Experimental test rig designed to analyse pumping station operation controlled by pressure at different key points

. The operation of pumps within a pumping station can be controlled by the Pressure Set Point (PSP) value compared to the reading of a pressure sensor placed either at the exit of the pumping station, on the main discharge pipe ( classical PSP method ), or at a critical/monitoring point within the water distribution network (termed further as remote-control PSP method ). We designed and built an experimental test rig, allowing to study both control methods, in the attempt to check the advantage of the remote-control PSP method , with respect to the classical PSP method , since governing equations show that they are equivalent. The design relies on the theoretical approach and numerical results presented within this paper.


Introduction
Excessive operating pressure in Water Distribution Networks (WDN) induces stress on pipes and thus increases the risk of leakages and bursts [1,2], while the energy consumed for pumping is wasted unnecessarily. The overall effect is a rise of the WDNs' operation, maintenance and repairing costs [3]. Conversely, insufficient pressure in WDN induces water demand shortage at the consumers. Therefore, water companies need to optimise the pressure in the supplied network, depending on the requested variable demand.
Generally, in a Pumping Station (PS), pumps operation is controlled by the pressure value measured by a pressure sensor on the PS main discharge pipe. The system pressure must be kept around a so-called Pressure Set Point (PSP) − a pressure level at which the pressure switch is adjusted to actuate on rising or falling pressure (a small adjustable pressure range includes that level). One can keep the PSP level by successively switching on or off certain pumps. Variable speed driven pumps allow smooth adjustments of pump speed in order to reach the PSP level [4,5]. Within the WDN design phase, the PSP value can be computed using hydraulic analysis; for an existing WDN, with predictable water consumption, the PSP value selection can rely on historical or archived data. The above control method, based on the PSP value compared to the reading of a pressure sensor placed at the exit of the pumping station, on the main discharge pipe, will be termed further as classical PSP method. Although simple and easy to implement, it was proved that the classical PSP method usually ensures an excessive pressure across the WDN, greater than the level requested by the end-users (consumers).
To avoid exceeding pressure at the end-users, one can choose to control the pumps operation within a PS with respect to a PSP value compared to the reading of a pressure sensor positioned at a critical point (monitoring node) within the WDN. For WDNs fed by multiple pumping stations, each PS can be controlled by its own PSP level, compared to pressure values measured at a critical point that is 100% fed by the considered PS [6,7]. The latter control method implies remote-control (i.e. real time data acquisition and transmission); thus, it will be termed further as remote-control PSP method. Due to additional costs attached to the remote-control, that method might be less appealing for water companies.
In order to verify if there is any advantage of the remote-control PSP method, with respect to the classical PSP method, we designed and built an experimental test rig, that can be adapted to study both control methods. This paper focuses on the theoretical approach and numerical results that allowed us to design the test rig.

Experimental test rig description
An experimental setup was designed and recently built, to test a pumping station in a pipe network, when the PS operation is controlled either by the classical PSP method, or by the remote-control PSP method. The above experimental facility is located at the University "Politehnica" of Bucharest [8]. The studied pumping station [9] is equipped with 3 parallel-coupled pumps (of vertical centrifugal type), each with variable frequency drive. Technical details related to that PS can be found in Dunca et al. [10]. The scheme of the test rig is presented in figure 1, where pumps are denoted as P1, P2 and P3. The pumping station is fed through a main suction pipe from the open tank R (with a capacity of 75 litres, open to the atmosphere). There are 3 consumers (placed at nodes 2931), each fed through a pipe that branches out from the main discharge pipe. A Throttle Control Valve (TCV) allows establishing a certain outflow rate at each consumer. In order to keep a constant water level in the tank, the flowing jets are redirected to the tank using a gutter channel (figure 2). The remaining flow rate (the difference between the pumped flow rate and the outflows at the consumers) enters the tank through the pipe 14. The downstream TCV placed on pipe 14 is used to control the flow rate in the hydraulic system.  Tables 1 and 2 contain the network geometrical data: length Lj and inner diameter Dj of pipes with j = 117, and elevation zj of 14 nodes with j = 1831. To simplify notations, for j = 13, we considered that the pipe j has 2 components: one attached to the suction side (s) and the other attached to the discharge side (d) of the pump (thus, the pipes with j = 13 are bordered each by the pair of nodes 19 & 24; 20 & 23; 21 & 22, as in figure 1). The speed factor  is the ratio between the actual speed and the nominal speed of the pump. The head − flow rate curve, and the efficiency curve of the pumps P1P3 running at a certain speed are defined as in [ for j = 13, with the pumping head H in meters, flow rate Q in m 3 /s and dimensionless efficiency . The cavitation curve of the pumps P1P3 running at their nominal speed is fitted by a polynomial regression curve: with the net positive suction head NPSH in meters, for the flow rate Q in m 3 /s.

