Artelys contribution to WP 1.4:
Pumping plan optimization of a Water
Distribution Network
1 - Context and challenge
1.1 - Context
Saving energy in water industry is a major issue. Indeed, about 8% of world electricity consumption
arises from water industry. Moreover, the total electricity consumption of the water and wastewater
sectors will grow globally by a predicted 33% in the next 20 years especially due to the increasing
number of people getting drinking water (world population increase combined to development of
the infrastructure).
On the operation side, 1/3 of total operation cost is energy, mostly electricity for pumping. For
example, the electricity consumption of all Swiss water networks is approximately 400 000 MWh per
year. Thus, minimizing the pumping costs becomes a major technical challenge in a context of
European electricity markets opening where electricity prices will become more dynamic in order to
ensure balance between production and consumption.
Furthermore, due to regulated water prices, increase of energy prices and commitment to promote
water demand reduction, business context is under stress and water companies need to preserve
their profit by increasing efficiency and especially energy efficiency.
In parallel, the virtual energy market is taking place. New actors are collecting opportunities to
reduce subsequently energy demand for short period of time at peak hours, or at the opposite to
absorb energy surplus in case of high production level (in particular from renewable sources) at low
demand hours. Value of these new services to the grid is very variable, depending on the energy
market dynamic prices.
Considering the large volume of water storage and the pumping capabilities, the water networks may
offer good opportunities for this business. To make this real, it is necessary to provide new decision
making tools and analytics, ensuring that the water demand will be entirely fulfilled, evaluating the
economic equation, and providing the best strategy to make the most of the virtual energy market.
1.2 - Objectives
In order to reach the goal of energy efficiency in water distribution network operations, we defined
two case studies:
1.2.1 - Case study 1
The objective of case study 1 consists in developing a decision support tool which will help a Water
Distribution Network operator to schedule the operation of pumps (start, stop, operation level) and
valves (opening, closing) on a horizon of 1 day function of:
 water demand forecast on each node of the network,

SPOT electricity price curve.
The decision support tool should be able to suggest the “best” schedule of pumps and valves (called
the “pumping plan”), in the sense of minimizing the cost of the electricity used to operate the network,
while respecting operational constraints such as:
 minimum and maximum level of water in the water tanks,
 minimum and maximum pressure at each nodes,
 minimum and maximum pressure on each pipes,
 regenerate water stock at the end of the day,
 etc,
and physical constraints such as conservation of mass and head loss.
1.2.2 - Case study 2
The objective of the case study 2 consists in adapting the tool developed as part of case study 1 in
order to evaluate, in day ahead, for each time step:


the potential of load shedding or overconsumption (MWh),
the corresponding additional cost resulting from the deviation to the optimal pumping plan.
2 - Problem modeling
2.1 - Introduction
In order to provide drinking water to customers, electrical pumps pump water from water sources
and inject water through pipes with an adequate pressure. Water tower can be seen as stocks able to
ensure pressure on the network by gravity. Water flow in the pipes lead to pressure losses.
Figure 1 - Schematic representation of a Water Distribution Network
2.2 - A network flow optimization problem
The water distribution network may be modeled as a non oriented graph 𝐺 = (𝑉, 𝐸) where:



Vertices 𝑉 represent:
o Customers nodes
o Water towers
o Sources
Edges 𝐸 represent:
o Pipes
o Pumps
o Valves
Flow represents:
o Water flows through the pipes
Thus, optimization of water supply network pumping plans can be seen as a network flow
optimization problem.
2.3 - A MINLP formulation
Formulation of the constraints is the same for the two case studies. We can consider two types of
constraints:


Operational constraints:
o satisfying client water demands ;
o ensure a sufficient pressure to the users ;
o regenerate water stocks.
Physical constraints:
o Flow conservation rule (Kirchhoff’s nodal rule)
o head loss constraints:
𝑄|𝑄|𝜆 = 𝐻𝑚 − 𝐻𝑛 , where 𝑄 is flow and 𝐻 pressure.
In the case study 1, objective function consists in minimizing the daily pumping costs according to an
hourly varying electricity price:
𝑇
𝑃
𝑚𝑖𝑛 ∑ ∑ 𝑃𝐶(𝑝, 𝑡). 𝑀𝑊ℎ_𝑝𝑟𝑖𝑐𝑒(𝑡)
𝑡=1 𝑝=1
with:




