World Environmental and Water Resources Congress 2016

Modeling Variable Speed Pumps for Optimal Pump Scheduling 1

Ruben Menke1,*; Edo Abraham1; and Ivan Stoianov1

Downloaded from ascelibrary.org by Imperial College London on 05/19/16. Copyright ASCE. For personal use only; all rights reserved.

Dept. of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, U.K. * Corresponding author. E-mail: [email protected]

Abstract Increasingly more variable speed drive pumps are installed in water distribution systems worldwide. However, the modeling of variable speed drives in such networks remains difficult, especially in the context of mathematical optimization for pump scheduling. For the problem of energy usage minimization, formulated as a mixed integer program, we propose a new simplified convex relaxation of the hydraulic characteristics of a variable speed pump. By using different model approximations for the power curve, as a function of pump speed and volumetric flow rate, we study the trade-off in the computational complexity and quality of solutions obtained. We use two small benchmark networks and real pump data from three different pumps to calculate operating schedules. We show that these schedules, are computed in a time suitable for operational usage, enable operation with lower operating cost compared to fixed speed pumps, and their sub-optimality can be computed.

1. Introduction Water distribution systems (WDS) consume a considerable amount of a cities energy (Bunn and Reynolds, 2009). Variable speed drive (VSD) pumps can be used to reduce the energy requirements through the increased variation in the performance. Lamaddalena and Khila (2012) show up to 35% reduction in energy cost in irrigation systems through the use of variable speed pumps. Compared to fixed speed pumps, optimization of variable speed drive pumps using an explicit mathematical programming formulation is computationally difficult due to the level of increased non-linearities. The pump performance is not described by a single curve but a range of curves that are usually given for a range of speeds as shown in Figure 1a. Therefore, the operating points of VSD pumps can be more efficiently adapted over a wide range as demand and other system requirements change (Simpson and Marchi, 2013). Previous efforts have constructed approximations for the functions describing the pump’s speed, the pumps hydraulic output and power consumption. The optimization of the pump schedule including the operating condition of the variable speed pump has

© ASCE World Environmental and Water Resources Congress 2016

199

Downloaded from ascelibrary.org by Imperial College London on 05/19/16. Copyright ASCE. For personal use only; all rights reserved.

World Environmental and Water Resources Congress 2016

Figure 1: Characteristic and power curves of a VSD pump, by example of the ETAline-100-100-210 (KSB Aktiengesellschaft, 2013) been approached with a range of meta-heuristics such as ant colony optimization (Hashemi et al., 2014), or genetic algorithms (Lingireddy and Wood, 1998). Other methods aim to minimize the specific energy consumption of the system and solve an idealized WDS evaluating a series of local optima to find a near global optimum (Bene and Hös, 2012). For cost analysis and optimization algorithms, Ulanicki et al. (2008) consider it necessary to have efficiency and power characteristic models with pump speed and number of pumps in a group as the model control variables. To enable this, there are refinements to the estimations of the pump state based on pump affinity laws (Simpson and Marchi, 2013) or changes to the hydraulic simulation method (Todini et al., 2007; Wu et al., 2012). For heuristic methods, such as genetic algorithms, which compute the hydraulic model’s response separately to the optimization step, the power consumption of the pumps for a given operating schedule are computed using an explicit non-linear model and a hydraulic solver. The main shortcoming of such approaches lies in their inability to find certifiably optimal schedules. For mathematical optimization methods, such as mixed-integer quadratic programming, the model must explicitly formulate the pump cost and hydraulic state. For a convex objective function the problem can be solved to global optimality by most modern solvers (Achterberg, 2009; IBM, 2009). We propose a method linking the power consumption to the flow rate and head difference created by the pump. The non-linear relations of power, head and flow and speed are approximated through a proxy variable and the resolution of the exact state is relegated to after the optimization of the schedule. We extend the mixed integer piece-wise linear approximation of the pump scheduling problem previously described in Menke et al. (2015) to also account for variable speed pumps. This paper is organized is follows. In section 2 we describe the mathematical model extension and compare quality of a range of approximations of objective

