<figure><img src="image_33.png" /><figcaption>Figure 5.10 Comparison of September (lower figures in each subfigure) and October (upper figures in each subfigure) day 1 data sets. The dashed lines are the minimum and maximum scenarios. The median scenario is the solid line. The actual realization of the inflows is the plus sign line.</figcaption></figure>

are represented in the same fashion. Again, Figures 5.10b and 5.10c show that for Chute-Savane and Lac-St-Jean, the actual inflows in October are very close to the median scenario, therefore there is no gain in using a stochastic optimization model, as the deterministic median scenario allows to obtain a good solution. For this unusual October case, solving the short-term unit commitment and loading problem with a median scenario is acceptable. This affirmation is to be used with caution as situations like these have a low probability of occurring. These results show that there is certainly a gain in using a stochastic model for the short-term hydropower optimization model, as relying on the median scenario offers a less robust solution than multiple scenarios.
## 5.6 Conclusion

This paper presents a stochastic short-term hydropower optimization method which emphasizes inflow scenario trees. Few papers looked specifically into stochastic short-term models and we extend the modeling presented in [91] to consider uncertain inflows. The optimization method considers inflow uncertainty, head variations and nonlinear and nonconvex relationship between discharge and power output. The scenario tree generation method first uses kernel density estimation to generate random values of inflows. Then, the path of nodes, from root to leaf, that minimizes the Wasserstein distance is found in the scenario tree and the corresponding nodes are updated using stochastic approximation. The process is repeated until the termination criterion, which is the convergence of the tree in Wasserstein distance, has been reached. A stability test has shown that the scenario tree generation method is consistent. A highlight of this method is that it uses all data available at each iteration to improve the values of the scenario tree nodes. The scenario trees are inputs to a two-phase optimization process. The first phase, loading problem, allows to find water discharge, volume and number of turbines working in each plant. The second phase, unit commitment, chooses the exact combination of turbines to use, to maximize energy production and penalize unit startups. A major feature of this modeling of the problem is that the water head is not neglected. For this paper, the models are tested on three hydropower plants. A rolling-horizon procedure is retained on a 31 day planning horizon. The stochastic solution is compared to the median scenario. Furthermore, fast computation time allows this method to be scaled in order to be applied in full to the Saguenay-Lac-St-Jean hydroelectric system. Future work based on this paper consists on investigating the complexity required in the scenario tree structure. Since a rolling-horizon framework is retained and that only the solution of the first-stage is kept, tests with scenario fans instead of scenario trees will be conducted.

## Notation

The following notation is used throughout the paper :

<table><tr><td>N</td><td>set of nodes</td></tr><tr><td>E</td><td>set of leaf nodes</td></tr><tr><td>K</td><td>set of scenarios</td></tr><tr><td>C</td><td>set of hydroelectric plants</td></tr><tr><td>u<sup>c</sup></td><td>number of hydroelectric plants upstream of plant c ∈ C</td></tr><tr><td>s ∈ {1, 2, ..., n<sub>i</sub><sup>c</sup>}</td><td>index of surfaces corresponding to number of active turbines with hydroelectric plant c and node i</td></tr></table>
<table>
  <tr>
    <td>l &isin; {1, 2, ..., n<sub>i</sub><sup>c</sup>}</td>
    <td>index of combinations with<br>hydroelectric plant c and node i</td>
  </tr>
  <tr>
    <td>t &isin; {1, 2, ..., T<sup>c</sup>}</td>
    <td>index of turbines of hydroelectric plant c</td>
  </tr>
  <tr>
    <td>&pi;<sub>j</sub><sup>c</sup></td>
    <td>probability of scenario j for plant c</td>
  </tr>
  <tr>
    <td>v<sub>i</sub><sup>c</sup></td>
    <td>volume of plant reservoir c at node<br>i (hm<sup>3</sup>)</td>
  </tr>
  <tr>
    <td>q<sub>i</sub><sup>c</sup></td>
    <td>water discharge at plant c and node<br>i (m<sup>3</sup>/s)</td>
  </tr>
  <tr>
    <td>&theta;</td>
    <td>start-up penalty for any turbine (MW)</td>
  </tr>
  <tr>
    <td>&beta;<sub>li</sub><sup>c</sup></td>
    <td>power generated by combination l &isin; n<sub>i</sub><sup>c</sup><br>at plant c and node i</td>
  </tr>
  <tr>
    <td>y<sub>si</sub><sup>c</sup> =</td>
    <td>
      <div style="display: flex; align-items: center;">
        <span style="font-size: 2.5em; line-height: 1;">{</span>
        <div style="margin-left: 1em;">
          <p>1 if surface s is chosen at node i</p>
          <p>for plant c</p>
          <p>0 otherwise</p>
        </div>
      </div>
    </td>
  </tr>
  <tr>
    <td>f<sub>lit</sub><sup>c</sup> =</td>
    <td>
      <div style="display: flex; align-items: center;">
        <span style="font-size: 2.5em; line-height: 1;">{</span>
        <div style="margin-left: 1em;">
          <p>1 if turbine t of combination l</p>
          <p>for plant c is active at node i</p>
          <p>0 otherwise</p>
        </div>
      </div>
    </td>
  </tr>
  <tr>
    <td>x<sub>li</sub><sup>c</sup> =</td>
    <td>
      <div style="display: flex; align-items: center;">
        <span style="font-size: 2.5em; line-height: 1;">{</span>
        <div style="margin-left: 1em;">
          <p>1 if combination l of plant c</p>
          <p>is chosen at node i</p>
          <p>0 otherwise</p>
        </div>
      </div>
    </td>
  </tr>
  <tr>
    <td>d<sub>ti</sub><sup>c</sup> =</td>
    <td>
      <div style="display: flex; align-items: center;">
        <span style="font-size: 2.5em; line-height: 1;">{</span>
        <div style="margin-left: 1em;">
          <p>1 if turbine t of plant c is started</p>
          <p>at node i</p>
          <p>0 otherwise</p>
        </div>
      </div>
    </td>
  </tr>
  <tr>
    <td>&chi;<sub>si</sub><sup>c</sup></td>
    <td>power for surface s at node i and plant c (MW)</td>
  </tr>
  <tr>
    <td>&Psi;<sub>s</sub><sup>Ac</sup>(q<sub>i</sub><sup>c</sup>, v<sub>i</sub><sup>c</sup>)</td>
    <td>power production function without spillage for surface s and plant c</td>
  </tr>
  <tr>
    <td>&Psi;<sub>s</sub><sup>Bc</sup>(q<sub>i</sub><sup>c</sup>, v<sub>i</sub><sup>c</sup>)</td>
    <td>power production function with spillage for surface s and plant c</td>
  </tr>
  <tr>
    <td>&delta;<sub>i</sub><sup>c</sup></td>
    <td>inflow of plant c at node i (m<sup>3</sup>/s)</td>
  </tr>
  <tr>
    <td>w<sub>i</sub></td>
    <td>duration of node i (h)</td>
  </tr>
  <tr>
    <td>&gamma;</td>
    <td>conversion factor from water discharge (m<sup>3</sup>/s) to (hm<sup>3</sup>/h)</td>
  </tr>
  <tr>
    <td>&Phi;<sub>j</sub><sup>c</sup>(v<sub>j</sub><sup>c</sup>)</td>
    <td>water-value function for plant c and scenario j</td>
  </tr>
  <tr>
    <td>&zeta;<sub>i</sub></td>
    <td>conversion factor to energy units (GWh)</td>
  </tr>
  <tr>
    <td>v<sub>min</sub><sup>c</sup></td>
    <td>minimal volume of plant c reservoir (hm<sup>3</sup>)</td>
  </tr>
  <tr>
    <td>v<sub>max</sub><sup>c</sup></td>
    <td>maximum volume of plant c reservoir (hm<sup>3</sup>)</td>
  </tr>
  <tr>
    <td>q<sub>min</sub><sup>c</sup></td>
    <td>minimum water discharge at plant c (m<sup>3</sup>/s)</td>
  </tr>
  <tr>
    <td>q<sub>max</sub><sup>c</sup></td>
    <td>maximum water discharge at plant c (m<sup>3</sup>/s).</td>
  </tr>
</table>
# CHAPITRE 6 ARTICLE 3 : SCENARIO TREE MODELING FOR STOCHASTIC SHORT-TERM HYDROPOWER OPERATIONS PLANNING

