, quality, and energy efficiency [32]. Research conducted by Gomes et al. [33] on energy efficiency in industrial processes and PID controllers demonstrates how an adequate control strategy together with a correct controller adjustment procedure can contribute to saving energy consumption.

Electrical Power and Control System
There are different types of motors to drive pumps, but the induction motor is one of the most widely used because of the advantages it presents compared with other electric machines [11,14,27,28]. The accelerated advance of microprocessor systems and power electronics has made it possible to position variable speed drives as the main technology for motion control in applications driven by induction motors [10,29]. These types of equipment reduce the input energy to the motor when it is not operating at full load, which allows energy savings and operational improvements in systems that previously used motors without any type of speed control.
As concerns control techniques, there are two broad categories used by variable speed drives: scalar control and vector control. The most widely used scalar control technique is voltage-frequency (V/f) control, and vector control includes oriented field control and direct torque control [28][29][30]. Scalar control is less precise and presents a limited response in torque compared with other control techniques; it is simple to implement, it presents lower losses in switching, reduced computational processing, and a lower cost of implementation [4,11,12,14,30]. Some of the most common applications driven by the variable speed drive induction motor are pumps, compressors, fans, and small machines that operate at high speeds, but also, these types of applications require a strategy of simple control and moderate precision, so the V/f control is the most feasible in these cases [6][7][8].
Automatic process controllers can also help improve the performance of electromechanical systems. The implementation of better monitoring, control, and optimization strategies improves the energy performance directly through the reduction of waste (very frequently associated with undue oscillations in the process) and indirectly through better maintenance practices [31]. The most widely used automatic process controller is the PID controller, present in more than 95% of control loops, constituting an important tool for improving productivity, quality, and energy efficiency [32]. Research conducted by Gomes et al. [33] on energy efficiency in industrial processes and PID controllers demonstrates how an adequate control strategy together with a correct controller adjustment procedure can contribute to saving energy consumption. Figure 4 shows the two cases to be modeled, simulated, and studied, which are made up of two pumps connected in parallel supplying flow and pressure to a water network (system) where there is a variable demand for flow and constant pressure needs to be maintained. When there are low flow rates, both cases operate with a single pump at variable speed. However, it is necessary to operate the two pumps in parallel if the flow increases; in this scenario, case one operates with one pump at a fixed speed and the other at a variable speed, and case two operates with both pumps at variable speed. In the discharge head of each pumping group, there is an ideal pressure sensor (simulating a manometer or pressure transducer) and an ideal flow sensor (simulating a flow meter). The models of the hydraulic, mechanical, electrical, and control components were obtained from Simulink's Simscape library in the R2019b version of MATLAB. These models are detailed below.

Study Cases for Modeling and Simulation
Energies 2022, 15, x FOR PEER REVIEW 6 of 22 Figure 4 shows the two cases to be modeled, simulated, and studied, which are made up of two pumps connected in parallel supplying flow and pressure to a water network (system) where there is a variable demand for flow and constant pressure needs to be maintained. When there are low flow rates, both cases operate with a single pump at variable speed. However, it is necessary to operate the two pumps in parallel if the flow increases; in this scenario, case one operates with one pump at a fixed speed and the other at a variable speed, and case two operates with both pumps at variable speed. In the discharge head of each pumping group, there is an ideal pressure sensor (simulating a manometer or pressure transducer) and an ideal flow sensor (simulating a flow meter). The models of the hydraulic, mechanical, electrical, and control components were obtained from Simulink's Simscape library in the R2019b version of MATLAB. These models are detailed below.     The mechanical component corresponds to an ideal source of angular velocity (from the induction motor) and an ideal torque load (towards the induction motor). The hydraulic component comprises a suction pipe and a discharge pipe. In addition, a check valve is installed in the discharge to isolate the pump when it goes out of operation. An ideal pressure sensor is connected to both sides, simulating a manometer or pressure transducer, and an ideal flow sensor is also used at the discharge, simulating a flow meter. The centrifugal pump is parameterized with the performance curve obtained from the datasheet provided by the manufacturer.
The characterization of the centrifugal pump is carried out using two one-dimensional search tables: (i) differential pressure vs. flow and (ii) brake power vs. flow-both at nominal angular velocity and flow density. To obtain the parameters at another angular velocity, the laws of affinity are used [34]. Equations (1)-(9) describe the complete model.

