1. The Problem Stochastic Process Simulations
Stochastic Modelling
for Hydro-Electric Reservoir Management
Team 1
Pacific Institute of Mathematical Sciences
Graduate Mathematical Modelling in Industry Workshop - 2016
August 13, 2016

2. The Problem Stochastic Process Simulations
Our Group!
• Ismail Hossain - University of Manitoba
• Clifford Allotey - University of Manitoba
• Farzaneh Jannat - University of Manitoba
• Weifei Ouyang - Shanghai Jiao Tong University
• Alfred Liu - University of Windsor
• Clint Seinen - University of Victoria
• Faisal Atakora - University of Manitoba

3. The Problem Stochastic Process Simulations
Our Group!
• Ismail Hossain - University of Manitoba
• Clifford Allotey - University of Manitoba
• Farzaneh Jannat - University of Manitoba
• Weifei Ouyang - Shanghai Jiao Tong University
• Alfred Liu - University of Windsor
• Clint Seinen - University of Victoria
• Faisal Atakora - University of Manitoba
Mentor: Dr. Fabian Bastin - University of Montreal

4. The Problem Stochastic Process Simulations
The Problem

5. The Problem Stochastic Process Simulations
Stems from Hydropower!

6. The Problem Stochastic Process Simulations
Introduction to Inflow Modeling
“The central element is a stochastic model for natural inflows”
(Pritchard, 2014)
Major complications
• How to model the seasonality effects?
• How to model the spatial and time correlations?

7. The Problem Stochastic Process Simulations
Inflows usually have positive serial relation.The model takes the
form:
Xt = Ft(Wt−1)
• Wt−1 is state variable at time t − 1, for univariate, we always
take Wt−1 = Xt−1,
for higher-order model take Wt−1 = (Xt−1, Xt−2, . . . , Xt−r )
Existing models
• Multivariate AR(1)
• Iterated function system
No current approach is totally satisfactory.

8. The Problem Stochastic Process Simulations
Quebec has a lot of dams!

9. The Problem Stochastic Process Simulations
4-Dam System!
Instead of all the dams in Quebec, we consider the following four
dam system

10. The Problem Stochastic Process Simulations
How We Attacked the Problem
• Analyzed the historical data
• to determine the extent of correlations and create foundation
for the stochastic process

11. The Problem Stochastic Process Simulations
How We Attacked the Problem
• Analyzed the historical data
• to determine the extent of correlations and create foundation
for the stochastic process
• formulate a stochastic process

12. The Problem Stochastic Process Simulations
How We Attacked the Problem
• Analyzed the historical data
• to determine the extent of correlations and create foundation
for the stochastic process
• formulate a stochastic process
• implement a simulation of the 4-dam system

13. The Problem Stochastic Process Simulations
Stochastic Model of Inflow
Xt − µt = ϕ(Xt−1 − µt−1) + εt
where,
• Xt : Inflow of water at time t (weeks)
• µt : Expected value of inflow at t
• ϕ : Auto-regression parameter (ϕ = -0.4589)
• Xt−1 : Inflow of water at time t − 1
• µt−1 : Expected value of inflow at time t − 1
• εt : Error term of week t that follows the standard normal
distribution

14. The Problem Stochastic Process Simulations
Let Xit be the inflows into the respective dams (i = 1, 2, 3, 4.)




X1t
X2t
X3t
X4t



 ∼ N(µt, Σt)
where Σt is covariance matrix at time t. With this setup the
spatial correlation in the data set will be reflected in the
simulation. However the temporal correlation is not guaranteed,
thus we incorporate the AR(1) model into our model.

15. The Problem Stochastic Process Simulations
Temporal Correlations

16. The Problem Stochastic Process Simulations
Generated Scenarios
Figure : Deviation from the Mean

17. The Problem Stochastic Process Simulations
Generated Scenarios
Figure : Dam 1

18. The Problem Stochastic Process Simulations
Generated Scenarios
Figure : Dam 2

19. The Problem Stochastic Process Simulations
Generated Scenarios
Figure : Dam 3

20. The Problem Stochastic Process Simulations
Generated Scenarios
Figure : Dam 4

21. The Problem Stochastic Process Simulations
Simulations
• SimJulia! Why?

22. The Problem Stochastic Process Simulations
Simulations
• SimJulia! Why?
• Good for discrete events
• Allows room for future complexities
• Potential for random “complications”, or breakdowns

23. The Problem Stochastic Process Simulations
Simulations
Prior to a realistic model containing ALL our lovely
components......

24. The Problem Stochastic Process Simulations
Simulations
Prior to a realistic model containing ALL our lovely
components......we prototype!
• start with an initial volume
• wait a week
• make a random decision on what to spill/store/run through
turbines
• update system with decision and random inflows according to:
Volnew = Volprev + inflows + Volupstream − Volturbine − Volspill

25. The Problem Stochastic Process Simulations
Simulations
Prior to a realistic model containing ALL our lovely
components...... we prototype!

26. The Problem Stochastic Process Simulations
Simulations
The Real Model!
Again, we start with an initial volume behind the reservoirs.

27. The Problem Stochastic Process Simulations
Simulations
The Real Model!
Again, we start with an initial volume behind the reservoirs.
Then create the scenario tree with 200 realizations of our
stochastic process.

28. The Problem Stochastic Process Simulations
Simulations
The scenario tree is then input into the C++ optimizer, which
then decides on how much water to run through the turbines or
how much is spilled.

29. The Problem Stochastic Process Simulations
Simulations
The scenario tree is then input into the C++ optimizer, which
then decides on how much water to run through the turbines or
how much is spilled.
Unfortunately.....

30. The Problem Stochastic Process Simulations
Simulations
But!

31. The Problem Stochastic Process Simulations
Simulations
But!
We know SimJulia can be used!

32. The Problem Stochastic Process Simulations
Simulations
But!
We know SimJulia can be used!
We know our stochastic method can be used to create the
necessary scenario tree format!

33. The Problem Stochastic Process Simulations
Simulations
But!
We know SimJulia can be used!
We know our stochastic method can be used to create the
necessary scenario tree format!
We know scenred2 and the C++ optimizer can be called from
with-in the SimJulia environment!

34. The Problem Stochastic Process Simulations
Simulations
But!
We know SimJulia can be used!
We know our stochastic method can be used to create the
necessary scenario tree format!
We know scenred2 and the C++ optimizer can be called from
with-in the SimJulia environment!
We have created a solid frame-work for future work on this project!

35. The Problem Stochastic Process Simulations
Thank You!