Cet article a été soumis à une revue avec comité d'arbitrage le 8 juillet 2016.

Le rapport technique suivant est publié :

S. Séguin, C. Audet et P. Côté. Scenario tree modeling for stochastic short-term hydropower operations planning. *Cahiers du GERAD*, (G-2016-54), 2016.

**Abstract** : The authors investigate the complexity needed in the structure of the scenario trees to maximize energy production in a rolling-horizon framework. Three comparisons, applied to the stochastic short-term unit commitment and loading problem are conducted. The first one involves generating a set of scenario trees built from inflow forecast data over a rolling-horizon. The second replaces the entire set of scenario trees by the median scenario. The third replaces the set of trees by scenario fans. The method used to build scenario trees, based on minimization of the nested distance, requires three parameters : number of stages, number of child nodes at each stage, and aggregation of the period covered by each stage. The authors formulate the question of finding the best values of these parameters as a Blackbox optimization problem that maximizes the energy production over the rolling-horizon. Numerical experiments on three hydropower plants in series suggest that using a set of scenario trees is preferable to using the median scenario, but using a fan of scenarios yields a comparable solution with less computational effort.

**Acknowledgments** : The authors would like to thank Marco Latraverse at Rio Tinto for providing data. Also, Alois Pichler for the scenario tree generation method and Stein-Erik Fleten for collaborating on the stochastic version of the model. Sara Séguin would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC), Fonds de recherche du Québec - Nature et Technologies (FRQNT) and Rio Tinto for their financial support.

## 6.1 Introduction

Short-term hydropower models are used on an operational basis to determine the production plan of an hydroelectric system. From these models, the reservoir volumes, water flows and the unit commitment of the turbines in operation are determined. Deterministic models [85]
in which there are no uncertainties, as well as stochastic short-term hydropower models [16]
have recently been proposed. In this paper, uncertain inflows are considered, based on the
authors previous work [89].

The province of Quebec, located in Canada, is a territory rich in lakes and rivers and 99 % of its energy production comes from hydropower. Multiple aluminium producers operate plants in this province since the electrolysis process used to extract aluminium from the bauxite requires immense amounts of electricity. Rio Tinto is a mining company that operates aluminium smelters in the Saguenay Lac-St-Jean region, located in the north of the province. They are also owners of a hydroelectric system that allows them to produce 90 % of the energy required for the operations of the aluminium plants. Therefore, it is in their interest to manage the hydroelectric system in an efficient manner, since they need to buy, from Hydro-Quebec at fixed priced contracts, the energy shortage to fully operate the smelters. In an operational context for Rio Tinto, the implementation of the decisions obtained from the short-term hydropower optimization models are to be taken without exact knowledge of daily inflows. Inflow scenarios are built based on a 7 day deterministic forecast of precipitations prepared by Environment Canada. The CEQUEAU [75] hydrological model is used to create inflow scenarios based on the hydraulics of the watershed and on the precipitation forecast. Multiple inflow scenarios are available and one way to treat uncertainty in an optimization model involves scenario trees and multistage stochastic programs. Many methods have been developed to generate scenario trees. Some of the most popular methods are moment-matching [57], scenario reduction [54], copulas [60], or minimization of the nested distance [82], for example. In this paper, the authors specifically use the minimization of the nested distance to build the inflow scenario trees. As there is no reduction in the number of scenarios at each iteration, all of the inflow scenarios are used to update the value of the scenario tree nodes. Also, the minimization of the nested distance implies that the first four moments, which are mean, variance, skewness and kurtosis are matched.

This method consists of two steps that are repeated until convergence of the nested dis-
tance is achieved. The first step uses kernel density estimation [87] to generate a new inflow
scenario that is close to the distribution of the available inflow scenarios. The second step
uses this newly generated scenario to update the values of the scenario tree nodes based on
the stochastic gradient descent method. This scenario tree generation method requires input
parameters : the number of stages, the number of nodes per stage and the aggregation of
each stage.

In a decision-making operational context, rolling-horizon schemes [74] are used by the hy-
dropower producers to implement the solutions of the stochastic short-term hydropower op-
timization models. As the inflow previsions are updated daily, a scenario tree is generated, then the optimization models are solved and the solutions of the first-day are implemented. Once the actual realizations of the inflows for the reservoirs are known at the end of the day, reservoir levels are updated with the realization of the inflows and the scenario tree generation process and optimization is repeated for the next day, with the new forecast. In some cases [17] the time window is decreased as information becomes available, and in other cases the time window moves forward in the horizon [110]. In [33], a test bed is used to compare four optimization algorithms. The forecasts are updated each time a decision is taken and the same methodology has been retained in this research.

As the scenario tree generation method [89] used in this paper requires input parameters, this work studies if there is an energy production gain when the scenario tree parameters are optimized in an objective of maximizing the energy production throughout the rolling-horizon. The rolling-horizon scheme, consisting of scenario tree generation and short-term hydropower optimization is embedded within a Blackbox optimization model. Blackbox optimization is used when the objective functions or the constraints of a problem can only be calculated through a computer code, as it is the case in this problem. Blackbox optimization methods have been applied successfully to many engineering problems [8]. In the field of hydrology, Blackbox optimization has been used to find the optimal locations for GMONs [4], which are devices used to measure snow water equivalent in remote areas of watersheds. Another study [73] proposes to calibrate the 23 parameters of the hydrological model HSAMI. It is used in the daily forecast of reservoir inflows and parameters relating to evapotranspiration, snowmelt, infiltration and percolation and finally the routing of surface runoff. An interesting study in reservoir management [46] compares the solution of the reservoir optimization problem with many reduced scenario trees to determine the effect of this reduction on the optimization solution. In the same sense, a thorough research [108] aims at finding the optimal level of scenario tree reduction to obtain the best performance for a hydropower reservoir management problem.

