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tank system. In (Fang et al., 2010), the problem of op-
timization of pumps operation in a multi tank system
is studied using complex genetic algorithms.
In (Ormsbee et al., 2009) three different explicit
formulations of the optimal pump scheduling problem
are presented. It takes into account the electric tar-
iff, but the decision variables define the start and stop
times of the different pumps in the system. The result-
ing optimization problem needs to be solved by non
linear algorithms, genetic algorithms, or semi heuris-
tic ad-hoc algorithms. This discrete explicit formula-
tion is not directly applicable if the system has some
valves that can reconfigure the network. In that case
is not sufficient to decide which pumps are started
or stopped at which time, but also the switching of
the valves, that changes the resulting inlet flow to the
tanks for a given pumps state. Furthermore, our ob-
jective is to use standard solvers to deal with the op-
timization problem. For that reason, in this paper, the
decision variables are explicit, but consist of the com-
bination of pumps and valves that must be active at
each instant, from the set of possible combinations.
The mathematical formulation proposed allows to use
standard parsers (as Yalmip) and solvers (as Mosek or
CBC), to solve the optimization problem.
In section II the problem of pumping system opti-
mal operation is described. In section III the mathe-
matical model of the problem is developed. In section
IV the basic optimization problem, derived from the
mathematical problem, is formulated. A more com-
plex optimization problem is formulated in section V
to reach more practical solutions, with a reduced num-
ber of pump commutations. Section VI shows the ap-
plication to a real pumping system and section VII
summarizes the main conclusions.
2 DESCRIPTION OF THE
PROBLEM
This paper deals with the problem of optimizing the
operation of a water pumping system a predefined
structure, with several wells and pumps, several tanks
and several ducts and valves. The objective of the
optimization is the minimization of the overall op-
erational cost. This cost is related to the individual
pumping energetic cost of each well (kWh per cubic
meter) and, especially, to the electric tariff that es-
tablish a different price depending on the time. The
automatic control system must decide which valves
and which pumps should be operated at every time
along the day to minimize the cost while fulfilling
some constraints.
The main constraint is to serve from each tank the
required daily water flow. This flow is time varying
and uncertain, but it can be predicted because it fol-
lows an approximate daily pattern. The other impor-
tant constraint is due to the size of the tanks, each one
having a maximum and minimum level that should
not be violated.
Other secondary constraints that may apply in-
clude a maximum number of daily starts and stops of
each pump, or a minimum running time once a pump
is switched on. Other more complex restrictions could
be related to the concentration of some pollutants in
the different wells. For example, the nitrate concen-
tration may be too high in a given well, so that, that
water must be mixed with the one from another well
to fulfil a concentration requirement. The constraint
could then be a minimum mixing ratio. This last con-
straint could be time varying, as the concentration of
the wells may change with time.
3 MATHEMATICAL
MODELLING OF THE
PROBLEM
First of all, the pumping system is assumed to have N
p
pumps (one in each well), N
t
tanks and N
v
valves. The
valves are used to connect or disconnect the pumps to
the tanks. The total number of possible combinations
is 2
N
p
+N
v
, but not all the combinations are possible
in a given system. Let us define as N
c
the number
of valid combinations of valves and pumps. In or-
der to formulate mathematically the problem, a bi-
nary matrix, M
c
, of size N
c
× (N
p
+ N
v
), is defined,
where each row represents one of the valid combina-
tions, and where the corresponding elements take the
value 1 or 0 depending on the active or inactive state
of the valves and the pumps in that combination.
Along the paper, a simple example will be used
to illustrate the proposed approach. The considered
pumping system has N
p
= 2 wells (pumps), N
v
= 1
valve and N
t
= 2 tanks. The tank 1 can be filled from
pump 1, no matter the state of the valve (with a flow
of 200 m
3
/h), and from pump 2 if the valve 2 is open
(with an additional flow of 100 m
3
/h), while the tank
2 can only be filled from pump 2 if the valve is closed
(with a flow of 80 m
3
/h).
Taking into account the physical limitations de-
scribed above, the matrix that defines de N
c
= 6 valid
combinations of pumps and valves in this example is
shown in table 1. The value X means that the resulting
flows are the same if the valve is closed (0) or open
(1).
For each combination of valves and pumps, there
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