© ASCE World Environmental and Water Resources Congress 2016

200

World Environmental and Water Resources Congress 2016

201

functions. In section 3 we describe the data of the pump and network examples used as well as the quality of the approximations. In section 4 we present the results and the discussion and thoughts on further work are summarized in section 5 and the conclusions are in section 6. Downloaded from ascelibrary.org by Imperial College London on 05/19/16. Copyright ASCE. For personal use only; all rights reserved.

2. Model formulation The optimization problem for scheduling WDS pumps for DR can be formulated as: minimize:Pump operation cost, (1) subject to:Hydraulic constraints of pumps and pipes mass balance of the system. The model described here is an extension of the convex approximation of the pump characteristics described in Menke et al. (2015) for optimal scheduling of fixed speed pumps. Here we consider VSD pumps and introduce an additional variable to describe the status of the VSD pump. It is given by: Δℎ − − (2) = where Δℎ is the head difference across the pump, q the flow through the pump and a, b, c the constants describing the quadratic fit to the characteristic curve at full speed. The continuous variable τ ∈ [0, 1] can be thought of as throttle or speed setting of the pump and is non-dimensional. The hydraulic state of a pump in (2) can be approximated using a piecewise linear model. For a given time step, a VSD pump ip connecting nodes J 1 and J 2 is constrained by: and + and + ℎ −ℎ ≤ (3) ⋮ + =1 : ℎ − ℎ ≤ Δℎ and =0 : =0 where

∈ 0,1 is the pump switch ON or OFF and

is the flow through the pump

and ℎ and ℎ are the hydraulic heads at the nodes J1 and J2. The linear coefficients ⋯ and ⋯ describe the five hyperplanes bounding the convex set approximating the feasible region of head and flow through the pump. ∆hub is an upper bound on the pressure head generated by the pump. These constraints are enforced using a big-M method as detailed in Menke et al. (2015) and Gleixner et al. (2012) and described below:

© ASCE World Environmental and Water Resources Congress 2016

World Environmental and Water Resources Congress 2016

−ℎ + ℎ −

,

−

,

≤

(1 −

)

−ℎ + ℎ −

,

−

,

≤

(1 −

)

,

⋮ −

,

≤

−ℎ + ℎ − 0≤ Downloaded from ascelibrary.org by Imperial College London on 05/19/16. Copyright ASCE. For personal use only; all rights reserved.

202

where M is a large constant,

≤

,

(4) (1 −

)

≤

is the upper limit on the flow rate through the pump.

Table 1: Relative residual norms of the best fit power estimates for a range of pumps and a range of power function approximations. The power curve approximation is an approximation of the power consumption given by (⋅) a function of the given terms. Power function Residual norms (%) approximation ETA NK NB ( , ℎ) 66.8 48.6 49.5 ( , ) 46.5 30.3 33.4 12.1 8.5 10.2 ( , , ) 24.3 14.6 12.4 ( , , ) 6 5.2 ( , , , ) 9.8

Power function Residual norms (%) approximation ETA NK NB 4.5 ( , , ℎ, ℎ ) 15.2 100 100 100 ( , ) 10.5 9.2 10.4 ( , , ) 66.3 62.9 54.9 ( , , ) 8.2 7.3 ( , , , ) 10.1