𝑇 is the number of time steps
𝑃 is the number of pumps
𝑃𝐶(𝑝, 𝑡) the consumption of the pump 𝑝 at time 𝑡
𝑀𝑊ℎ_𝑝𝑟𝑖𝑐𝑒(𝑡) the 𝑀𝑊ℎ price at time 𝑡
In the case study 2, objective function consists in maximizing the benefit of changing the initial
pumping plan according to an adjustment offer at the time 𝑡 ∗ :
𝑃
𝑚𝑎𝑥 ∑(𝑃𝐶𝑖 (𝑝, 𝑡 ∗ ) − 𝑃𝐶(𝑝, 𝑡 ∗ )). 𝐴𝑑𝑗𝑢𝑠𝑡𝑚𝑒𝑛𝑡_𝑝𝑟𝑖𝑐𝑒
𝑝=1
𝑇
𝑃
+ ∑ ∑(𝑃𝐶𝑖 (𝑝, 𝑡) − 𝑃𝐶(𝑝, 𝑡)). 𝑀𝑊ℎ_𝑝𝑟𝑖𝑐𝑒(𝑡)
𝑡>𝑡 ∗ 𝑝=1
with 𝐴𝑑𝑗𝑢𝑠𝑡𝑚𝑒𝑛𝑡_𝑝𝑟𝑖𝑐𝑒 the proposed adjustment price.
Thus, in both case studies, our problem have:






A linear objective function
Binary variables
o Pump states (on/off)
o Valve states (on/off)
Continuous variables
o Flows 𝑄
o Heads 𝐻
Nonlinear constraint (head loss constraints)
Linear constraints
Variables bounds
It brings the problem into the mixed discrete/continuous optimization field. The problem is a Mixed
Integer NonLinear Problem (MINLP). A class of optimization which are really difficult to solve.
3 - Solving method
3.1 - Problem decomposition
Due to the complexity of the problem, direct solving methods are ineffective. Moreover, as the
problem is nonlinear and non convex, there is no guarantee on optimality.
That is why we decomposed the solving procedure in 2 steps:
1. Firstly, the problem is linearized and solved with a MIP solver (Xpress Optimizer) to
determine:
 Starts and stops of pumps
 Opening and closing of valves
2. Then, all binary variables are fixed, in the nonlinear model, with the results of the previous
optimization. The resulting continuous nonlinear problem is solved with an interior point
method (KNITRO).
3.2 - Linear relaxation
Linearized the problem consists in relaxing the nonlinear head loss constraints.
It consists in replacing the equation:
𝑄|𝑄|𝜆 = 𝐻𝑚 − 𝐻𝑛
By the following formulation:
𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑜𝑟. 𝜆 = 𝐻𝑚 − 𝐻𝑛
The difficulty consists in finding a good estimator which will preserve the distinction of the flow in
the pipes.
3.3 – Solution quality control
The decomposition of the problem is interesting insofar the linear relaxation solving give us a lower
bound of the problem. The quality of the lower bound is directly linked to the quality of the
relaxation.
Figure 2 - Iterative solving method
As it can be seen in Figure 2, a loop has been implemented in the solving procedure in order to allow
the user to control the quality of the solution.
4 – Results
4.1 - Test water distribution network
Our demonstrator has been tested on the water distribution network of Birkerod, a small town
located in Denmark.
This network consists of:
 2 pumps with 2 operating speeds,
 2 water sources,
 1 water tower,
 1 valve,
 178 client nodes,
 180 pipes.
4.2 - Results on case study 1
The objective of the test case 1 was to minimize the daily pumping costs according to the electricity
SPOT price from the European Power Exchange (EPEX) market.
Total computation time is around 5 minutes.
We observed 3 network operation modes:
Mode 1
The two pumps are not working. The water
tank ensure pressure on the network.
Mode 2
The pump number 2 (on the south of the
network) is the only one which is working. The
valve (between the north and the south) is
open and there is a small water flow to the
water tank.
Pump 2 supply to the south of the network.
There is a big water flow to the water tank.
Mode 3
The two pumps are working. The valve
between the north and the south is closed.
Pump 1 supply to the north of the network.
The following figure shows the correlation between operations mode and electricity price (Mode 1 is
in red, mode 2 in yellow and mode 3 in green).
As we can see, mode 1 occurs when the electricity price is high and mode 3 occurs when the
electricity price is low.
The obtained solution (price of the pumping plan) is 286.7 euros and lower bound is 283.3 euros.
That mean that the deviation with the optimal solution is less than 1%.
Results on case study 2
The objective of the test case 2 was to maximize the profit of changing the pumping plan obtained
previously according to an adjustment offer of 47.35 euros/MWh between 7 am and 8 am.
As we can see in the following figure, the new pumping plan (in red) save electricity at 8 am contrary
to the initial pumping plan (in blue). The consumption is reported at the cheapest hour (5 pm).
Pump Electricty
consumption (kWH)
Pump 1
600
400
Plan initial
200
Plan ajusté
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Due to the change of the pumping plan, the water level in the water tower will decrease at 8 am.
However, the water level will remain the same from 5 pm.
Water level in the
water tower
Water Tower
4
3.5
Plan initial
3
Plan ajusté
2.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24