The value of hydrological information inmultireservoir system operation

Thèse

Jasson Piña Fulano

Doctorat en Génie des EauxPhilosophiæ doctor (Ph. D.)

Québec, Canada

© Jasson Piña Fulano, 2017

The value of hydrological information inmultireservoir system operation

Thèse

Jasson Piña Fulano

Sous la direction de:

Amaury Tilmant, directeur de recherchePascal Côté, codirecteur de recherche

Résumé

La gestion optimale d’un système hydroélectrique composé de plusieurs réservoirs est un problème

multi-étapes complexe de prise de décision impliquant, entre autres, (i) un compromis entre les consé-

quences immédiates et futures d’une décision, (ii) des risques et des incertitudes importantes, et (iii)

de multiple objectifs et contraintes opérationnelles. Elle est souvent formulée comme un problème

d’optimisation, mais il n’existe pas, à ce jour, de technique de référence même si la programmation

dynamique (DP) a été souvent utilisée. La formulation stochastique de DP (SDP) permet la prise en

compte explicite de l’incertitude entourant les apports hydrologiques futurs. Différentes approches

ont été développées pour incorporer des informations hydrologiques et climatiques autres que les ap-

ports. Ces études ont révélé un potentiel d’amélioration des politiques de gestion proposées par les

formulations SDP. Cependant, ces formulations sont applicables aux systèmes de petites tailles en

raison de la célèbre « malédiction de la dimensionnalité ».

La programmation dynamique stochastique duale (SDDP) est une extension de SDP développée dans

les années 90. Elle est l’une des rares solutions algorithmiques utilisées pour déterminer les politiques

de gestion des systèmes hydroélectriques de grande taille. Dans SDDP, l’incertitude hydrologique

est capturée à l’aide d’un modèle autorégressif avec corrélation spatiale des résidus. Ce modèle

analytique permet d’obtenir certains des paramètres nécessaires à l’implémentation de la technique

d’optimisation. En pratique, les apports hydrologiques peuvent être influencés par d’autres variables

observables, telles que l’équivalent de neige en eau et / ou la température de la surface des océans. La

prise en compte de ces variables, appelées variables exogènes, permet de mieux décrire les processus

hydrologiques et donc d’améliorer les politiques de gestion des réservoirs. L’objectif principal de ce

doctorat est d’évaluer la valeur économique des politiques de gestion proposées par SDDP et ce pour

diverses informations hydro-climatiques.

En partant d’un modèle SDDP dans lequel la modélisation hydrologique est limitée aux processus

Makoviens, la première activité de recherche a consisté à augmenter l’ordre du modèle autorégressif et

à adapter la formulation SDDP. La seconde activité fut dédiée à l’incorporation de différentes variables

hydrologiques exogènes dans l’algorithme SDDP. Le système hydroélectrique de Rio Tinto (RT) situé

dans le bassin du fleuve Saguenay-Lac-Saint-Jean fut utilisé comme cas d’étude. Étant donné que ce

système n’est pas capable de produire la totalité de l’énergie demandée par les fonderies pour assurer

pleinement la production d’aluminium, le modèle SDDP a été modifié de manière à considérer les

iii

décisions de gestion des contrats avec Hydro Québec. Le résultat final est un système d’aide à la

décision pour la gestion d’un large portefeuille d’actifs physiques et financiers en utilisant diverses

informations hydro-climatiques. Les résultats globaux révèlent les gains de production d’énergie

auxquels les opérateurs peuvent s’attendre lorsque d’autres variables hydrologiques sont incluses dans

le vecteur des variables d’état de SDDP.

iv

Abstract

The optimal operation of a multireservoir hydroelectric system is a complex, multistage, stochastic

decision-making problem involving, among others, (i) a trade-off between immediate and future con-

sequences of a decision, (ii) considerable risks and uncertainties, and (iii) multiple objectives and

operational constraints. The reservoir operation problem is often formulated as an optimization prob-

lem but not a single optimization approach/algorithm exists. Dynamic programming (DP) has been

the most popular optimization technique applied to solve the optimization problem. The stochastic

formulation of DP (SDP) can be performed by explicitly considering streamflow uncertainty in the

DP recursive equation. Different approaches to incorporate more hydrologic and climatic information

have been developed and have revealed the potential to enhance SDP- derived policies. However, all

these techniques are limited to small-scale systems due to the so-called curse of dimensionality.