The paper compares three scenario trees approaches to solve the stochastic short-term unit commitment and loading problem (SSTUL). The first one uses a Blackbox optimization solver to identify the set of scenario trees that maximizes energy production over the rolling-horizon, using a methodology similar to that proposed in [12] in which Blackbox optimization is used to tune algorithmic parameter values. The second one is much simpler, both conceptually and in terms of computational effort, as it only uses the median scenario. The third one contains a fan of a limited number of representative scenario trees. Numerical experiments in an operational context with real data suggest that the computational effort invested in finding the best set of scenario trees outperforms the median scenario approach, but scenario
fans produce comparable results in less time.

The paper is organized as follows. Section 6.2 presents the methodology, more precisely the stochastic short-term unit commitment and loading problems and the rolling-horizon scheme used to validate the optimization models. Section 6.3 introduces the concept of Blackbox optimization and exposes the formulation of the rolling-horizon scheme as a blackbox. A case study is presented in Section 6.4 and numerical experiments are shown in Section 6.5. Final remarks are given in Section 6.6.

## 6.2 Methodology

The present section gives a high-level description of the main building blocks necessary for this work. Each section describes the input and output of each of them, and the authors voluntarily avoid presenting the technical specificities of each block, but provide references for detailed descriptions.

### 6.2.1 Stochastic Short-Term Unit Commitment And Loading Problem

This research targets the stochastic short-term unit commitment and loading problem. In hydropower operations planning, this model is used on an operational basis to dispatch water available for production between the turbines and power plants that compose the system. The model considers head-dependency in the power production functions, efficiencies of each turbines and turbine startups are penalized. Without presenting the mathematical modeling of the problem, as it is not the scope of the present research, the contents of this model is explained for the reader to understand its outcome. In a recent work, the authors have also considered stochastic inflows in the reservoirs [89]. A scenario tree is used to represent the uncertain inflows and therefore, the SSTUL model is solved for each scenario tree node. Refer to paper [91] for more details on the formulation. The SSTUL optimization model consists of two optimization phases. The first phase, namely the loading problem, maximizes the expected energy production and reservoir volumes, water flows and total number of turbines working are determined. The second phase, namely the unit commitment determines the exact combination of turbines working to maximize energy production and penalize unit startups.

### 6.2.2 Rolling-Horizon Scheme

In the context of real operations of a hydropower system, forecasts of the inflows are updated daily and are denoted $\phi_h$, where $h$ is the day in the rolling-horizon. Therefore, it is necessary to
take an immediate decision, without exact knowledge of future inflows. By the end of the day, the actual inflows that occurred throughout the day are known and the reservoir volumes are updated. This rolling-horizon scheme is then repeated for each day of the planning horizon.

The entire process is composed of five main components and can be viewed on Fig 6.1. The first one is called the *scenario tree generator (STG)* and it takes as input the scenario tree parameters *α* and *κ*, which are respectively the number of child nodes per stage and the aggregation of each stage, as well as the fixed parameter *D* representing the number of stages of the scenario tree and finally, the daily inflow forecasts denoted *φ*<sub>h</sub>. As its name indicates, it produces a scenario tree that becomes the input of the second component namely the *stochastic short-term unit commitment and loading problem (SSTUL)*, which consists of two elements. The *loading problem* (LP) is a nonlinear mixed-integer program that determines the water discharge, reservoir volume and number of active turbines at each node for each power plant. This information is fed to the *unit commitment problem (UC)*, a linear integer program and it determines the exact combination of turbines to use in order to maximize the total energy production at each node and penalize turbines start-up. The output of the *SSTUL* is the solution to the *SSTUL* optimization problem and is used to compute the energy generation for the first node in the scenario tree, denoted *E(h)*, in which turbine startups are penalized. From here, the reservoir levels are updated with the actual realization of the inflows in the component named *reservoir level update*. These 5 components are iteratively solved for each time period but only the solution for the first node of the scenario tree is retained every day, allowing to build a release policy for the whole rolling-horizon. The output of the rolling-horizon, namely *RH(α, κ, D)* is the summation of the energy generation *E(h)* for each day *h* of the rolling-horizon, more precisely the first node energy generation every day with startups penalized and the value of the remaining water in the reservoirs at the end of the rolling-horizon.

<figure><img src="image_34.png" /><figcaption>Figure 6.1 Rolling-horizon scheme. These steps are repeated for each day *h* of the rolling-horizon.</figcaption></figure>

The objective is to maximize the total energy production over the whole rolling-horizon, by
evaluating the energy production for the first stage solution of each day in the rolling-horizon, penalizing turbine startups and considering the value of the remaining water in the reservoirs, using water-value functions.

### 6.2.3 Parameters Of The Scenario Tree

The overall energy estimated over the rolling-horizon depends on the way that the scenario trees are generated. Generation of these trees depend on a set of three parameters. The main objective of the present work is to tune the values of these parameters so that the overall energy is maximized. The authors formulate this question as a Blackbox optimization problem : find the scenario tree parameter values that maximize the energy production. This section describes in more details the scenario tree parameters.

The scenario tree generating method requires input parameters : number of stages, number of child nodes per stage as well as aggregation of each stage. In addition, it requires as input the inflow scenarios denoted by $\phi$. As the effect of the number of stages is investigated, it is kept as a varying parameter. As for the number of nodes per stage and aggregation, they are treated as Blackbox optimization variables. For a given stage, all nodes must have the same number of child nodes. Since the inflow forecast is for 30 days, it is necessary to aggregate the days into stages in the scenario tree in order to reduce the number of optimization variables. The first stage is not aggregated and hence is not an optimization variable.

Let $D$ be the number of stages, $\alpha_q$ the number of child nodes, for each stage $q \in \{1, 2, \dots, D-1\}$ and $\kappa_r$ is the number of days aggregated in stage $r \in \{2, 3, \dots, D\}$. Fig. 6.2 illustrates a scenario tree with 4 stages, with $\alpha = (1, 2, 2)$ and $\kappa = (2, 12, 15)$. With an inflow forecast of 30 days, the following is to be satisfied : $\sum_{r=2}^{4} \kappa_r = 29$, as it is the case on Fig. 6.2.