=
(1) The mechanical component corresponds to an ideal source of angular velocity (from the induction motor) and an ideal torque load (towards the induction motor). The hydraulic component comprises a suction pipe and a discharge pipe. In addition, a check valve is installed in the discharge to isolate the pump when it goes out of operation. An ideal pressure sensor is connected to both sides, simulating a manometer or pressure transducer, and an ideal flow sensor is also used at the discharge, simulating a flow meter. The centrifugal pump is parameterized with the performance curve obtained from the datasheet provided by the manufacturer.

Hydropneumatic Tank Model
The characterization of the centrifugal pump is carried out using two one-dimensional search tables: (i) differential pressure vs. flow and (ii) brake power vs. flow-both at nominal angular velocity and flow density. To obtain the parameters at another angular velocity, the laws of affinity are used [34]. Equations (1)-(9) describe the complete model. Figure 6 shows the model of the hydropneumatic tank, consisting of an accumulator tank with precharged gas, a connection pipe between the tank and the water network, and a blocking valve. The accumulator tank comprises a chamber with precharged gas and a chamber with fluid, which is connected to a hydraulic system. The process is mathematically modeled with Equations (10)- (15) [35]. Figure 6 shows the model of the hydropneumatic tank, consisting of an accumulator tank with precharged gas, a connection pipe between the tank and the water network, and a blocking valve. The accumulator tank comprises a chamber with precharged gas and a chamber with fluid, which is connected to a hydraulic system. The process is mathematically modeled with Equations (10)- (15) [35].

Hydropneumatic Tank Model
At t = 0, the initial condition is VF = Vi, where Vi is a parameter that can be entered initially.

Piping System Model
The water network, or piping system, is shown in Figure 7. In this model, there is the main pipeline, consumption valves that simulate the variable flow demand, and a discharge tank. The pipe is modeled with a circular cross section, taking into consideration only the resistive property of the pipe. The Darcy equation is At t = 0, the initial condition is V F = V i , where V i is a parameter that can be entered initially.

Piping System Model
The water network, or piping system, is shown in Figure 7. Figure 6 shows the model of the hydropneumatic tank, consisting of an accumulator tank with precharged gas, a connection pipe between the tank and the water network, and a blocking valve. The accumulator tank comprises a chamber with precharged gas and a chamber with fluid, which is connected to a hydraulic system. The process is mathematically modeled with Equations (10)- (15) [35].
At t = 0, the initial condition is VF = Vi, where Vi is a parameter that can be entered initially.

Piping System Model
The water network, or piping system, is shown in Figure 7. In this model, there is the main pipeline, consumption valves that simulate the variable flow demand, and a discharge tank. The pipe is modeled with a circular cross section, taking into consideration only the resistive property of the pipe. The Darcy equation is In this model, there is the main pipeline, consumption valves that simulate the variable flow demand, and a discharge tank. The pipe is modeled with a circular cross section, taking into consideration only the resistive property of the pipe. The Darcy equation is used to calculate the pressure loss due to the friction, and the Haaland approximation is used to calculate the friction factor in the turbulent regime. In the transition of regimens, from laminar to turbulent, the friction factor is determined by linear interpolation. As a result, the section of pipe is modeled according to Equations (16)-(18) [36].

Induction Motor Model
To simulate the induction motor, the asynchronous machine model is used. This block makes it possible to model three-phase induction motors of the winding rotor, squirrel cage, or double squirrel cage type. The stator and rotor windings are "Y" connected with internal neutral. The electrical component of the machine is represented by a six-order state-space model (for double squirrel cage machine) or a fourth-order state-space model (for single squirrel cage machine), and the mechanical part is modeled by a second-order system [37]. In Figure 8, the equivalent circuit of the electrical system of the asynchronous double cage machine is shown under the reference frame dq. The electrical system is governed by Equations (19)- (34). used to calculate the pressure loss due to the friction, and the Haaland approximation is used to calculate the friction factor in the turbulent regime. In the transition of regimens, from laminar to turbulent, the friction factor is determined by linear interpolation. As a result, the section of pipe is modeled according to Equations (16)-(18) [36].