Stochastic Dual Dynamic Programming (SDDP), an extension of the traditional SDP developed in the

90ies, is one of the few algorithmic solutions used to determine the operating policies of large-scale

hydropower systems. In SDDP the hydrologic uncertainty is captured through a multi-site periodic au-

toregressive model. This analytical linear model is required to derive some of the parameters needed to

implement the optimization technique. In practice, reservoir inflows can be affected by other observ-

able variables, such snow water equivalent and/or sea surface temperature. These variables, called

exogenous variables, can better describe the hydrologic processes, and therefore enhance reservoir

operating policies. The main objective of this PhD is to assess the economic value of SDDP-derived

operating policies in large-scale water systems using various hydro-climatic information.

The first task focuses on the incorporation of the multi-lag autocorrelation of the hydrologic variables

in the SDDP algorithm. Afterwards, the second task is devoted to the incorporation of different exoge-

nous hydrologic variables. The hydroelectric system of Rio Tinto (RT) located in the Saguenay-Lac-

Saint-Jean River Basin is used as case study. Since, RT’s hydropower system is not able to produce

the entire amount of energy demanded at the smelters to fully assure the aluminum production, a

portfolio of energy contacts with Hydro-Québec is available. Eventually, we end up with a decision

support system for the management of a large portfolio of physical and financial assets using various

hydro-climatic information. The overall results reveal the extent of the gains in energy production that

the operators can expect as more hydrologic variables are included in the state-space vector.

v

Contents

Résumé iii

Contents vi

List of Tables vii

List of Figures viii

Acknowledgments ix

Preface x

Introduction 10.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Specific objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 Literature Review 41.1 The reservoir operation problem . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 The main solution strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Methods 122.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Optimization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Stochastic Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Hydrologic information and large scale systems . . . . . . . . . . . . . . . . . . 152.6 Stochastic Dual Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . 172.7 Climatological information as exogenous variable . . . . . . . . . . . . . . . . . 252.8 Hydropower scheduling and contract management . . . . . . . . . . . . . . . . 252.9 SDDPX formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Case study 313.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 The Gatineau River Basin and hydropower system . . . . . . . . . . . . . . . . . 313.3 The Saguenay-Lac-Saint Jean River Basin and Rio Tinto system . . . . . . . . . 33

4 Overview of Results 384.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

vi

4.2 Incorporation of multi-lag autocorrelation . . . . . . . . . . . . . . . . . . . . . 384.3 Incorporation of exogenous variables . . . . . . . . . . . . . . . . . . . . . . . 434.4 Joint optimization of physical and financial assets . . . . . . . . . . . . . . . . . 45

Conclusion 50

Future work 52

Bibliography 53

A Paper I: Horizontal Approach to asses the Impact of Climate Change on WaterResources Systems 60

B Paper II:Optimizing multireservoir system operating policies using exogenous hy-drologic variables 72

C Autoregressive modeling 88C.1 Multisite periodic autoregressive model MPAR . . . . . . . . . . . . . . . . . . 88

vii

List of Tables

3.1 Gatineau hydropower system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Rio Tinto hydro-power system characteristics . . . . . . . . . . . . . . . . . . . . . 35

4.1 Extrapolation techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 SDDP formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3 Average annual results - Differences with respect to the SDDP(1) model . . . . . . . 454.4 Average annual results - Differences with respect to the SDDP(1) model . . . . . . . 47

C.1 Autoregressive parameters φp,t and polynomial characteristic roots ui . . . . . . . . . 95

viii

List of Figures

1.1 Illustration of reservoir system optimization as sequential decision process. Modifiedfrom Labadie (2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Decision tree in reservoir operation problem (adapted from Pereira et al. (1998)) . . 51.3 Reservoir optimization classification (Ahmad et al., 2014) . . . . . . . . . . . . . . . 51.4 SDP principle when maximizing the sum of immediate and future benefits functions 61.5 Future benefit function - FBF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Dynamic programing Optimization procedure. Adapted from Labadie (2004) . . . . 91.7 SDP and SDDP principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.8 SDDP and exogenous variables principles . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Multistage decision problem scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Piecewise linear approximation of FBF function Ft+1 . . . . . . . . . . . . . . . . . 172.3 Example reservoir system configuration and connectivity matrix (adapted from Labadie