Theoretical approach
To design the test rig from figure 1, we will assume that all 3 pumps run at their nominal speed, and all valves are fully open, namely the TCV (of DN50 size) on pipe 14 and the TCVs (of DN25 size) at nodes 2931. At each consumer (j = 2931), the requested demand is Qr = 0.2710 -3 m 3 /s = 0.972 m 3 /h (it corresponds to an economic velocity [4] at the valve of DN25 size). The available water demand Qa is computed with a pressuredemand relationship [11][12][13], as: where p (in meters of water column) is the available pressure at the consumer, pref is the reference gauge pressure (requested to ensure the outflow Qr) and pmin is the minimal gauge pressure for which an outflow exists. For the studied test rig, pref = 2 mWC and pmin = 0 mWC. The available pressure head is ) where H is the nodal head (in meters),  is the water density and g is the gravity. Head losses on pipes are computed with the Darcy-Weisbach formula, as 2 j j Q R for j = 117, where R is the hydraulic resistance (in s 2 /m 5 ) and Q is the flow rate (in m 3 /s). The fiction factor is computed using the Swamee and Jain formula [14]. Minor losses will be considered only at the TCV on pipe 14, as 2 14 Q R v , where the local hydraulic resistance Rv depends on the minor loss coefficient . When that valve is fully open (as for the present design assumptions), then  = 0; in section 4, that valve will be partially closed ( > 0). To simplify the hydraulic analysis, any other minor losses, as well as the inlet and outlet kinetic terms on pipes, will be further neglected (for the studied test rig, these assumptions, valid for Lj/Dj > 200 [4], fit only the main suction pipe with j = 5 and the main discharge pipe with j = 10).
Further, the nonlinear system (7) was solved in GNU Octave [4,15], using the built-in function fsolve (also available in MATLAB [16]); fsolve requests a usersupplied initial solution (starting guess), denoted here by a column vector w0 of 31 components w0j. It turned out that the above nonlinear system was not at all sensitive to the initial solution. The final solution w of the system (7), obtained for w0 with all components equal to zero (w0j = 0, for j = 131), is presented in table 3.
In table 3, flow rates wj = Qj for j = 117 are presented both in m 3 /s and m 3 /h. The last 3 flow rate values equal the corresponding available demand in nodes 29 to 31: Q15 = Qa29; Q16 = Qa30 and Q17 = Qa31. All 3 outflow rates are smaller than the imposed requested demand Qr − this is consistent with the available pressure heads at the consumers, which are below the reference pressure head of 2 m.
Based on the parameters attached to the duty point of each pump (table 4): computed flow rate Qj, pumping head HPj (1) and pump efficiency j (2) for j = 1, the power Pj of each pump, for j = 13 can be computed, as: The total power consumed for pumping is PT = 1117 W. The duty points of the pumps are located towards the end of the head − flow rate curve (1), where the efficiency drops to 37%. Obviously, the operation of pumps at full capacity within the designed network is not the best choice, so in the future, it will be suitable to conduct most of the experimental tests with the discharge valve on pipe 14 partially closed (  0). For the computed pumped flow rates Qj (j = 13), one must verify if the pumps operate with or without cavitation when running at nominal speed.
The net positive suction head available at the pump inlet is defined as [4]: In (9)  The non-cavitating operating condition [4], written as j r j a NPSH NPSH  (11) for j = 13, is fulfilled for all pumps. Thus, for the designed configuration of the test rig, the operation of pumps at full capacity is safe, falling in the noncavitating regime (for the considered pumping station, the nominal speed is the maximum speed reached by the pumps). It's worth mentioning that decreasing the diameter of the main suction pipe (with index j = 5) is not an option, because the increase of the head losses on the suction circuit leads to cavitation in pumps: e.g. if the pipe 5 is of DN40 size, with an inner diameter D5 = 29.2 mm, the solution of the system (7) gives Q1 = 0.000845, Q2 = 0.000845 and Q3 = 0.000846 m 3 /s, as well as a minimal head H21 = -9.02 m near the inlet of pump P1; the NPSH available at the pump inlet (9) gives negative values.