Induction Motor Model
To simulate the induction motor, the asynchronous machine model is used. This block makes it possible to model three-phase induction motors of the winding rotor, squirrel cage, or double squirrel cage type. The stator and rotor windings are "Y" connected with internal neutral. The electrical component of the machine is represented by a sixorder state-space model (for double squirrel cage machine) or a fourth-order state-space model (for single squirrel cage machine), and the mechanical part is modeled by a secondorder system [37]. In Figure 8, the equivalent circuit of the electrical system of the asynchronous double cage machine is shown under the reference frame dq. The electrical system is governed by Equations (19)-(34).  V qs = R s i qs + dϕ qs /dt + ωϕ ds T e = 1.5p ϕ ds i qs − ϕ qs i ds (25) ϕ qs = L s i qs + L m i qr (26) ϕ ds = L s i ds + L m i dr (27) ϕ qr1 = L r1 i qr1 + L m i qs Equations (35) and (36) apply to the mechanical component.
The induction motor is used in two connections: with a direct connection to the grid (frequency and nominal speed) and with a connection to a variable speed drive (frequency and variable speed). Figure 9a shows the model of the induction motor with a direct connection to the grid, and in Figure 9b is the model for a variable speed drive made up of an induction motor and a frequency inverter.  (35) and (36) apply to the mechanical component.
The induction motor is used in two connections: with a direct connection to the grid (frequency and nominal speed) and with a connection to a variable speed drive (frequency and variable speed). Figure 9a shows the model of the induction motor with a direct connection to the grid, and in Figure 9b is the model for a variable speed drive made up of an induction motor and a frequency inverter.  The variable speed drive is modeled with the block Drive AC2 "Space Vector PWM VSI Induction Motor Drive", which contains an asynchronous and a PWM-VSI variable frequency drive. The asynchronous machine is configured as a three-phase induction motor, and its variable frequency converter is made up of a three-phase rectifier, DC link, three-phase inverter, and control system, as seen in Figure 10a. Both the inverter and the rectifier of the block AC2 are based on the Universal Bridge, which allows the modeling and simulation of static converters using forcibly switched devices (GTO, IGBT, MOSFET) and naturally switched power electronic devices (diodes or thyristors) connected in a bridge configuration [38]. In this work, a rectifier made up of power diodes and an inverter based on IGBTs were selected.

Variable Speed Drive Model
The variable speed drive is modeled with the block Drive AC2 "Space Vector PWM VSI Induction Motor Drive", which contains an asynchronous and a PWM-VSI variable frequency drive. The asynchronous machine is configured as a three-phase induction motor, and its variable frequency converter is made up of a three-phase rectifier, DC link, three-phase inverter, and control system, as seen in Figure 10a. Both the inverter and the rectifier of the block AC2 are based on the Universal Bridge, which allows the modeling and simulation of static converters using forcibly switched devices (GTO, IGBT, MOSFET) and naturally switched power electronic devices (diodes or thyristors) connected in a bridge configuration [38]. In this work, a rectifier made up of power diodes and an inverter based on IGBTs were selected.  Regarding the speed controller, the Drive AC2 has a closed loop speed control, made up of the classic V/f scalar control plus a slip compensator. Figure 10b shows the block diagram of the speed controller, which is fed back with a tachogenerator connected to the motor shaft. The slip compensator, based on a PI controller, corrects the speed error in the motor. The constant V/f ratio block allows the fundamental component of the voltage to the motor to have the desired amplitude and frequency. This is why it provides the voltage and frequency reference to the PWM generator [39]. Figure 10c shows the block diagram of the PWM generator based on space vector modulation (SVM), whose mission is to generate the control pulses for the IGBTs [39]. Figure 11 shows the block diagram of the pumping process control system. It has a PI pressure controller as an external loop and a slip compensator as an internal loop. Feedback for each control loop is achieved with a pressure transmitter and a tachometer, respectively. In practice, variable speed drives contain a PLC that provides digital inputs and outputs to connect discrete signals, analog inputs to receive sensor signals, analog outputs to control actuators, blocks to program PID algorithms, and specialized macros for pump control. Regarding the speed controller, the Drive AC2 has a closed loop speed control, made up of the classic V/f scalar control plus a slip compensator. Figure 10b shows the block diagram of the speed controller, which is fed back with a tachogenerator connected to the motor shaft. The slip compensator, based on a PI controller, corrects the speed error in the motor. The constant V/f ratio block allows the fundamental component of the voltage to the motor to have the desired amplitude and frequency. This is why it provides the voltage and frequency reference to the PWM generator [39]. Figure 10c shows the block diagram of the PWM generator based on space vector modulation (SVM), whose mission is to generate the control pulses for the IGBTs [39]. Figure 11 shows the block diagram of the pumping process control system. It has a PI pressure controller as an external loop and a slip compensator as an internal loop. Feedback for each control loop is achieved with a pressure transmitter and a tachometer, respectively. In practice, variable speed drives contain a PLC that provides digital inputs and outputs to connect discrete signals, analog inputs to receive sensor signals, analog outputs to control actuators, blocks to program PID algorithms, and specialized macros for pump control.