(2004)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Normalized Convex-Hull approximation at Passes-Dangereuses power station- Rio

Tinto system, Quebec, Canada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Backwards openings and the approximation of the FBF (adapted from Tilmant and

Kelman (2007)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.6 The ARX model structure (adapted from Ljung (1999)) . . . . . . . . . . . . . . . . 252.7 Schematization of energy trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.8 Schematic SDDPX toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1 Gatineau River Basin and hydropower system . . . . . . . . . . . . . . . . . . . . . 323.2 Weekly average (µ) and standard deviation (σ ) of inflow series - Gatineau River Basin 333.3 Rio Tinto hydropower system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4 Weekly average and standard deviation of inflow series - Saguenay-Lac-St-Jean River

Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.5 Schematic of minimum and maximum weekly levels at Lac-Saint Jean . . . . . . . 363.6 Weekly average and standard deviation of Snow Water Equivalent and precipitation -

Saguenay-Lac-St-Jean River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1 Draw-down refill cycle Baskatong and Cabonga Reservoirs. Lag-1 SDDP (Left panel)and multi-lag SDDP(p) (right panel) . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 (a) Statistical distribution of annual spillage losses (b) spillage deviation respect SDDP(1)(c) Statistical distribution of annual energy production (d) energy production deviationwith respect SDDP(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Gatineau and Du Lievre river basins . . . . . . . . . . . . . . . . . . . . . . . . . . 41

ix

4.4 Annual energy generation (a) cumulative distribution functions (b) relative differencesbetween the distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5 Baskatong drawdown-refill cycle - Climate change and current conditions . . . . . . 434.6 Accumulated drawdown-refill cycle Passes Dangereuses and Lac-Saint-Jean reservoirs 454.7 Statistical distribution of the annual spillage losses (left panel) and the marginal value

of water (right panel) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.8 Statistical weekly distribution of energy purchases from the portfolio of contracts . . 484.9 % of difference in the power efficiency respect SDDP(1) formulation for both config-

uration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

C.1 Mean and standard deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93C.2 (1) Partial correlogram PACF (2) Mean square error MSE (3) AIC (4) BIC . . . . . . 95C.3 Cabonga and Baskatong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96C.4 Paugan and Chelsea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

x

Acknowledgments

I would like to acknowledge various people who have been part of this adventure.

Firstly, I would like to thank my advisor, Amaury Tilmant for guiding and supporting me over the

years of my Ph.D study. I thank his time, patience and motivation in all the time of research and

writing of this thesis.

I thank also my co-supervisor, Pascal Côté for his kind support, knowledge and generosity.

I would like to thank my thesis committee members, Professors François Anctil, Fabian Bastin and

Marcelo Oliveros, for all of their valuable comments.

Special thanks to my friends in these latitudes: Nicolas and his patience with my French; Charles, his

chocolates and technical discussions; Thibaut and his patience with my Spanish. I would also like to

thank Charles-Hubert for his support and motivation. Thanks to Alex, Hector, Coraline, Bruno, Béné,

Sara, Diane, Maria Natalia, who made this experience more enjoyable.

No podría olvidar mi amada familia que desde la distancia fue un apoyo incondicional. Mil palabras

de agradecimiento por estar junto a mi en todos mis proyectos: gracias Pa, Ma, Adri y Santi. También

a mis amigos que siempre me acompañaron desde Colombia con memes, fotos, videos: Jenny, Moni,

Raquel, Sary, Yiyi, Stiwi, Sergio, Oscar, Maria Cristina...

xi

Preface

This PhD thesis presents the research carried out between January 2014 and August 2017 at the De-

partment of Civil and Water Engineering (Université Laval-UL). This research took place within the

framework of a NSERC-CRD grant with Rio Tinto (RT), under the supervision of Prof. Amaury

Tilmant (UL) and co-supervision of Dr. Pascal Côté (RT). The thesis is based on the following publi-

cations/presentations:

Paper I J. Pina, A. Tilmant, F. Anctil. Horizontal approach to assess the impact of climate change

on water resources systems. Journal of Water Resources, Planning and Management. Published

2016: Pina et al. (2016). http://ascelibrary.org/doi/abs/10.1061/(ASCE)WR.1943-5452.

0000737.

