```University of Sao Paulo
Department of Electrical and Computer Engineering
Intelligent Techniques Laboratory
Energy Cost Optimization in Water Distribution
Systems Using Markov Decision Processes
Paulo T. Fracasso, Frank S. Barnes and Anna H. R. Costa
Agenda
• Anatomy of Water Distribution Systems
• Problem relevancy
• Markov Decision Process
• Modeling a Water Distribution System as an MDP
• Monroe Water Distribution System
• Experiment results
• Conclusions
1
Water distribution system
• It is a complex system composed by pipes, pumps and other
hydraulic components which provide water supply to consumers.
Focus of
my work
2
Problem relevancy
• About 3% of US energy consumption (56 billion kWh) are used
for drinking water (Goldstein and Smith, 2002).
\$2 billion/year
3
Source: Electric Power Research Institute,1994.
Markov Decision Process - MDP
MDP is a model for sequential decision making in fully observable
environments when outcomes are uncertain.
Advantages of MDP compared to other techniques:

Real world – operates in uncertain and dynamic domains

Planning – generates control policies to sequential decisions

Optimal solution – guarantees to achieve a higher future payoff

Discrete domains (state and action)

Course of dimensionality
4
Markov Decision Process - MDP

MDP is defined as a tuple
S , A, T , R
where:
 S is a discrete set of states (can be factored in Nv features):

 
S   1 ,...,  N S  (  1 ,...,  1
1
NV
),..., (  N S ,...,  N SV )
N
1

 A is a discrete set of actions:
A
 A
 S
 T is a transition function where T
: S  A  S  0 ,1
T  ,  ,  '   P  s t 1   ' | s t   , a t   
 R is a reward function where R : S  A 
R  ,    rt | s t   , a t   
5

Markov Decision Process - MDP


Solving an MDP consists in finding a policy  , which is defined
as a mapping from states to actions, s.t.  : S  A
Bellamn’s equation allows to break a dynamic optimization
problem into simpler sub-problems:
V

   R  ,       T  ,   ,  '  V   '
 ' S

The optimal value of the utility is:
V

*
   max
A


*






R

,



T

,

,

'

V

'



 ' S


The optimal policy are the actions obtained from

    arg max  R  ,    
A
*

 T  ,  ,  '  V
 ' S
*

 '

V
*
  :
Water Distribution System modeled as an MDP

Topology of a typical water distribution system:

States (everything that is important to control):

Time – range:
discrete:


T  T
min
T  T
Tank level – range:
min

max
,T
H  H
discrete:
7
,T
S  T , H 1 ,..., H N H

min
min
  T ,..., T
,H
H  H
min
max
max


,H
min
  H ,..., H
max


Water Distribution System modeled as an MDP

Actions (what you can manipulate): A  U 1 ,..., U N 
U
U  0 ,1

Triggered directly:

Associated with a VFD – range:
U  0 ,1 
discrete: U

Transition function (how the system evolves):
H t  1   f  H ( t ), D ( t ), A ( t ) 


Calculated by EPANET
Reward function (how much an action cost):
BC
8
 0 ,  U ,..., 1

Consumption:

Demand: C D
CC 

 OP
  Pw ( t )   T  POP 


 ma x Pw ( t )   PDM
BC
R  CC  C D
FP
 Pw ( t )   T  P
FP




Demand
Water Distribution System modeled as an MDP
Electrical power
Final result:
Markov
Decision
Processes
Control policy
Energy price
schema
9
Constraints
Understand MDP results

Control policy:
 States variables: everything
that is important to control
(tank level and time)
Tank level
 Maps state variables into a set
of actions
 Set of actions: what you can
manipulate (pumps)
 Indicates controllability (avoid
black region)
 Correlated to demand curve
10
Time
Understand MDP results

Controller:
 Actions are based just on tank
level and time
Pump trigger
 Easy to implement and fast
to run in PLC (lookup table)
Tank level
 Uses control policy map to
produce actions
11
Time
Monroe Water Distribution System

Characteristics:
 11 pumps
 1 storage tank
 4 pressure monitoring
 40k people served
 182 miles of pipes
 Diameters varying
from 2 to 42 inches
12
Monroe Water Distribution System

Demand curve (during summer season):
Average: 6 700 GPM
Minimum: 4 188 GPM
Maximum: 8 389 GPM

13
Pressure restrictions (in PSI):

J-6: 65 ≤ P ≤ 70
▪ J-131: 45 ≤ P ≤ 55

J-36: 50 ≤ P ≤ 60
▪ J-388a: 40 ≤ P ≤ 90
Monroe Water Distribution System

Pumps (E2, E3, E4, E5, E6, E7, W8, W9, W10, W11 and W12):

Energy price schema:
14

On-peak (09:00 – 20:59): \$0.04014/kWh

Off-peak (21:00 – 08:59): \$0.03714/kWh

Demand (monthly): \$13.75/kW
MDP apply to Monroe WDS
Mathematical model:
 Set of states:
S  T , H
 Set of actions:
 where
T  0 , 24 
and
H  1, 33 . 25 
A  U 1 ,..., U 11 
 Transition function:
H ( t  1)  f  H ( t ), d ( t ), A ( t ) 
 Reward function:
 21:00
R ( t )  30   T   Pw ( t )  \$ on  peak 
 t  9:00

Pw
(
t
)

\$
 max  Pw ( t )   \$ demand

off  peak 
t  21:00

9 :00
Data flux diagram:
.INP FILE
15
EPANET
DLL
MATLAB
h ( t  1)
Pw ( t )
MDP results in Monroe WDS

Expected electrical power :
E5 and E7 consume 144.3kW
16
W11, E2 and E6 consume 320.4kW
MDP results in Monroe WDS

Number of activated pumps (27 possibilities):
on={E2,E6}
17
on{E5,E7}
on={W12,E3,E4,E5}
on={E2,E3,E4,E5}
MDP results in Monroe WDS


obtained from historical data (July 6th, 2010)

75% of WTP consumption is considered to be used in pump

One day is extrapolated to one billing cycle (30 days)

Both approaches started in the same level (19.3 ft)
Energy expenses
Off-peak energy [\$/month]
On-peak energy [\$/month]
Demand [\$/month]
Total [\$/month]
18
3 210.57
3 750.78
3 836.25
10 797.60
MDP
2 608.32
3 768.51
3 603.67
9 980.50
Difference
-23.1%
+0.5%
-6.5%
-8.2%
Conclusions

MDP avoids restrictions (level, pressure, and pumps) and reduces
expenses with energy

To reduce energy consumption is different to reduce expenses with
energy (demand is the biggest villain)

Summer season imposes small quantity of feasible actions

Verify if it is possible to reduce the number of pump combination

MDP policy is easy to implement in a non-intelligent device (PLC)
19
Contact
PAULO THIAGO FRACASSO
paulo.fracasso@usp.br
Av. Prof. Luciano Gualberto, trav.3, n.158, sala C2-50
CEP: 05508-970 - São Paulo, SP - Brazil
Phone: +55-11-3091-5397
20
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