The objective function to be minimized approximates the operating cost of the pumps, which are a function of the energy consumption and the cost of energy. The power consumption of a variable speed pump is a non-linear, non-convex function as the example in Figure 1b shows. Using a least squares method we fit linear and quadratic approximation to 10000 points describing the true power consumption for given, volumetric flow rate, pressure head, pump speed and τ values. The quality of the fits are assessed by comparing the residual norms of the least squares fit and summarized in Table 1. It shows that the linear approximations with the lowest residual norm is the function ( , ) , followed by ( , ℎ) and ( , ) is the least good approximation. When adding quadratic terms the performance of the ( , , , ) giving the closest approximations improves dramatically with ( , , ) and ( , , ). The ( , , , ), approximation, followed by ( ), ( ) ( ) , , ℎ, ℎ , , and , , are considerably worse. approximations This analysis indicates that the introduction of the τ parameter can provide accurate approximations of the pump state, that are at least as good as those based on the speed of the pump, in most cases better. As expected from the inspection of Figure 1, an approximation based on head h and flow alone is inferior to one based on flow and or . Using ( , , , ) as an example, the objective function for the pump schedule optimization becomes:

© ASCE World Environmental and Water Resources Congress 2016

World Environmental and Water Resources Congress 2016

(⋅) ≔ where

, ,

,

+

is the number of pumps,

, ,

,

203

+

, ,

,

+

, ,

the number of time steps and

(5)

,

, ,

⋯

, ,

Downloaded from ascelibrary.org by Imperial College London on 05/19/16. Copyright ASCE. For personal use only; all rights reserved.

are the coefficients of the power fit for pump during time step The optimisation problem is formulated with the other network components approximated as described in Menke et al. (2015). We use a piece wise linear approximation with five linear pieces to approximate the head loss across a pipe.

3. Case Study The networks investigated are a hydraulic model similar to the simplified system used by Bene and Hös (2012) and the popular Van Zyl benchmark network shown in Figure 2. The networks are modified and adapted for this investigation. The pumps in the Van Zyl network are replaced with “Grundfos-NK-100-250” and “Grundfos-NB-150-500” variable speed pumps. The storage tank in the simple network example is changed to a finite size of 70m diameter and 20 m elevation, and the pump station is fitted with three “ETAline-100-100-210” from KSB (Grundfos A/S, 2016c,b; KSB Aktiengesellschaft, 2013). The pumps were selected using the manufacturers web based selection tools, selecting the best possible pumps from the standard range available by the manufacturer (KSB Aktiengesellschaft, 2016; Grundfos A/S, 2016a). We investigate the computed operating cost of the WDS using variable and fixed speed pumps. The fixed speed pumps are the same pumps as the variable speed pumps but with the speed range fixed to the maximum speed of the pump.

Figure 2: The networks investigated in the case study These networks are used to show the changes in computational effort for a pump schedule optimization, the changes in modeled operating cost and the accuracy with which the optimization model predicts the operating cost obtained from a full hydraulic simulation. The computational effort of the power consumption approximation in the

© ASCE World Environmental and Water Resources Congress 2016

World Environmental and Water Resources Congress 2016

Downloaded from ascelibrary.org by Imperial College London on 05/19/16. Copyright ASCE. For personal use only; all rights reserved.

objective function is tested by assessing the convergence behavior of the branch and bound method (Menke et al., 2015). To assess the optimization method: 1. The computational effort is estimated by recording the time taken and the convergence of the relative optimality gap given by: − ∗ (6) Gap = ∗ where is the best feasible solution’s operating cost and ∗ is the best lower bound. The gap reported directly by the solver as relative global optimality gap. 2. The schedule that the optimization software evaluates to is passed to a hydraulic simulation to compute the power consumption using the best non-linear approximation for the pump power and a null space hydraulic solver for the WDS (Abraham and Stoianov, 2015). This provides , a more accurate estimate of the operating cost for a given schedule. 3. The error in estimating the actual operating cost compared to those predicted by the optimization method is given by: Δ = − (7)