Oral presentation at AGU Fall Meeting 2015: A spatial extrapolation approach to assess the impact

of climate change on water resource systems. https://agu.confex.com/agu/fm15/webprogram/

Paper62891.html

Paper II J. Pina, A. Tilmant, P. Côté. Optimizing multireservoir system operating policies using

exogenous hydrologic variables. Journal of Water Resources Research. Published: Pina et al. (2017).

http://onlinelibrary.wiley.com/doi/10.1002/2017WR021701/abstract

Oral Presentation at EWRI-2017: Optimizing multireservoir system operating policies using exoge-

nous hydrologic variables. https://eventscribe.com/2017/ASCE-EWRI/fsPopup.asp?Mode=

presInfo&PresentationID=254515

Oral Presentation at EGU-2017: Valuing physically and financially-induced flexibility in large-scale

water resources systems. http://meetingorganizer.copernicus.org/EGU2017/EGU2017-9687.

pdf

xii

Introduction

Reservoirs are essential for domestic and industrial uses, irrigated agriculture, energy production,

etc. Even though the reservoir operation problem has been studied for decades, its solution remains

challenging. Reservoir operating policies specify the amount of water to be released during a given

stage (time period), or the target storage to be reached at the end of that stage. The complexity of the

problem lies in the interdependence that exists between the immediate and the future consequences

of a release decision. In other words, a balance must be found between outflow, i.e. depleting the

reservoir, and keeping the water in storage for future uses. Since the future inflows are uncertain, the

problem is essentially stochastic. Furthermore, when hydroelectric systems are analyzed, the problem

is nonlinear because the hydropower production function is proportional to the product between the

head (storage) and the releases through the turbines.

The examination of the scientific literature reveals that reservoir operation is often formulated as an op-

timization problem but that no single optimization approach/algorithm exists. Dynamic programming

(DP) and its extensions have been extensively used to solve the reservoir operation problem. The basic

idea behind DP is to decompose the complex problem in a collection of simpler subproblems which

are then solved recursively. The fact that DP can be expanded to account for the hydrologic stochas-

ticity is also an interesting feature when dealing with reservoir operation problems. The stochastic

extension, called Stochastic DP (SDP), performs an optimization on all discrete combinations of the

state variables (storage and hydrologic). Using interpolation techniques, these optimal solutions are

generalized to other points of the state-space domain. In an attempt to better describe the hydrologic

processes, different approaches and extensions of SDP, such as Sampling SDP (SSDP) (Kelman et al.,

1990) and Bayesian SDP (BSDP) (Karamouz and Vasiliadis, 1992) have been developed. However,

those improvements quickly hit the wall: as the traditional DP-based solution strategy relies on the

discretization of the state-space, the problem becomes quickly intractable due to the so-called curse of

dimensionality. Since the computational effort increases exponentially with the number of state vari-

ables, the researchers and practitioners were left with a inevitable trade-off between system complexity

(the number of individual reservoirs) and hydrologic complexity (the number of hydrologic processes

that can be considered). System complexity is desirable to identify synergies between power stations

and to avoid the difficulties associated with aggregation/disaggregation techniques. Hydrologic com-

plexity, on the other hand, should ultimately yield better release policies by reducing the uncertainty

regarding future inflows.

1

This thesis attempts at removing this trade-off using an alternative technique that is not affected by the

curse of dimensionality: Stochastic Dual Dynamic Programming (SDDP). In SDDP the optimal solu-

tion is extrapolated to the rest of the state-space domain from a limited number of discrete points that

are carefully sampled (whereas traditional techniques rely on a dense grid covering the domain and

the solutions are interpolated). SDDP has been used in hydropower-dominated systems such as Nor-

way (Rotting and Gjelsvik, 1992; Mo et al., 2001; Gjelsvik et al., 2010), South and Central America

(Pereira, 1989; Homen-de Mello et al., 2011; Shapiro et al., 2013), New Zealand (Kristiansen, 2004).

The SDDP algorithm also constitutes the core of generic hydro-economic models that have been used

to analyze a variety of policy issues in river basins: e.g. Euphrates-Tigris River basin (Tilmant et al.,

2008), the Nile River basin (Goor et al., 2011), the Zambezi River basin (Tilmant and Kinzelbach,

2012) or in Spain (Pereira-Cardenal et al., 2016; Macian-Sorribes et al., 2016).