The function that characterizes the scenario tree generation is compactly denoted by :

$$STG(\alpha, \kappa, \phi), \qquad (6.1)$$

where $\alpha \in \mathbb{N}^{D-1}$ is the number of child node per stage, $\kappa \in \mathbb{N}^{D-1}$ is the aggregation of each stage and $\phi$ is the collection of inflow scenarios. The scenario tree generation returns a scenario tree determined by the parameters $\alpha$ and $\kappa$, the collection of inflow scenarios $\phi$ and a value of inflow for each reservoir and scenario tree node is returned.
<figure><img src="image_35.png" /><figcaption>Figure 6.2 A scenario tree with $D = 4$ stages</figcaption></figure>

### 6.2.4 Loading And Unit Commitment Problems

The loading problem takes as input a scenario tree determined by the parameters $\alpha$, $\kappa$ and $\phi$ obtained with $STG(\alpha, \kappa, \phi)$. The loading problem returns the water flows, reservoir volumes and number of turbines working, for every plant and scenario tree node. The loading problem maximizes the energy production in the first stage and expected energy production in the future, subject to water balance constraints and hydropower production functions selection. The entire model is presented in Section 4.2 of [89].

The function that characterizes the loading problem is compactly denoted by :

$$LP(STG(\alpha, \kappa, \phi)). \qquad (6.2)$$

The loading problem is a nonlinear mixed-integer problem, but the relaxation of binary variables is sufficient [91] to obtain an integer solution over certain conditions. In this specific case, the energy demand is not considered.

The unit commitment problem takes as input the output of the loading problem, more precisely the water flows, reservoir volumes and number of turbines working, for every plant and scenario tree node. It also requires the scenario tree structure as input, determined with $STG(\alpha, \kappa, \phi)$. The unit commitment problem consists of maximizing the energy production in the first stage and expected energy production in the future and penalizes startups of turbines, subject to turbine selection and turbine startups constraints. The unit commitment problem returns the turbines working for every plant and scenario tree node. The entire
model is presented in Section 4.3 of [89].

The function that characterizes the unit commitment problem is compactly denoted by :

$$UC(LP(STG(\alpha, \kappa, \phi))). \qquad (6.3)$$

The unit commitment problem is a linear integer program.

## 6.3 Blackbox optimization

This section introduces the concept of Blackbox optimization. Then, the formulation of the rolling-horizon scheme as a Blackbox optimization problem is exposed. Finally, the validation of the formulation of the problem as a blackbox is presented.

### 6.3.1 Blackbox Optimization Concept

Blackbox optimization targets problems in which the objective function and/or constraints can only be computed through a computer simulation. Blackbox optimization problems are often nonsmooth, nonconvex and discontinuous. The mesh adaptive direct search method (MADS) [10] is designed for Blackbox optimization and has been successfully applied to many engineering problems.

MADS discretizes the state space by defining a mesh whose coarseness is adjusted at the end of each iteration. The algorithm consists of two steps that are repeated until a predefined stopping criterion is reached. The first step, namely the *search* step evaluates different points that lie on this mesh with the aim of finding a better solution than the current best. This step is flexible and may be tailored to specificities of the problem. The second step, namely the *poll* step is mandatory as the convergence properties of MADS relies on it. A positive spanning set of directions is determined and if a better solution than current best is found, it is set as best solution. During the different iterations, the mesh size is reduced when the algorithm fails to improve the solution and increased when a new best solution is found. The NOMAD software [63], which is the implementation of the MADS method is used to solve the Blackbox optimization model.

### 6.3.2 Blackbox Formulation Of The Rolling-Horizon Scheme

The scenario tree parameters $\alpha$ and $\kappa$ that maximize the energy production throughout the rolling-horizon are to be identified. A blackbox that takes a tree structure as input, and
solves the whole rolling-horizon scheme, which contains the scenario tree generation and the SSTUL optimization problem is defined. It returns the total energy produced throughout the rolling-horizon.

## Mathematical formulation

The scenario tree parameters $\alpha$ and $\kappa$ are the optimization variables provided as input to the blackbox. The output of the blackbox is the total energy produced, considering startup penalties and value of remaining water in the reservoirs in the rolling-horizon scheme. The SSTUL optimization variables, which are water discharges, reservoir volumes and turbines working are internal to the blackbox and are transparent to the Blackbox optimization problem.

Therefore, the Blackbox optimization problem is to maximize total energy produced throughout the rolling-horizon scheme, with startups penalized and value of the water remaining in the reservoirs. The output of the blackbox is compactly denoted by :

$$ RH(\alpha, \kappa, D), \qquad (6.4) $$

where $D$ is fixed and is the number of stages in the scenario tree.

The objective function given by Eq.(6.5) is to maximize energy production throughout the rolling-horizon, considering startup penalties and value of remaining water in the reservoirs. It relates to the unit commitment problem and since this solution relies on the output of the loading problem, problems UC and LP are to be solved in order to obtain the value of $RH(\alpha, \kappa, D)$. In other words, it is necessary to solve the SSTUL problem when solving the Blackbox optimization problem.

### Problem BB:

$$ \max_{\alpha, \kappa \in \mathbb{N}^{D-1}} RH(\alpha, \kappa, D) \qquad (6.5) $$

$$ \text{subject to :} \qquad \sum_{r=2}^{D} \kappa_r = 29, \qquad (6.6) $$

$$ 2 \le \prod_{q=1}^{D-1} \alpha_q \le 50, \qquad (6.7) $$

$$ \alpha_q, \kappa_r > 1, \quad \forall q \in \{1, 2, \dots, D-1\}, \forall r \in \{2, 3, \dots, D\}. \qquad (6.8) $$

The main difficulty of the problem above resides in evaluating $RH(\alpha, \kappa, D)$. The computational time required to evaluate the blackbox increases with the complexity of the scenario trees, rather than with the number of optimization variables. For example, for a 5-stage sce-
nario tree and $\boldsymbol{\alpha} = (1, 4, 3, 3)$, a total of 52 nodes, a single evaluation of $RH(\boldsymbol{\alpha}, \boldsymbol{\kappa}, D)$ takes 32 minutes, on a given data set, while a 6-stage scenario tree and $\boldsymbol{\alpha} = (2, 5, 5, 1, 1)$, a total of 162 nodes, takes 1 hour. Moreover, there is only a difference of 2 optimization variables between these scenario tree structures.

Constraints (6.6) are to ensure the aggregation of the scenario tree stages are equivalent to the inflow forecast, which is 30 days in this paper. Since the first stage is not aggregated then it is forced to 1. The upper and lower bounds on the number of scenarios are given by (6.7). Nonnegativity is enforced with constraints (6.8). The number of stages $D$ is kept as a varying parameter in the numerical experiments in order to evaluate its effect on the results. Hence, it is fixed for every experiment.