4. Results The numerical results are summarized in Table 2. It shows that for both networks, the simulated operating cost, which are obtained from hydraulic simulations using the schedule found in the optimization, are lower when using VSD pumps compared to fixed speed pumps. The differences in operating cost range between 3–19%. The computational time to find the optimal schedule are up to 20 × greater in the case of the variable speed pumps for the Simple network, while in the Van Zyl network the solution does not converge to the optimal solution at all. However, the solution found after the time limit of 300s is up to 19% better than that of the fixed speed pump operation. The power function approximation ( , , ) has a considerably higher computational effort than the other formulations for no significant improvements in estimated operating cost accuracy or in residual norms in the power curve fits compared to ( , , ) and ( , , , ) . The accuracy of the estimate of the operating cost obtained from the optimization solver is assessed by comparing it to the operating cost computed when hydraulically simulating the nonlinear WDS with the given schedule; Table 2 shows an unexpected spread. The residual norms describing the accuracy of the fits in Table 1 do not provide an accurate prediction of the accuracy of the models in pump schedule optimization.

© ASCE World Environmental and Water Resources Congress 2016

204

World Environmental and Water Resources Congress 2016

Table 2: Summary of the results, showing time taken to converge to global optimality, simulated operating cost and difference between estimated and simulated operating cost, relative to simulated cost. Where the solver failed to converge the relative optimality gap at the time limit of 300s is given.

Downloaded from ascelibrary.org by Imperial College London on 05/19/16. Copyright ASCE. For personal use only; all rights reserved.

Power function Simple Network (Figure 2a) Van Zyl Network (Figure 2b) approximation Time (s) Cost ( ) (£) Δ (%) Time/Gap Cost ( ) (£) Δ (%) * ( , ) 8.56 15822 17.0 34 % 20191 2.1 * 9.02 15822 1.4 35 % 19867 20.7 ( , , ) * 28.83 16311 2.0 46 % 22198 18.6 ( , , ) * 7.29 15945 6.0 35 % 20229 22.8 ( , , , ) 0.44 16364 3.5 26.4 s 24917 1.8 ( , ) (FSD) * Converged to the optimality gap given by equation (6) in the time limit. The convergence of the optimization method when using different VSD models is highlighted in Figure 3. It shows the rapid initial convergence of the solution as well as the relative optimality gap of the solution obtained. Of particular interest are intermediate solutions found during the optimization and the operating cost obtained when simulating their operation. For example the intermediate solutions for the optimization of the operation with ( , , , ) are summarized in Table 3.

Figure 3: Convergence behavior of the schedule optimization using the OF formulations as summarized in Table 2. The convergence graphs show slower times and convergence rates than those reported in the Table 2. This is due to the additional computational effort and inefficiencies introduced through capturing the convergence behavior of the solver.

© ASCE World Environmental and Water Resources Congress 2016

205

World Environmental and Water Resources Congress 2016

206

Table 3: Operating cost of intermediate solutions as simulated by a hydraulic model.

Downloaded from ascelibrary.org by Imperial College London on 05/19/16. Copyright ASCE. For personal use only; all rights reserved.

Time (s) Gap (%) Operating cost (£)