Process Control System
The controller parameters Kp, Ti, and Td, can be determined by applying various analytical or experimental techniques [40].

Results and Discussion
For the simulations, 120 gpm @ 35 m and 3500 rpm was considered, with a threephase induction motor of 5 HP, 220 Vac, 60 Hz, and two poles. The study cases were contrasted with each other and with the base case. The base case consisted of operating the pumps at a fixed speed through a direct connection to the electrical grid, requiring a hydropneumatic tank and an on-off control with hysteresis. For the base case, a hydropneumatic tank of 80 gal was considered. To reduce the on/off frequency of the pump, the capacity of the hydropneumatic tank must be increased, or the hysteresis band must be widened, which can make the application more expensive and affect the quality of the The controller parameters K p , T i , and T d , can be determined by applying various analytical or experimental techniques [40].

Results and Discussion
For the simulations, 120 gpm @ 35 m and 3500 rpm was considered, with a three-phase induction motor of 5 HP, 220 Vac, 60 Hz, and two poles. The study cases were contrasted with each other and with the base case. The base case consisted of operating the pumps at a fixed speed through a direct connection to the electrical grid, requiring a hydropneumatic tank and an on-off control with hysteresis. For the base case, a hydropneumatic tank of 80 gal was considered. To reduce the on/off frequency of the pump, the capacity of the hydropneumatic tank must be increased, or the hysteresis band must be widened, which can make the application more expensive and affect the quality of the service because of low pressures. When upgrading the traditional on-off control for a regulatory control with variable speed drive, certain advantages and disadvantages were observed, which are detailed below.

Dynamic Response
Two tests were carried out to study the dynamic response: -Test 1: In low flow conditions with a pump operating at variable speed. This operation was the same in both study cases; -Test 2: In high flow conditions with two pumps operating in parallel. In case one, one pump operated at a fixed speed and the other at variable speed, and in case two, both pumps were at variable speed.
In both tests, the pressure set point was set at 40 psi, and the acceleration/deceleration ramps of the variable speed drive were set at 3600 rpm/s. The hydropneumatic tank capacity was reduced to 10 gal, which is an 87.5% reduction from the base case. The system responses were for step-type disturbances produced by sudden changes in the consumption valve. These disturbances were ideal and sufficiently drastic since, in a real system, every valve has a response time, even those of the quick opening type.
The results of test 1 are presented in Figure 12, where the dynamic response of the pressure, flow, and slip is shown, under three scenarios: with slip compensation and a hydropneumatic tank (Figure 12a), no-slip compensation (Figure 12b), and without the hydropneumatic tank ( Figure 12c). As can be seen, in the three scenarios, it was possible to keep the pressure constant at 40 psi in view of the variable flow demand. When it was operated with a hydropneumatic tank (Figures 11b and 12a), the pressure overshoot did not exceed 10% of the set point; when the tank was removed (Figure 12c), the overshoot exceeded 50%. This shows that it is not advisable to completely eliminate this element since it contributes to the damping of pressure transients due to sudden changes in demand and prevention of water hammer [41], in addition to improving the energy efficiency of the system when operating at low flow rates [42]. The action of the slip compensator is also observed. In Figure 12a, slip compensation is active, while in Figure 12b, it is inactive. As observed, a slight improvement in the transient response is achieved (Figure 12a), but this improvement was not representative. service because of low pressures. When upgrading the traditional on-off control for a regulatory control with variable speed drive, certain advantages and disadvantages were observed, which are detailed below.