Fig. 6.3 illustrates this process. A scenario tree structure with parameters $\boldsymbol{\alpha}$ and $\boldsymbol{\kappa}$ is given to the blackbox. The rolling-horizon scheme is then launched. For every day of the rolling-horizon, scenario trees are built based on the forecast $\phi_h$, then the SSTUL is solved, the first-day solution is kept, reservoir levels are updated with the realization of inflows and the process is repeated for the whole rolling-horizon. Then, the actual energy production and expected future production of the remaining water in the reservoirs for the whole rolling-horizon is calculated, with startups of turbines penalized and is denoted $RH(\boldsymbol{\alpha}, \boldsymbol{\kappa}, D)$. The optimization solver calls the blackbox simulation with various values of the parameters $\boldsymbol{\alpha}$ and $\boldsymbol{\kappa}$. Each time the solver collects the output $RH(\boldsymbol{\alpha}, \boldsymbol{\kappa}, D)$ and from it, it produces new inputs to the blackbox. The solver terminates when it can no longer improve the solution.

<figure><img src="image_36.png" /><figcaption>Figure 6.3 The Blackbox optimization solution approach. Index $h$ is the day in the rolling-horizon framework.</figcaption></figure>
### 6.3.3 Blackbox Validation

The scenario tree generation method applied in this paper is stochastic, which means that for the same set of scenario tree parameters, the inflow values on the nodes of the scenario tree differ from one evaluation to the other. Therefore, the objective function value $RH(\alpha, \kappa, D)$ of the blackbox differs for the same structure of scenario trees. The blackbox is validated to ensure that trying to optimize the scenario tree parameters is relevant. Since the scenario tree generation method is stochastic, the state space changes dynamically while the solver is searching for scenario tree parameters values. Therefore, it is important to assess that the variation of the state space is negligible compared to converging towards a solution. A fixed scenario tree structure is used and the blackbox is evaluated 100 times with the same set of scenario tree parameters, on a given data set.

Results of this validation are shown on Fig. 6.4. The number of evaluations of the blackbox corresponding to a given range of objective function values are represented on a histogram. The following values were used for the parameters : $D = 5$, $\alpha = (1, 1, 2, 1)$ and $\kappa = (3, 6, 8, 12)$. One can see that energy production is very similar at every evaluation of the blackbox, the standard deviation is 7.4622 GWh, compared to a 7.0478 ×10⁵ GWh mean. The values near the mean confirm that the scenario tree generation method is valid, as evaluations of the blackbox with the same scenario tree parameters lead to similar values every time.

<figure><img src="image_37.png" /><figcaption>Figure 6.4 Validation of the blackbox. For the same scenario tree structure, histogram of the energy production in GWh.</figcaption></figure>
Exhaustive enumeration versus Blackbox optimization

To assess the pertinence of using a Blackbox optimization problem to find the scenario
tree structure that maximizes the energy production on the rolling-horizon, a comparison
between the solution found with the Blackbox optimization and the global optimal solution
is performed.

<figure><img src="image_38.png" /><figcaption>Figure 6.5 Convergence of Blackbox optimization for a 5-stage scenario tree</figcaption></figure>

A 5-stage scenario tree with fixed aggregation of stages $\kappa = (3, 6, 8, 12)$ is chosen since the state space is relatively small. The problem contains only integer optimization variables, therefore it is possible to do an exhaustive enumeration of the domain. Considering constraints on minimal and maximal number of scenarios, there is a total of 331 points in the state space. Total computational time to run the evaluations cumulates to 27 hours. The best solution found is a tree with a structure $\alpha = (5, 1, 2, 3)$ and the value of the objective function is 804.63 TWh. Since the objective function is evaluated at every possible combination of $\alpha$ in the domain, this is the global optimal solution.

The Blackbox optimization solver NOMAD is used to find the scenario tree that maximizes the energy production throughout the rolling-horizon. The convergence is shown in Fig. 6.5. The optimization requires 57 evaluations, but the best solution is found after 23 evaluations. The time to conduct the optimization is 7 hours. The solution found by NOMAD is a tree with structure $\alpha = (4, 5, 1, 2)$ and the objective is 804.63 TWh, which is identical up to the second decimal to that found by the exhaustive enumeration procedure. The figures shows clearly that in less than 10 evaluations the objective function is substantially increased.

For this 5-stage scenario tree case, the computational time required by NOMAD is approxi-
mately one fourth of that of the enumeration strategy. The difference between the objective
values obtained with the optimization and enumerating all the state space is negligible. The-
refore, optimizing the tree structure with NOMAD allows to find an acceptable solution. Since
the scenario tree generation method is stochastic, the objective function value varies for the
same scenario tree structure, so it is difficult to assess if NOMAD found the global solution
or not, but there is an increase in the objective function when the structure of the tree is
optimized, as shown in Fig. 6.5.

The starting point led to an objective function of 804.46 TWh while the best solution found is 804.63 TWh. The difference between these two values is 0.17 TWh, which is much greater than the standard deviation of 7.4 GWh and confirms that the maximum found by NOMAD truly is a maximum and not induced by noise.

**6.4 Case study**

This sections presents the case study used to validate the optimization models presented in
this research.

**6.4.1 Hydropower System**

<figure><img src="image_39.png" /><figcaption>Figure 6.6 Hydroelectric system studied. Squares represent power plants and triangles reservoirs.</figcaption></figure>

The optimization method presented in this paper is tested on a portion of a hydroelectric
system that belongs to Rio Tinto and is shown on Fig. 6.6. It is located in Saguenay in the
province of Québec, Canada. The sub-system consists of 4 reservoirs and 3 power plants in
series, namely Chute-du-Diable (CD), Chute-Savane (CS), Lac-St-Jean (LSJ) and Isle-Maligne
(IM). Travel time of the water between plants is neglected. The installed capacity for these
3 power plants is about 950 MW. The power plants and reservoirs characteristics are shown in Table 6.1.

<table><caption>Table 6.1 Reservoir and power plants characteristics</caption><thead><tr><th>Reservoir and/or power plant</th><th>Nb. turbines</th><th>Reservoir capacity (hm³)</th></tr></thead><tbody><tr><td>CD</td><td>5</td><td>452</td></tr><tr><td>CS</td><td>5</td><td>119</td></tr><tr><td>LSJ</td><td>-</td><td>5596</td></tr><tr><td>IM</td><td>12</td><td>171</td></tr></tbody></table>

## 6.4.2 Structure Of The Data

Six data series are available for numerical experiments. There are 31 days in the rolling-horizon scheme and every day has a 30-day inflow forecast. Fig. 6.7 shows an example of scenario trees for the first-day of the rolling-horizon. The scenario trees, in black, are built from the inflow scenarios, which are shown in grey. Every day, inflow scenarios are received for the three reservoirs, then a scenario tree generation method, based on the minimization of the nested distance [81] is used to generate scenario trees approximating the multi-variate distribution of the inflow forecasts.

