Interior point methods specialized to optimal pump operation costs ofwater distribution networks

Aline Lima1,∗ and Aurelio Oliveira1

1 Universidade Estadual de Campinas, IMECC - DMA, Caixa Postal 6065, CEP 13083-859 - Campinas - SP - Brazil.

In the water distribution problem the loss is important and the objective function must consider it combined with pump costs.The problem becomes complex because the loss in each branch is as a nonlinear function of water outflow. The objective ofthis work consists in solving the water distribution problem using interior point methods and to exploit the particular structureof the problem and the specific matrix sparse pattern of the of the resulting linear systems. The interior point methods showto be robust, achieving fast convergence in all instances tested.

1 Mathematical Model

The mathematical model for the water distribution problem of a threshed net is given by:

Minimize αn∑

i=1

kiqai + βctp (1)

s.a. Aq = Ep − d, Tq = 0 (2)At(H + h) − Kq = 0, Hj ≥ Pminj (3)

qmin ≤ q ≤ qmax, pmin ≤ p ≤ pmax (4)

where the equations in (2) represent the distribution network, the equations in (3) represent the minimum pressures in nodes[1], and the equations on (4) represent the limits of outflow and pressure of the supplying system. The objective function (1)considers the water loss in each branch thus searching the solution of least global losses. The pumping costs also is consideredand both objectives are combined through the weights α and β.

In this work, we will ignore the equation that enforces the minimum pressure on nodes (3), in order to obtain a similarmodel of an electrical power generating DC system, for which already exists a full development [2].

2 Primal-Dual Method Applied to the Optimal Pump Operation Cost of Water Distri-bution Networks

The primal-dual interior point methods consist on applying Newton’s method on the optimality conditions for the problemabove which is given by the primal and dual feasibility and the complementarity conditions. For problem (1 - 4) this approachresults in the following linear system:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Adq − Edp = d1 + Ep − AqTdq = d2 − Tqdq + dsq = qmax − q − sq

dp + dsp = pmax − p − sp

Btdy + dzq − dwq − Kdq = k1 − Bty − zq + wq + Kq−Etdy + dzp − dwp = c + Ety − zp + wp

Zqdq + Qdzq = µe − QZqeZpdp + Pdzp = µe − PZp

Wqdsq + Sqdwq = µe − SqWqeWpdsp + Spdwp = µe − SpWpe.

(5)

∗ Corresponding author E-mail: amlima@ime.unicamp.br

PAMM · Proc. Appl. Math. Mech. 7, 2060021–2060022 (2007) / DOI 10.1002/pamm.200700306

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

This system can have its dimension significantly reduced through variable elimination, resulting in the following smaller linearsystem:

(B̃D̃−1q B̃t + D̃)dy = r, (6)

which can be solved in efficient way, by exploiting the sparse matrix structure of the problem, and leading to the interior pointmethod search directions.

Therefore, the primal-dual interior point methods can be summarized as follows:

Consider x: the set of primal variables; and t: the set of dual variables; np: dimension of vector x; σ ∈ (0, 1) and τ ∈ (0, 1).Given x0 and t0 an initial interior point. For k = 1, 2, . . . do:Compute the residuals residuals.Compute the search directions dk

x and dky .

Compute the step length αkp e αk

d .Update the variables:xk+1 = xk + αk

pdkx

tk+1 = tk + αkddk

t

Until convergence is achieved.

3 Numerical Results and Conclusions

The described method 3 was implemented using MATLAB, and it was tested in a real water distribution system. Several testsare performed varying the weights α and β and the pumping costs.

The results show that the interior point methods are robust, achieving fast convergence in all instances tested, withoutpresenting numerical instability.

Acknowledgements This research was sponsored by the Brazilian Council for the Development of Science and Technology (CNPq).

References[1] W. F. Curi and M. B.M. Firmino, Prehdim - projeto de redes hidraulicas com dimensionamento malhado, Anais do XXV Iberian

Latin-American Congress on Computational Methods in Engineering - CILAMCE, Recife - PE, 1-15, (2004), (in Portuguese).[2] A. R.L Oliveira, S. Soares and L. Nepomuceno, Optimal active power dispatch combining network flow and interior point approaches,

IEEE Transactions on Power Systems, 18(4), 1235-1240 (2003).

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ICIAM07 Contributed Papers 2060022