In the traditional SDDP formulation, the hydrologic uncertainty is captured through a multi-site pe-

riodic autoregressive model (MPAR). This model is required to analytically derive the extrapolating

functions, and to synthetically generate the different scenarios for the simulation phase of the algo-

rithm. Recent works such us Lohmann et al. (2015), Pritchard (2015), Poorsepahy-Samian et al.

(2016), and Raso et al. (2017) reveal a particular interest in improving the built-in hydrologic model.

This research work follow this trend but focuses on the incorporation of various hydro-climatic infor-

mation into SDDP. To achieve this, additional, exogenous hydrologic state variables must be included

in the state vector, and the built-in hydrologic model must be extended to a multi-site autoregressive

model with exogenous variables (MPARX). This in turn requires that the analytical formulations of

the extrapolating functions be adjusted to accommodate the new exogenous hydrologic variables.

0.1 Objectives

The main objective is to assess the economic value of SDDP-derived operating policies in large-scale

water systems for various hydro-climatic information.

0.2 Specific objectives

1. Develop various analytical formulations for the hydrological model embedded in the SDDP

algorithm, and the respective formulation to determine the approximation of the benefit-to-go

functions.

2. Evaluate the economic performances of the multireservoir system associated with the hydrolog-

ical models developed in (1).

3. Assess the impact of alternative hydrologic information on the management of a hydropower

portfolio including physical (hydropower plants) and financial assets (contracts) .

2

To address the challenges posed by these research objectives, three main activities are developed: 1)

Incorporation of multi-lag autocorrelation 2) Incorporation of exogenous variables and 3) Joint opti-

mization of physical and financial assets.

1. Incorporation of multilag autocorrelation

During this activity, the available Markovian SDDP toolbox (Tilmant et al., 2008), in which

the hydrologic uncertainty is captured by a multi-site lag-1 autoregressive model, is modified

to take into account multiple lags. The built-in hydrologic multisite periodic autoregressive

model of order p (MPAR(p)) is capable of analyzing different hydrologic series, estimating

the parameter of the model, selecting the order p of the periodic model, and generating the

set of hydrologic information required to implement the SDDP algorithm. The mathematical

formulation to couple the higher order MPAR(p) model with the SDDP toolbox is presented

in Paper I. Therein, an assessment of climate change scenarios on a large scale hydropower

system system using the modified toolbox is presented.

2. Incorporation of exogenous variables

In this activity, a hydrologic model capable of processing various hydrologic and climatic infor-

mation is coupled with the optimization model. The built-in model now consists on a MPAR(p)

with lag-b exogenous variables MPARX(p,b). The mathematical formulation to derive the ex-

trapolating functions based on the MPARX model is presented in Paper II.

3. Joint optimization of physical and financial assets

For this activity, a portfolio of financial assets (i.e. sale/purchase energy contracts) is included

in SDDP. A new state variable, accounting for the amount of energy remaining in the contracts,

is included in the state-space vector. The mathematical formulation is presented in Paper II.

Therein, a joint optimization of physical (reservoir and plants) and financial (energy contracts)

assets is presented.

0.3 Outline

The present thesis summarizes the main findings of the two research papers. Chapter 1 presents a

literature review on the reservoir operation problem. Chapter 2 describes the solution strategy to

optimize release policies for large multireservoir systems. Chapter 3 presents different case study

where the various new formulations are applied, and in chapter 4 the main findings are summarized.

In chapter 5 can be found the conclusion remarks and in the appendix the research papers.

3

Chapter 1

Literature Review

1.1 The reservoir operation problem

The operation of a multi-reservoir system is a complex, multistage, stochastic decision-making prob-

lem involving, among others, (i) a trade-off between immediate and future consequences of a release

decision, (ii) considerable risks and uncertainties and (iii) multiple objectives and operational con-

straints (Oliveira and Loucks, 1997). The complexity of the problem lies in the interdependence that

exists between the immediate and the future consequences of a decision at a given stage. In other

words, a balance between storage and release decisions must be found at each stage (Figure 1.1).