<figure><img src="image_40.png" /><figcaption>Figure 6.7 Scenario trees built from inflow forecast scenarios</figcaption></figure>

## 6.5 Computational experiments

Two types of tests for the optimization of the scenario trees are conducted on the data series available. First, the aggregation of the stages is fixed arbitrarily and only the structure of the
scenario tree is optimized, for different values of stages. Therefore, the Blackbox optimization
variables $κ$ are fixed in the mathematical model **BB**. For a 5-stage scenario tree, there are
4 Blackbox optimization variables $α$, which are the structure of stages 2, 3, 4 and 5. For a
9-stage scenario tree, there are 8 Blackbox optimization variables. Second, the structure and
the aggregation of the stages is optimized, for different values of stages. For a 9-stage scenario
tree, there are 16 Blackbox optimization variables, more precisely $α$ are the structure and $κ$
the aggregation of each stage.

The solver IPOPT [105] is used for the loading problem, XPRESS [2] for the unit commitment problem and NOMAD [63] for the Blackbox optimization problem.

### 6.5.1 Deterministic Optimization With The Median Scenario

In this section, results of the SSTUL with optimized scenario tree parameters are compared
to a deterministic optimization where only the median scenario in the forecast is used. First,
results with optimization of the structure only and a fixed aggregation and second, results
with optimization of the structure and aggregation. Optimizing the SSTUL with a median
scenario comes down to solving a deterministic model. For the producer, comparing the
solution obtained from the stochastic model gives a value on the interest of investing time to
solve a stochastic model.

Basically, the same rolling-horizon scheme for the stochastic method and the deterministic
optimization is used. For the stochastic method, a scenario trees based on the inflow forecast
is generated and the SSTUL model is solved. For the median scenario method, the median
scenario out of all of the inflow scenarios is chosen and a deterministic model is solved.

**Optimization of the structure only**

The solution obtained with the SSTUL model and optimization of the structure of the scenario
tree is compared to the solution obtained when solving the SSTUL with the median scenario.

Results are presented with a boxplot for every number of stages and all 6 test cases, as seen
on Fig. 6.8a. As an example, take the boxplot for 5-stages. The lower box represents the
first quartile and the top of the box represents the third quartile. The band in the middle of
the box represents the median. The bottom of the line linking the whisker to the box is the
minimum of the data and the top of the line linking the box to the whisker is the maximum
of the data.

For stages ranging from 5 to 9, the boxplot of the difference in energy between the stochastic
solution, for which the scenario trees structure is determined with Blackbox optimization, and
<figure><img src="image_41.png" /><figcaption>(a) Optimization of the structure of the scenario (b) Optimization of the structure and aggregation of the scenario tree</figcaption></figure>

Figure 6.8 Boxplots of the difference in energy production between stochastic and deterministic optimization

the median scenario is shown. A positive value indicates the stochastic method produces more energy. As explained previously, since the scenario tree generation method is stochastic, two standard deviations are required for the difference in energy to be valid and not blackbox noise, more precisely ±16 GWh, and are shown on Fig. 6.8a with the pair of horizontal dotted-lines.

Fig. 6.8a illustrates that, with an arbitrarily aggregation of the stages, optimizing the structure of the scenario tree leads to similar results for every given number of stages. Therefore, the number of stages does not have an important influence on the energy production in the SSTUL problem.

## Optimization of the structure and the aggregation

The solution obtained with the SSTUL model and optimization of the structure and the aggregation of the scenario tree is compared to the solution obtained when solving the SSTUL with the median scenario.

Fig. 6.8b illustrates the results for the 6 test cases. For a number of stages varying from 5 to 9, the difference in energy production between the solution obtained using scenario trees and the solution obtained using the median scenario, a boxplot is displayed.

Fig. 6.8b reveals that optimizing the scenario tree structure and aggregation increases the difference between the stochastic solution and median scenario given a higher number of stages. Therefore, the aggregation of the stages is an important parameter in the scenario
tree generation method, as the difference in energy increases when the number of stages increases. The best median is obtained with a 5-stage scenario tree, but the 9-stage scenario tree produces a bigger difference in energy production when using scenario trees instead of using the median scenario when solving the SSTUL. Therefore, it is an important parameter since it is difficult to conclude on the best number of stages to use.

### 6.5.2 Scenario Fans

Relying on the median scenario to implement a decision is highly risky. Often, in hydro-power optimization, scenario fans are used in the optimization models since they present a compromise between investing a lot of time and effort in generating the trees and solving a stochastic multi-stage model and solving a deterministic model. This section presents, for different number of stages, comparisons with different numbers of scenarios in a scenario fan fashion, as shown in Fig. 6.9. The same scenario tree generation method that was used previously is used to generate the trees, given different values of stages and number of scenarios. For the aggregation, the value obtained during the Blackbox optimization of the scenario tree structure and aggregation was used, as it is the one that maximizes energy production. For example, a structure $\alpha = (1, 25, 1, 1)$ is given as input to the STG to obtain a scenario fan of 25 scenarios and 5 stages.

<figure><img src="image_42.png" /><figcaption>Figure 6.9 Scenario fan with *x* scenarios and *D* stages</figcaption></figure>

Multiple tests were conducted on the different test cases. For 5, 6, 7, 8 and 9 stages, scenarios fans of 3, 6, 9, 15 and 25 scenarios were used as input to the SSTUL model. As explained previously, since the scenario tree generation method is stochastic, two standard deviations are required for the difference in energy to be valid and not blackbox noise, more precisely $\pm 16 \, \text{GWh}$, and are shown on Fig. 6.10 with the pair of horizontal dotted-lines. Results are presented in Fig. 6.10. The difference in energy between the energy produced with the solution obtained using scenario trees and the solution obtained using scenario fans decreases when the number of scenarios increase. Therefore, solving the SSTUL with a scenario fan containing
more than 15 scenarios results in the same objective function and does not justify the use of a complex scenario tree structure. Standard deviation for 15 scenarios is 30.6973 GWh and for 25 scenarios is 26.4338 GWh. Moreover, a Student's t-test performed on the distributions of 15 and 25 scenarios reject the null hypothesis, therefore demonstrating that the difference in energy is significant and not caused by blackbox noise.

<figure><img src="image_43.png" /><figcaption>Figure 6.10 Boxplots of the difference in energy production between stochastic optimization of the structure and the aggregation of the scenario tree compared to scenario fans</figcaption></figure>

These results demonstrate that using scenario fans instead of complex scenario tree structures leads to good results and allows to find the solution in a satisfying computational time effort. Also, the scenario fan results show that the distribution of the total volume of inflows is preserved if the scenario tree generation method does not alter the scenarios, therefore by using, in this case, more than 15 scenarios. Moreover, in a rolling-horizon, the decision that is taken at the first node every day is mostly influenced by the total volume of inflows, rather than by the structure of the scenario trees that model the distribution of the inflows.