Dynamic Response
Two tests were carried out to study the dynamic response: -Test 1: In low flow conditions with a pump operating at variable speed. This operation was the same in both study cases; -Test 2: In high flow conditions with two pumps operating in parallel. In case one, one pump operated at a fixed speed and the other at variable speed, and in case two, both pumps were at variable speed. In both tests, the pressure set point was set at 40 psi, and the acceleration/deceleration ramps of the variable speed drive were set at 3600 rpm/s. The hydropneumatic tank capacity was reduced to 10 gal, which is an 87.5% reduction from the base case. The system responses were for step-type disturbances produced by sudden changes in the consumption valve. These disturbances were ideal and sufficiently drastic since, in a real system, every valve has a response time, even those of the quick opening type.
The results of test 1 are presented in Figure 12, where the dynamic response of the pressure, flow, and slip is shown, under three scenarios: with slip compensation and a hydropneumatic tank (Figure 12a), no-slip compensation (Figure 12b), and without the hydropneumatic tank ( Figure 12c). As can be seen, in the three scenarios, it was possible to keep the pressure constant at 40 psi in view of the variable flow demand. When it was operated with a hydropneumatic tank (Figures 11b and 12a), the pressure overshoot did not exceed 10% of the set point; when the tank was removed (Figure 12c), the overshoot exceeded 50%. This shows that it is not advisable to completely eliminate this element since it contributes to the damping of pressure transients due to sudden changes in demand and prevention of water hammer [41], in addition to improving the energy efficiency of the system when operating at low flow rates [42]. The action of the slip compensator is also observed. In Figure 12a, slip compensation is active, while in Figure 12b, it is inactive. As observed, a slight improvement in the transient response is achieved ( Figure  12a), but this improvement was not representative.  In Figure 13, the results of test 2 are presented, where the dynamic response of several hydraulic and mechanical variables of interest is observed. Figure 13a shows the results of the first case study, and Figure 13b the second case study. In both cases, very similar transient responses are observed for pressure and flow rate, and in a steady state, there were practically no differences. The dynamic response of the flow rate, total dynamic head, suction pressure, discharge pressure, angular velocity, and mechanical torque of each pump train is also appreciated. It is noticeable that in the pumps that operated at variable speeds, the angular speed and the mechanical torque increased and decreased with the flow demand. This action is what produces power savings. In Figure 13, the results of test 2 are presented, where the dynamic response of several hydraulic and mechanical variables of interest is observed. Figure 13a shows the results of the first case study, and Figure 13b the second case study. In both cases, very similar transient responses are observed for pressure and flow rate, and in a steady state, there were practically no differences. The dynamic response of the flow rate, total dynamic head, suction pressure, discharge pressure, angular velocity, and mechanical torque of each pump train is also appreciated. It is noticeable that in the pumps that operated at variable speeds, the angular speed and the mechanical torque increased and decreased with the flow demand. This action is what produces power savings.  Figure 14 shows the waveforms of the currents and voltages during the induction motor starting process. Figure 14a corresponds to the direct start that occurred in the base case, through a direct connection to the electrical grid, a process that lasted approximately 200 ms (12 cycles). It is observed that the starting current was approximately 8.5 times the steady state current. On the other hand, Figure 14b shows the starting process through the acceleration ramp of 3600 rpm/s provided by the variable speed. In this second case, it is observed that the starting current did not exceed three times the current in steady state (Figure 14b), therefore this transient current was reduced by 64.7% compared with the  Figure 14 shows the waveforms of the currents and voltages during the induction motor starting process. Figure 14a corresponds to the direct start that occurred in the base case, through a direct connection to the electrical grid, a process that lasted approximately 200 ms (12 cycles). It is observed that the starting current was approximately 8.5 times the steady state current. On the other hand, Figure 14b shows the starting process through the acceleration ramp of 3600 rpm/s provided by the variable speed. In this second case, it is observed that the starting current did not exceed three times the current in steady state (Figure 14b), therefore this transient current was reduced by 64.7% compared with the base case (Figure 14a). The frequency inverter smooths the starting and stopping (Figure 14b) which prevents some unwanted electrical, mechanical, and hydraulic effects produced in sudden starts and stops, such as water hammer and pressure peaks [43], vibrations in the pipes [44], voltage drops in the motor and the electrical grid [45], or mechanical stress that reduces the useful life of the motor [46]. 14b) which prevents some unwanted electrical, mechanical, and hydraulic effects produced in sudden starts and stops, such as water hammer and pressure peaks [43], vibrations in the pipes [44], voltage drops in the motor and the electrical grid [45], or mechanical stress that reduces the useful life of the motor [46]. When a pump operates at variable speed, the waveforms of the voltages and currents at the input and output of the drive are as shown in Figure 15a. The line voltage and current waveforms, for cases one and two, are presented in Figure 15b,c, respectively. For these simulations, a short-circuit power of the source of 450 kVA and a ratio X/R = 10 were considered. Distortions in current and voltage waveform are observed, which can be evaluated with a THD indicator. When a pump operates at variable speed, the waveforms of the voltages and currents at the input and output of the drive are as shown in Figure 15a. The line voltage and current waveforms, for cases one and two, are presented in Figure 15b,c, respectively. For these simulations, a short-circuit power of the source of 450 kVA and a ratio X/R = 10 were considered. Distortions in current and voltage waveform are observed, which can be evaluated with a THD indicator. 14b) which prevents some unwanted electrical, mechanical, and hydraulic effects produced in sudden starts and stops, such as water hammer and pressure peaks [43], vibrations in the pipes [44], voltage drops in the motor and the electrical grid [45], or mechanical stress that reduces the useful life of the motor [46]. When a pump operates at variable speed, the waveforms of the voltages and currents at the input and output of the drive are as shown in Figure 15a. The line voltage and current waveforms, for cases one and two, are presented in Figure 15b,c, respectively. For these simulations, a short-circuit power of the source of 450 kVA and a ratio X/R = 10 were considered. Distortions in current and voltage waveform are observed, which can be evaluated with a THD indicator.