Stage

t

Stage

T

Inflow1

Stage

1

Release1

Storage1

Benefits1

Storage2

State transition

Inflowt Releaset

Storaget

Benefitst

Storaget+1

InflowT ReleaseT

StorageT

BenefitsT

StorageT+1

Allocation decision

State Returns

Figure 1.1: Illustration of reservoir system optimization as sequential decision process. Modified fromLabadie (2004)

Since the future inflows are uncertain, the problem is essentially stochastic. At each stage of the

decision process, reservoir operators face the future hydrologic uncertainty and the decision made can

affect the availability of the resource and thus the future benefits (Figure 1.2).

This decision making problem has been studied for several decades and a state-of-art reviews can be

4

Operating consequences

Future hydro conditions

Decision

Reservoir

Use reservoir

Dry Deficit

Wet ok

Do not use reservoir

Dry ok

WetSpillage losses

Figure 1.2: Decision tree in reservoir operation problem (adapted from Pereira et al. (1998))

found in Yeh (1985), Labadie (2004), Rani and Moreira (2009) and more recently in Ahmad et al.

(2014). The examination of the scientific literature reveals that reservoir operation is often formulated

as an optimization problem but that no single optimization approach/algorithm exists.

Reservoir Operation

Linear Programming

(LP)

Network Flow programming

Interior Point Method

Non-Linear Programming

(NLP)

Sequential Linear Programming

Sequential Quadratic

Programming

Method of Multiplier

Generalized Reduced

Gradient Method

Dynamic Program (DP)

Deterministic DP

Stochastic DP

Computational Intelligence (CI)

Fuzzy Set Theory

Artificial Neural Network

Evolutionary computer

Figure 1.3: Reservoir optimization classification (Ahmad et al., 2014)

In the stochastic case, however, two techniques exist: Implicit Stochastic Optimization (ISO) and

Explicit Stochastic Optimization (ESO). ISO methods are actually deterministic methods which use

a large number of historical or synthetically generated hydrological scenarios to derive optimal op-

5

eration policies: they use optimization, regression and simulation techniques to refine promising op-

eration rules iteratively. ESO formulations, on the other hand, require the explicit representation of

probabilistic streamflows or other uncertain parameters such us energy and water demands, spot prices

for energy, etc. (Labadie, 2004). To handle the complexity of the reservoir operation problem, a tem-

poral decomposition approach is required (Zahraie and Karamouz, 2004). This hierarchical approach

relies on a chain of optimization models for long, mid and short term planning horizons. The solution

strategies presented herein focus on the mid-term hydropower scheduling, which seeks to determine

optimal weekly release policies. As we will see in the next section, even though various optimization

techniques are available to solve the mid-term reservoir operation problem (Figure 1.3), this research

relies on extension of Dynamic Programming.

Optimal decision

Future benefits function

Immediate benefitsfunction

End-of-period storage

Immediate + future benefits functions

Be

nef

its

$

Marginal water values

Figure 1.4: SDP principle when maximizing the sum of immediate and future benefits functions

1.2 The main solution strategies

Along with linear programming and nonlinear programming, dynamic programming and its exten-

sions have been extensively studied. Dynamic programming (DP), first introduced by Bellman (1957),

solves the problem by breaking the multistage problem into simpler one-stage subproblems, which are

6

then solved recursively. With this principle in mind, the objective function of the one-stage optimiza-

tion problem becomes the sum of the immediate and future benefits from system operation (Figure

1.4). As we can see, the immediate benefits decrease as the end-of-period storage increases as less

water is available for immediate uses. At the same time, future benefits increase as more water is kept

in storage. The derivatives of the immediate and future benefit functions correspond to the immediate

and future marginal water values respectively (Tilmant et al., 2008) and, at the optimal solution, both

values are identical. The marginal value of water indicate what the operator would be willing to pay

to get an additional unit of water in a particular reservoir and at a given time of the year.

To account for the hydrologic uncertainty, DP can be expanded by adding hydrological variables in the

state vector. Consequently, the release decisions are now function of the storage and the hydrologic

state variable (e.g. previous or current inflow). This stochastic DP (SDP) formulation, often referred

to as a Markov decision process, explicitly considers the streamflow lag-1 correlation found in the

flow records; the recursive equation uses the fact that inflow during any given time period is related to

the previous one by a conditional probability (Yeh, 1985).