Control methods based on PSP
In section 1 we highlighted two methods used to control the pumps operation based on the Pressure Set Point, namely: • classical PSP method (commonly implemented by the water companies); • remote-control PSP method (more expensive than the classical one, due to additional costs).
Theoretical aspects attached to the above control methods will be checked in this section, starting from the nonlinear system of equations (5), which was derived in section 3 for the experimental test rig, in the assumption that all pumps run at their nominal speed and all valves in the system are fully open.

Classical PSP method
The studied pumping station (PS) is equipped with a control panel with back-light display, function, navigation and operating keys. An adjustable PIDcontroller integrated in the software of the control panel [9] ensures that the system pressure remains constant, around the PSP level.
The standard operation of that PS relies on the classical PSP method: it is based on a pressure sensor on the discharge side of the PS, namely on the pipe 8 ( figure  3), placed under a membrane pressure vessel (MPV). For a proper functioning of the PS, the air pressure in the membrane switch vessel must be adjusted to a correct value, which is 50 kPa lower than the switch-on pressure on the waterside [9].
The pumping station is presented in figure 3, where the network components were labelled as in figure 1.
Taking into account the nodal head values computed in nodes 2224 (see table 3), and the elevation of 0.53 m at those nodes, the discharge pressure can be computed: p22  p23  p24 = 1.24 bar. Based on the pressure sensor accuracy (0.03 bar), we will assume further that the PSP is compared to the pressure value of node 24, located at the PS exit. To verify how the PS responds to the classical PSP method, the system of equations (5) will be changed, based on the following assumptions: a head value at node 24 will be imposed, so H24 will be no more the 24th unknown of the system (5). It will be replaced by another unknown (let's say) the speed factor of pump P3. Thus, 3 will become the 24th unknown of the system (5), while 1 = 2 = 1 will be still considered. To solve the resulting system of equations, unknowns will be denoted as wj with j = 131, as follows: Thus, the functions f3, f9 and f10 of the system (7) will be changed.
A value of H24 will be retrieved from previous data [10].
Dunca et al. [10] obtained experimental results on the same pumping station, controlled by the same classical PSP method, but with the PS connected to a short closed-loop hydraulic circuit. We are interested now in selecting H24 values fitting the PS capabilities, if possible, values greater than the ones computed in table 3 − we would like to operate as close as possible to the pumps best efficiency point (53.7% [9]), not at some duty points of low efficiency as in table 4.
Among the 63 sets of data available from [10], we will focus only on 3 sets (see table 5), with 1 = 2 = 1 and 3  0. For those sets, the PSP values (in bar) will be equivalent here to the pressure p24, allowing us to compute the corresponding head: H24 = (p24/(g) + z24). The total pumped flow rate established in [10] at the PS exit (equivalent to Q10 in the present paper) is added in table 5. The above flow rate was modified in [10] by throttling a discharge valve, with a certain value of the minor loss coefficient . The control valve was of the same type and size as the TCV of pipe 14.  (13) attached to the classical PSP method, by imposing a high value of the head H24 (as in table 5), it is compulsory to decrease the total pumped flow rate with respect to Q10  0.003 m 3 /s from table 3. Thus, a minor loss coefficient  > 0 at the TCV on pipe 14 will be set, which will increase the hydraulic resistance in the function f14 of (13), with respect to (7).
The numerical results, obtained after solving the system (13) for H24 = 25.835 m (PSP = p24 = 2.45 bar) and  = 513 (bigger than the value from table 5), are presented in table 6. The system of equations (13) was solved for 4 pairs of the input values {H24; }, starting with the same initial solution w0 as for (7). Near the 3 head values from table 5, the head value of 18.695 m (which was the minimal value tested in [10]) was additionally added. For all cases, pumps run without cavitation. The computed results are synthesised in table 7, namely the speed factor 3, the total discharge flow rate Q10, the head in node 28 (at the end of the main discharge pipe), pumps efficiency values j (j = 13) and the total power consumed for pumping, PT.
Obviously, by imposing such high values for H24, the available pressure at each consumer is greater than the reference pressure, so according to (4), all three outflows equal the requested demand.