Starting Current and Harmonic Distortion
(a)  Table 1 presents a summary of the total harmonic distortion (THD), an indicator that can be obtained using the Fast Fourier Transform (FFT), a tool that is part of Simulink's specialized "powergui" block. The noncontrolled rectifier bridge was responsible for the harmonic distortion on the line side, and the inverter IGBT bridge produced the motor side harmonic distortion. Harmonic pollution generates various unwanted effects on the electrical grid and the loads connected to it [47][48][49] and also damages the induction motor [50,51]. However, the switching of power semiconductors also produces the reflected wave phenomenon, although this was not observed in the simulation. The reflected wave is amplified if the distance between the frequency inverter and the motor increases, generating overvoltages harmful to the conductors and the induction motor and also damaging overcurrents for the frequency inverter [52,53]. The harmonic distortion and reflected wave can be mitigated by implementing filters [49,53]. Table 2 shows the mechanical power required for the base case and the two study cases and also shows the percentage of mechanical power savings of the two study cases with respect to the base case. Eight operating scenarios defined by the opening percentage of consumption valves one and two were evaluated. With the torque and angular velocity data, the mechanical power was calculated using Equation (8).  Table 1 presents a summary of the total harmonic distortion (THD), an indicator that can be obtained using the Fast Fourier Transform (FFT), a tool that is part of Simulink's specialized "powergui" block. The noncontrolled rectifier bridge was responsible for the harmonic distortion on the line side, and the inverter IGBT bridge produced the motor side harmonic distortion. Harmonic pollution generates various unwanted effects on the electrical grid and the loads connected to it [47][48][49] and also damages the induction motor [50,51]. However, the switching of power semiconductors also produces the reflected wave phenomenon, although this was not observed in the simulation. The reflected wave is amplified if the distance between the frequency inverter and the motor increases, generating overvoltages harmful to the conductors and the induction motor and also damaging overcurrents for the frequency inverter [52,53]. The harmonic distortion and reflected wave can be mitigated by implementing filters [49,53]. Table 2 shows the mechanical power required for the base case and the two study cases and also shows the percentage of mechanical power savings of the two study cases with respect to the base case. Eight operating scenarios defined by the opening percentage of consumption valves one and two were evaluated. With the torque and angular velocity data, the mechanical power was calculated using Equation (8). In high-demand scenarios (two pumps in parallel), the mechanical power was reduced to 19.95% for case one and 28.29% for case two. With decreasing demand, both cases operate with a single pump at variable speed, and mechanical power savings of up to 47.23% were obtained compared with the base case at a fixed speed. This analysis did not consider the losses of the motor or the frequency inverter, since under nominal conditions, electric motors are more efficient than pumps, and frequency inverters more efficient than motors; However, the efficiency of the motor can change with variations in load torque and speed [54][55][56]. The efficiency of the electrical system with a variable speed drive can be reduced by up to 8% with respect to the efficiency of the system with direct connection to the grid, but this reduction is not significant in relation to the energy savings achieved [56].