It is important to note that when using scenario fans, only the first node solution is relevant since the rest of the stages is biased, induced by the deterministic fashion of the scenario fan. When using scenario trees, non-anticipative constraints appear at each stage, therefore all of the inflow values contained in the scenario tree are relevant and could be useful in another context.
## 6.6 Conclusion

In this paper, the authors have presented an innovative method to determine if complex scenario tree structures are required when solving the short-term unit commitment and loading problem (SSTUL) with uncertain inflows used in a real-time decision making context. Multistage stochastic programs are used to solve the SSTUL problem and the modeling of the problem allows to account for head-dependency as well as limited turbine restarts. In an operational context, a rolling-horizon scheme is used to implement the solutions. Every day, forecasts of inflows are available. From these, a scenario tree is constructed to represent the distribution of the inflows, then the optimization models are solved. The solution is implemented and the reservoir volumes are updated once the real realization of the inflows are known. Only the solution for the first node every day is retained, as new forecasts become available and that the process is repeated. The scenario tree generation method that is used requires input parameters, which are the number of stages, child node per stage as well as the aggregation of each stage.

To measure the benefits of using complex scenario tree structures, the whole rolling-horizon was modeled as a Blackbox optimization model, to find the scenario tree parameters that maximize the energy production throughout the rolling-horizon, for different values of the number of stages. Results are compared with scenario fans and they show that the decision taken at the first node every day is mostly influenced by the volume of inflows, rather than the structure of the scenario trees, which means that using scenario fans leads to good results and requires less computational time.
## 6.7 Notation

The following symbols are used in this paper :

$D$ = number of stages;

$E(h)$ = energy produced per day $h$ with startups penalized;

$h$ = index of the rolling-horizon $h = (1, 2, \dots, 31)$ (days);

$LP(STG(\alpha, \kappa, \phi))$ = loading problem function;

$q$ = index of the number of child nodes per stage $r = (1, 2, \dots, D-1)$;

$r$ = index of the aggregation per stage $r = (2, 3, \dots, D)$;

$RH(\alpha, \kappa, D)$ = blackbox function;

$STG(\alpha, \kappa, \phi)$ = scenario tree generation function;

$UC(LP(STG(\alpha, \kappa, \phi)))$ = unit commitment function.

$\alpha_q$ = number of child nodes per stage $q$;

$\kappa_r$ = aggregation of each stage $r$ (days); and

$\phi_h$ = inflow scenarios for every day $h$ in the rolling-horizon.
# CHAPITRE 7 DISCUSSION GÉNÉRALE

Ce chapitre expose les travaux, soit les trois objectifs ayant mené à cette thèse.

Le premier objectif est de modéliser le problème de chargement et de répartition optimal, aussi dit à court terme, puis de le résoudre en ne considérant aucune incertitude dans le modèle d'optimisation, plus spécifiquement en développant un modèle déterministe. Le modèle à court terme est complexe car plusieurs inter-dépendances existent entre les variables du problème. La puissance produite par une turbine est une fonction non linéaire du débit turbine et de la hauteur de chute nette. Or, la hauteur de chute nette est aussi une fonction non linéaire qui dépend, entre autres, du débit total. Chaque turbine a une efficacité différente et de plus, il est important de limiter les démarrages des turbines, car ils causent une usure prématurée de l'équipement. Ce premier modèle déterministe a mené à une modélisation du problème qui considère toutes les caractéristiques énumérées précédemment. Pour ce faire, une modélisation par combinaisons de turbines est proposée, et un algorithme de programmation dynamique permet de construire des surfaces de puissance maximale pour chaque centrale avant toute optimisation. Une méthode d'optimisation en deux phases permet de résoudre le modèle à court terme. La première phase, soit le problème de chargement, est un modèle non linéaire mixte en nombres entiers, et la deuxième phase, soit le problème de répartition, est un modèle linéaire en nombres entiers. Les résultats numériques démontrent que, comparativement à des décisions opérationnelles historiques, le modèle performe bien. De plus, une comparaison avec un modèle non linéaire mixte en nombres entiers en une phase est effectuée et le modèle en deux phases proposé permet d'obtenir des solutions réalisables en un temps de calcul très court, ce qui n'est pas le cas avec le modèle en une phase.

Le second objectif reprend la modélisation déterministe proposée, mais est étendue afin de considérer des apports incertains dans les réservoirs. Une méthode de génération d'arbres de scénarios, qui sont une représentation discrète des scénarios d'apports, est retenue afin de modéliser différents scénarios d'apports dans le modèle d'optimisation stochastique. La méthode consiste à minimiser la distance imbriquée à l'aide d'une méthode en deux étapes. La première étape génère un nouveau scénario d'apports en échantillonnant la distribution des scénarios d'apports et la deuxième étape modifie les valeurs d'apports dans les noeuds de l'arbre de scénarios en utilisant une méthode de descente de gradient stochastique. Ces étapes sont répétées jusqu'à ce que la convergence de la distance soit atteinte. L'équivalent déterministe d'un modèle stochastique, c'est-à-dire qu'un modèle déterministe est résolu à chaque noeud de l'arbre de scénarios, est utilisé pour la résolution. Un horizon roulant permet
de valider la méthode d'optimisation. Puisque les prévisions d'apports sont mises à jour quotidiennement, un arbre de scénarios est généré, le modèle d'optimisation est résolu, la solution du premier noeud seulement est implémentée, les volumes des réservoirs sont mis à jour avec la vraie réalisation des apports, puis le tout est relancé lorsque les prévisions sont mises à jour. Les résultats numériques démontrent que le modèle stochastique permet d'augmenter la production d'énergie, comparativement à l'utilisation du scénario médian des apports.

Le troisième objectif étudie la complexité nécessaire des arbres de scénarios utilisés dans le modèle d'optimisation stochastique. La méthode de génération d'arbres de scénarios nécessite des valeurs de paramètres d'entrée, plus précisément, le nombre d'étapes, le nombre de noeuds fils par étape ainsi que l'aggrégation de chaque étape. Puisqu'un horizon roulant est utilisé et que seule la solution du premier noeud est implémentée, nous proposons une méthode pour déterminer si un arbre de scénarios complexe est nécessaire ou si l'utilisation d'un peigne de scénarios est suffisant pour obtenir une bonne solution dans le modèle d'optimisation à court terme stochastique. Pour ce faire, un modèle d'optimisation boîte noire est proposé pour trouver les valeurs des paramètres d'entrée de la méthode de génération des arbres de scénarios qui permet de maximiser l'énergie produite de l'horizon roulant au complet. Les résultats numériques démontrent que, dans ce cas précis, des peignes de scénarios sont suffisants pour résoudre le modèle d'optimisation dans un temps de calcul court.
# CHAPITRE 8 CONCLUSION ET RECOMMANDATIONS

Ce chapitre conclut les travaux ayant mené à cette thèse. Les limitations des solutions proposées sont exposées, ainsi que des améliorations futures à ces travaux de recherche.