0.70 79 58426

1.95 77 49936

3.87 53 19561

4.76 47 19248

6.57 44 19198

12.54 995.72 39 18 18715 16792

5. Discussion Comparable to previous findings (Lamaddalena and Khila, 2012; Lingireddy and Wood, 1998), we find lower operating cost for the networks operated with variable speed pumps that the networks operated with the same pumps as fixed speed pumps. The proposed formulation for the objective function does not describe the pump state using flow rate and operating speed, but instead uses a scalar term to explicitly describe the operating point of the pump. This shifts the non-linearities of the description outside the objective function of the optimization problem and enables the optimization problem to be formulated with linear constraints and a linear or quadratic objective function. The quality of the fit representing the accuracy with which a linear or quadratic formulation of the approximation can represent the actual power consumption. The real data is best represented by a cubic fit, which yields a problem with highly non-linear constraints. The slow convergence and insufficient accuracy of the ( , , ) approximation, could be related to the difference in magnitude of the coefficients of the approximation. The coefficients for and differ by (3) − (4). While solvers generally scale variables to minimize the performance impact a strongly changing gradient has on the convergence behavior, this cannot be completely compensated as it concerns only one variable. The mixed integer formulation of the optimization problem is only an approximation of the optimization problem formulated in (1), which is a non-linear problem and cannot be solved to provable global optimality for most cases. The branch and bound algorithm finds solutions with known levels of sub-optimality compared to the global solution of the convex mixed integer problem posed. However, for many applications, such as comparing performance and operating cost under different operating conditions, this is sufficient. The simulated operating cost of the intermediate schedules found during convergence of the solver are close to the finally found globally optimal solution. This affirms that the selected branch and bound method, while struggling to confirm global optimality in a short time frame, is suitable to find a pump schedule suitable for operations and comparisons of operations for further research. To develop this work further, the computational effort and performance need to be investigated on larger and more complex networks. Not all pumps used in a WDS can be equipped with infinitely finely adjustable pulse width modulation frequency

© ASCE World Environmental and Water Resources Congress 2016

World Environmental and Water Resources Congress 2016

converters. This introduces rounding or setting adjustments not accounted for in this optimization method. Further investigations could show what additional cost this would imply and how this can be accounted for in the optimization method.

6. Conclusions Downloaded from ascelibrary.org by Imperial College London on 05/19/16. Copyright ASCE. For personal use only; all rights reserved.

To solve the pump scheduling problem with mathematical optimization methods, we propose a new approximation to for the power consumption models of VSD pumps. We then formulate the optimization problem as a mixed integer problem with only linear constraints and a linear or a convex quadratic objective function. Using state-of-the-art solver, the problem is solved to global optimality for a small test case network and demonstrates that when the solution is interrupted before convergence, the sub-optimal solution found is better than the best operational performance with fixed speed pumps and often close to the global optima.

Acknowledgments Ruben Menke is funded by the Grantham Institute and Climate-KIC. Dr Edo Abraham is supported by the NEC-Imperial “Big Data Technologies for Smart Water Networks” project. Ruben Menke, Dr Edo Abraham and Dr Ivan Stoianov are members of the InfraSense Labs at Imperial College London.

References Abraham, E. and Stoianov, I. (2015). Sparse Null Space Algorithms for Hydraulic Analysis of Large-Scale Water Supply Networks. Journal of Hydraulic Engineering. doi:10.1061/(ASCE)HY.1943-7900.0001089 Achterberg, T. (2009). SCIP: solving constraint integer programs. Mathematical Programming Computation, 1(1):1–41. ISSN 1867-2949. doi:10.1007/ s12532008-0001-1 Bene, J. G. and Hos, C. J. (2012). Finding Least-Cost Pump Schedules for Reservoir Filling with a Variable Speed Pump. Journal of Water Resources Planning and Management, 138(6):682–686. ISSN 0733-9496. doi:10.1061/ (ASCE)WR.19435452.0000213 Bunn, S. M. and Reynolds, L. (2009). The energy-efficiency benefits of pumpscheduling optimization for potable water supplies, IBM Journal of Research and Development, doi:10.1147/JRD.2009. 5429018 Gleixner, A., Held, H., Huang, W., and Vigerske, S. (2012). Towards globally optimal operation of water supply networks. Numerical Algebra, Control and Optimization, (4):695–711. ISSN 2155-3289. doi:10.3934/naco.2012.2.695

© ASCE World Environmental and Water Resources Congress 2016

207

World Environmental and Water Resources Congress 2016

Downloaded from ascelibrary.org by Imperial College London on 05/19/16. Copyright ASCE. For personal use only; all rights reserved.