Investment Costs
CAPEX is higher in solutions with variable speed drive compared with traditional solutions at a fixed speed. However, the improvement in performance allows savings during the operation and maintenance of the system that increase profitability considering the entire life cycle. Table 3 presents, as a summary, the evaluation criteria of the cases studied.  Table 3 shows that both studied cases develop a better performance compared with the base case in terms of process quality and energy efficiency. In addition, both cases affect the quality of the electrical energy because of the harmonic pollution they produce and require a higher CAPEX compared with the base case.

Conclusions
In this research, two cases were studied to achieve a constant pressure system in a pumping station consisting of two centrifugal pumps in parallel, powered by electricity, and subjected to variable flow demand. The analysis was numerical in nature with modeling and simulation of systems carried out in MATLAB/Simulink R2019b. In the first case, one pump was operated at a fixed speed and the other at a variable speed, while in the second case, both pumps operated at a variable speed. These case studies were compared with each other and also compared with the base case where both pumps operated at a fixed speed. The results showed that with variable speed drives, it is possible to have a pressure regulating control system, achieving constant pressure in the face of variable flow demands, which is not achieved in the base case. In addition, variable speed drives smooth the current during acceleration and deceleration, leading to energy savings during startup and preventing unwanted electrical, mechanical, and hydraulic problems that are generated in sudden starts and stops (base case). Both study cases managed to reduce the required power by up to 47% compared with the base case. The second case obtained first place in energy efficiency by considering a variable speed drive in each pump. The harmonic distortion produced by frequency inverters was observed both on the line side and on the motor side. The harmonic distortion on the line side was higher in the second case, having two frequency inverters in parallel. This phenomenon can produce serious problems in the electrical grid, third-party equipment connected to the grid, and in induction motors, but it can be treated by implementing filters. The developed model is useful to study other control strategies that allow raising the degree of quality and efficiency of the pumping system, as well as harmonic filtering systems to improve the quality of energy.

Conflicts of Interest:
The authors declare no conflict of interest. Diameter of the hydraulic pipe L Geometric length of the pipe L eq Total equivalent length by local resistances r

Abbreviations and Symbols
Height of the roughness on the internal surface of the pipe v Kinematic viscosity of the fluid R s , L 1s Stator leakage resistance and inductance L m Magnetization inductance L s Total inductance of the stator V qs , i qs Stator voltage and current, referred to the q axis V ds , i ds Stator voltage and current, referred to the d axis ϕ qs , ϕ ds Stator fluxes referred to the q axis and the d axis R r1 , L lr1 Leakage resistance and inductance of rotor cage 1 R r2 , L lr2 Leakage resistance and inductance of rotor cage 2 L r1 , L r2 Total inductance of cage 1 and 2 of the rotor i qr1 , i qr2 Current of cage 1 and 2 of the rotor, referred to q axis i dr1 , i dr2 Current of cage 1 and 2 of the rotor, referred to d axis ϕ qr1 , ϕ dr1 Fluxes of the rotor cage 1 referred to q y and d axes ϕ qr2 , ϕ dr2 Fluxes of the rotor cage 2 referred to q y and d axes Control signal e(t) Control error