Remote-control PSP method
In section 1, we mentioned that the classical PSP method ensures an excessive pressure across the WDN − that statement is sustained by the results from tables 6 and 7.
The remote-control PSP method is supposed to reduce the pressure in the hydraulic system. To verify how the PS responds to the remote-control PSP method, the PSP will be compared to the pressure value in node 28 (the last node of the system). The nodal head is: H28 = (p28/(g) + z28). The procedure from subsection 4.1 will be repeated, pointing this time to H28. Accordingly, the system (5) will be altered assuming that the head H28 is an input (imposed) value, while 3 becomes the 28th unknown in (5). To solve the new system of equations, the following notations will be used:  input values from  table 7, namely the head H28 and the coefficient , will give the same results as the one obtained for the classical PSP method. That conclusion was predictable, since all 3 nonlinear systems of equations (7), (13) and (15) describe the same flowing phenomenon, based on the same physical laws. The only difference is the crossover of some input and output variables. Thus, for a given set of variables {3; ; H24; H28}, leading to a proper operation of the test rig, the same solution (flow rates and nodal heads distributions) will be obtained by inserting {3; } in (7), or {; H24} in (13), or {; H28} in (15). Nevertheless, the system (15) attached to the remotecontrol PSP method can be solved, for input nodal heads H28 (let's say) at least half smaller than the values from table 7. One computation was performed for the PSP equal to the reference pressure: p28 = pref = 2 mWC = 0.19 bar. The computed results are synthesised in table 8. Here, pumps also run without cavitation.

Discussions
All numerical results obtained in this paper (in section 3, as well as in subsections 4.1 and 4.2) are verifying, from the mathematical point of view, each system of equations (7), (13) and (15), as pointed above.
The variation of the computed values of the nodal head H24 with respect to the total discharge flow rate Q10, and the variation of the total power PT consumed for pumping with respect to Q10 are plotted in figures 4 and 5, respectively (a total of 9 points are plotted, issued from tables 3, 7 and 8).
Within a pumping station where pumps are working in parallel coupling, the parallel operation point [4,10] is defined by the pair of parameters, which consist of the nodal head at the PS exit (in our case, H24 in node 24) and the total pumped flow rate (in our case, Q10). As one can see in figure 4, the parallel operation points follow a decreasing trend, with increasing total discharge flow rate (it is the natural trend of the parallel coupling head − flow rate curve).  According to the curve plotted in figure 5, the parallel operation points follow an increasing trend of the power, with increasing pumped flow rate.
From the theoretical study conducted in this paper, the following concluding remarks emerge: • the nonlinear systems of equations (7), (13) and (15) describe the physics of the same hydraulic process, thus they provide the same solution for a given set of variables {3; ; H24; H28}, where different 2 out of 4 variables are inserted as input parameters in those systems, while the remaining 2 variables are unknowns; • since all 3 nonlinear systems of equations (7), (13) and (15) are determined, the solver [16] converges for any starting guess, e.g. for a (non-physical) null initial solution; • mathematically, one can test with various sets of variables {3; ; H24; H28}, but the studied pumping station has some limitations (protections) for low pumping heads (where pumps run at low efficiency), as well as for low rotational speeds. Previous tests [10] were  m in table 8), so those pressure-deficient consumers receive a smaller outflow than the requested one; • within the present study, all 3 pumps are working at nominal speed or near the nominal speed; the presented theoretical approach can be extended to the case where a single pump, or only 2 pumps are running.

Conclusions
In this paper, two methods used to control the pumps operation in pumping stations were termed as: classical PSP method and remote-control PSP method.
This paper focuses on the theoretical approach and numerical results linked to the design of an experimental test rig, intended to verify in the future if there is any advantage of the remote-control PSP method, with respect to the classical PSP method. Commonly, water companies implement the classical method, since the remote-control method implies additional costs.
The experimental facility, designed as presented here, already exists [8]. It consists of a pumping station with 3 parallel-coupled variable speed driven pumps (previously studied [10]), and a new piping network with 3 consumers (each with its throttling control valve, allowing to set different outflow rates). The present theoretical study was conducted assuming that all 3 pumps are working at nominal speed or near it, that valves are fully open at all consumers, and that the final discharge valve can be throttled from fully open to partially closed (through different minor loss coefficient values).
From the mathematical point of view, the nonlinear systems of equations describing the operation of the above test rig show that both PSP control methods are equivalent (so, one cannot claim that one is better than the other). Further experiments on the above test rig will highlight advantages and disadvantages of both control methods. Nevertheless, the present theoretical study offers enough data, useful to properly conduct the future experimental work.