Grundfos A/S (2016a). Guided Selection. URL http://product-selection. grundfos.com/guided-selection.html?qcid=44705762, last accessed:12/01/2016 Grundfos A/S (2016b). NB 150-500 Technical data. http://product-selection. grundfos.com/productdetail.html?from_suid=14519180226080292600399314006 7&pumpsystemid=70828135&qcid=71291116, last accessed: 06/01/2016 Grundfos A/S (2016c). NK 100-250 Technical data. http://product-selection. grundfos.com/product-detail.html?from_suid=145191864337605400 500126055283&pumpsystemid=70831142&qcid=71291761, last accessed: 06/01/2016 Hashemi, S. S., Tabesh, M., and Ataeekia, B. (2014). Ant-colony optimization of pumping schedule to minimize the energy cost using variable-speed pumps in water distribution networks. Urban Water Journal, 11(5):335–347. ISSN1573062X. doi:10.1080/1573062X.2013.795233 IBM (2009). V12. 1: User’s Manual for CPLEX, URL ftp://public.dhe.ibm.com/software/websphere/ilog/docs/optimization/cplex/ps_usr mancplex.pdf, last accessed: 12.01.2016 KSB Aktiengesellschaft (2013). Etaline/Etaline-R Type Series Booklet 1146.52/04– EN01.0. Pegnitz, Germany. http://www.ksb.com/propertyblob/1593570/ data/etaline, last accessed: 04/01/2016 KSB Aktiengesellschaft (2016). KSB EasySelect. URL http://www.ksb.com/ksben/pumpselection/KSB-EasySelect/, last accessed: 12/01/2016 Lamaddalena, N. and Khila, S. (2012). Energy saving with variable speed pumps in ondemand irrigation systems. Irrigation Science, 30(2):157–166. ISSN 0342-7188. doi:10.1007/s00271-011-0271-7 Lingireddy, S. and Wood, D. J. (1998). Improved Operation of Water Distribution Systems Using Variable-Speed Pumps. Journal of Energy Engineering, 124(3):90–103. ISSN 0733-9402. doi:10.1061/(ASCE)0733-9402(1998) 124:3(90) Menke, R., Abraham, E., Parpas, P., and Stoianov, I. (2015). Approximation of System Components for Pump Scheduling Optimization. Procedia Engineering, 119:1059–1068. ISSN 18777058. doi:10.1016/j.proeng.2015.08.935 Simpson, A. R. and Marchi, A. (2013). Evaluating the Approximation of the Affinity Laws and Improving the Efficiency Estimate for Variable Speed Pumps. Journal of Hydraulic Engineering, 139(12):1314–1317. doi:10.1061/(ASCE)HY.19437900.0000776 Todini, E., Tryby, M. E., Wu, Z. Y., and Walski, T. M. (2007). Direct computation of Variable Speed Pumps for water distribution system analysis, Water Managment Challenges in Global Change, Taylor & Francis, London

© ASCE World Environmental and Water Resources Congress 2016

208

World Environmental and Water Resources Congress 2016

Downloaded from ascelibrary.org by Imperial College London on 05/19/16. Copyright ASCE. For personal use only; all rights reserved.

Ulanicki, B., Kahler, J., and Coulbeck, B. (2008). Modeling the Efficiency and Power Characteristics of a Pump Group. Journal of Water Resources Planning and Management, 134(1):88–93. ISSN 0733-9496. doi:10.1061/(ASCE)07339496(2008)134:1(88) Van Zyl, J. E., Savic, D. A., and Walters, G. A. (2004). Operational optimization of water distribution systems using a hybrid genetic algorithm. Journal of Water Resources Planning and Management, 130(2):160–170. doi:10.1061/(ASCE) 0733-9496(2004)130:2(160) Wu, Z. Y., Tryby, M. E., Todini, E., Walski, T. M., Hartell, W. R., and Culin, K. L. (2012). System and method for modeling and simulating water distribution and collection systems including variable speed pumps. URL http://www.google .com/patents/US8265911, last accessed: 12/12/2015

© ASCE World Environmental and Water Resources Congress 2016

209