Les modèles d'optimisation et les modélisations proposées dans ces travaux se basent sur certaines hypothèses. D'abord, la modélisation déterministe qui est proposée est une méthode d'optimisation en deux phases. La première phase résout un modèle non linéaire mixte en nombres entiers. Or, nous avons démontré que la matrice des coefficients des contraintes est totalement unimodulaire, ce qui implique que la résolution de la relaxation linéaire des variables binaires permet d'obtenir une solution entière. Cette propriété fait en sorte qu'il est possible de résoudre un modèle non linéaire continu afin d'obtenir une solution entière, au lieu d'un modèle non linéaire mixte en nombres entiers. En pratique, cela signifie que les temps de calculs sont grandement réduits. Seules les contraintes hydriques sont considérées dans cette modélisation, alors qu'en pratique, il est important d'inclure les contraintes de demande en énergie, surtout dans le contexte des alumineries. Les salles de cuves nécessitent énormément d'énergie pour fonctionner et lorsqu'un problème survient et qu'une salle de cuve est brusquement arrêtée, la gestion du réseau hydroélectrique est à revoir dans l'immédiat. De plus, des contrats avec Hydro-Québec obligent Rio Tinto à fournir une demande en énergie de leur part dans un délai très court. L'inclusion de ces contraintes dans la modélisation proposée ne respecte plus l'hypothèse que la matrice des coefficients des contraintes est totalement unimodulaire et un modèle continu ne peut plus être utilisé pour résoudre la première phase de la méthode d'optimisation. Or, dans un contexte opérationnel, elles sont essentielles. La perte de l'hypothèse que la matrice des coefficients des contraintes est totalement unimodulaire implique la résolution d'un problème difficile à résoudre s'il n'est pas relaxé. La solution simple à ce problème est de résoudre un problème non linéaire mixte en nombres entiers, mais la taille du problème fait en sorte qu'il est très difficile à résoudre et que le temps de résolution est trop élevé. La modélisation proposée dans cette thèse devra être revue ou, du moins, la méthode de résolution.

Les fonctions de valeur de l'eau, qui valorisent l'eau restant à la fin de l'horizon de planification dans les réservoirs, sont très importantes en optimisation court terme puisqu'elles empêchent l'optimisation de vider le réservoir. Pour cette thèse, seule une fonction de valeur de l'eau pour le réservoir du Lac-St-Jean, qui est le plus gros en terme de volume, est utilisée. Deux constatations découlent de cette approche :

1. Les fonctions valeur de l'eau utilisées pour le réservoir du Lac-St-Jean sont obtenues par
un algorithme de programmation dynamique stochastique qui résout un modèle d'op-
timisation moyen terme. Les fonctions obtenues dépendent du volume du réservoir du
Lac-St-Jean et dans le modèle à moyen terme, sept discrétisations des apports naturels
sont effectuées. Les fonctions sont utilisées dans la thèse, mais il n'y a aucune valida-
tion des fonctions. L'impact de la sensibilité de ces fonctions, c'est-à-dire le nombre de
discrétisations des apports, les dépendances entre réservoirs et autres, sur la solution
de l'optimisation court terme n'est pas étudié.

2. Pour les réservoirs Chute-du-Diable et Chute-Savane, une contrainte de réservoir plein à la fin de l'horizon de planification est imposée pour empêcher de vider les réservoirs. En pratique, ces réservoirs nécessitent aussi des fonctions valeur de l'eau, mais pour les besoins de la thèse cette hypothèse permet d'obtenir des résultats réalistes.

Ainsi, la complexité nécessaire dans le modèle d'optimisation à moyen terme pour obtenir une bonne solution au modèle d'optimisation à court terme doit être étudiée. Par exemple, plusieurs variables hydrologiques peuvent être utilisées dans les modèles à moyen terme et la façon dont elles affectent la solution du modèle à court terme n'est pas claire.

Les prévisions d'apports qui sont utilisées dans cette thèse sont basées sur les prévisions de précipitations déterministes d'Environnement Canada. L'historique des prévisions est ensuite utilisé pour construire des scénarios de précipitations, basé sur les vraies réalisations de précipitations selon les prévisions. Un modèle hydrologique permet finalement d'obtenir des scénarios d'apports de réservoirs. Les prévisions sont très importantes afin d'obtenir de bons résultats dans les modèles d'optimisation à court terme. Seule cette méthode de génération d'apports est utilisée dans cette thèse et il serait intéressant de valider d'autres méthodes pour générer ces apports afin de quantifier leur impact sur la solution du modèle à court terme.

Dans le même ordre d'idées, une seule méthode de génération d'arbres de scénarios est utilisée dans cette thèse. Plusieurs méthodes de génération d'arbres de scénarios existent et il s'agit d'un domaine de recherche en soi. Puisque le sujet de cette thèse n'est pas d'étudier les méthodes de génération d'arbres de scénarios mais d'utiliser une méthode flexible permettant de générer des structures d'arbres différentes, nous avons utilisé une seule méthode. Il serait intéressant de comparer les résultats obtenus dans cette thèse avec la génération d'arbres par minimisation de la distance imbriquée avec, par exemple, SCENRED, une méthode de réduction pour la génération d'arbres de scénarios.

Concernant l'optimisation des paramètres des arbres de scénarios, des comparaisons avec d'autres solveurs d'optimisation boîte noire permettraient de valider la performance de NO-MAD. Dans la thèse, nous avons montré que l'utilisation de peignes de scénarios est suffisante
pour obtenir de bons résultats et que le temps de calcul est grandement diminué. Par contre, si un autre solveur permet d'obtenir des résultats dans un temps beaucoup plus court, d'autres tests pourraient être effectués. Dans la méthodologie proposée, l'algorithme d'optimisation boîte noire génère une nouvelle structure d'arbre de scénarios optimisée pour chaque série de données. Par contre, en pratique, les réalisations d'apports ne sont pas connues à l'avance et le choix de la structure d'arbres de scénarios est complexe, puisqu'elle doit être choisie avant de connaître les réalisations d'apports. Une piste de solution pourrait être de minimiser la variance de la fonction objectif de plusieurs structure d'arbres, par exemple.

Évidemment, dans le but d'avoir un modèle stochastique à court terme opérationnel, il est nécessaire d'ajouter les centrales hydroélectriques manquantes au modèle. Cela nécessite le calcul des surfaces de puissance maximale avec un algorithme de programmation dynamique, puis d'ajuster un modèle polynomial aux surfaces créées afin de les utiliser comme fonction objectif dans le modèle d'optimisation à court terme. Aussi, dans cette thèse, le temps de déplacement de l'eau entre les centrales est négligé puisqu'elles sont situées à proximité. Cependant, avec l'ajout de toutes les centrales, le temps de déplacement doit être considéré puisque certaines centrales sont éloignées.

Finalement, pour les ingénieurs de Rio Tinto qui éventuellement utiliseraient ces modèles d'optimisation, il serait intéressant de développer un outil d'analyse post-optimale pour les mises hors tension des groupes. Cet outil pourrait assister les ingénieurs lorsque des modifications au plan de production doivent être effectuées. Le but ne serait pas de développer un plan de maintenance des turbines, mais plutôt un outil d'analyse de l'impact des changements de turbines sur la production d'énergie et l'alimentation des alumineries.
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