End-of-period storage

Ben

efit

s $

Stochastic variable

Figure 1.5: Future benefit function - FBF

SDP solves the problem by replacing the continuous domain by a grid, and by solving the one-stage

DP optimization problem at each grid point. These optimal solutions are then generalized the rest of

the domain using interpolation techniques (Figure 1.5). Since this optimization is performed condi-

tionally on all discrete combinations of the state variables, this discrete approach is limited to small

scale problems with no more than four state variables. To illustrate the so-called curse of dimen-

sionality of a four-dimensional problem, let us imagine that the state variables are discretized in 10

values. Then, the one-stage optimization must be evaluated over a grid of 10 × 10 × 10 × 10 points

7

(104). Hence, the computational effort increases exponentially with the number of state variables as

10#variables.

Various strategies for dealing with the dimensionality issue associated with DP have been proposed

in the literature. For example, Turgeon and Charbonneau (1998), Saad et al. (1996), Archibald et al.

(1997) and Archibald et al. (2006) use aggregation- disaggregation techniques to reduce the scale of

the problem. Bellman and Dreyfus (1962) suggest the Dynamic Programming Successive Approxima-

tion (DPSA), which decomposes the multidimensional problem into a sequence of one-dimensional

problem by optimizing over one state variable at a time. Then releases are explicitly obtained from the

mass balance equation as a function of specified beginning and ending storage. Incremental Dynamic

Programming (IDP) and Discrete Differential Dynamic Programming (DDDP) address the dimension-

ality problem by restricting the state space to a corridor around a current given solution (Karamouz

et al., 2003). The methods are highly sensitive to initial storage trajectories and the discretization

intervals must be carefully selected to provide accurate solutions at a reasonable computational time

(Labadie, 2004). Although, these efforts reduce computation time, the curse of dimensionality is not

removed.

Along with the curse of dimensionality, the representation of streamflow persistence and hydrologic

forecasting information is an important issue when applying SDP: better streamflow foresight is ex-

pected to improve reservoir operation because it allows time for better decision making (Georgakakos,

1989). The generation of scenarios and forecast information for water resources management appli-

cations relies on the use of different stochastic hydrologic models (Pagano et al., 2004; Gelati et al.,

2010). Various approaches are available to forecast reservoir inflows, from regression relationships be-

tween inflows and climate observed data (e.g. snowpack, soil moisture, fall and winter precipitation),

to models properly initialized with climate forecast (e.g. Ensemble Streamflow Prediction (ESP) fore-

casts, forecast of ENSO, downscaled numerical climate model forecasts, etc.)(Anghileri et al., 2016;

Georgakakos, 1989). The National Weather Service’s (NWS) ESP procedure (Day, 1985) produces

streamflow forecast in the form of multiple hydrographs, and the forecast of ENSO are currently

available up to a year or more in advance (Gelati et al., 2014).

To exploit the potential value of these forecasts, different extensions of SDP have been developed. For

example, Kelman et al. (1990) proposed a sampling SDP (SSDP) that captures the complex temporal

and spatial structure of the streamflow process through the use of a large number of sample stream-

flows scenarios, instead of assuming that inflow stochasticity in SDP follows a probability density

function. Karamouz and Vasiliadis (1992) developed the Bayesian SDP (BSDP) which uses Bayesian

decision theory to incorporate new information by updating the transition probabilities; in BSDP nat-

ural and forecast uncertainties are both included in the model.

The advantage of using different hydrologic variables in SDP formulations has been presented in sev-

eral references. Bras et al. (1983) presented the introduction of real time forecast with an adaptive

control technique where flow transition probabilities and system objective are continuously updated.

8

Infl

ow

s

Time

DP

Statistical Frequency

analysis

SDP

Probability distributions

FBF

SSDP-BSDP

Statistical Frequency

analysis

Probability distributions

Historical series or

stochastic stream

generation model

Simulation model

Hydrologic Forecasting information

Hydrologic persistenceMonte

Carlo

ESOISO

Fore

cast

in

form

ati

on

Time

`

Figure 1.6: Dynamic programing Optimization procedure. Adapted from Labadie (2004)

Stedinger et al. (1984) developed a SDP model which employs the best inflow forecast of the cur-

rent period to define the policy. Georgakakos (1989) discussed the value of streamflow forecasts in

reservoir operation. Kim and Palmer (1997) compared the performance of the BSDP formulation

and three alternative SDP models, when the seasonal flow forecast and other hydrologic information

are included in the state vector. Faber and Stedinger (2001) and Kim et al. (2007) employed ESP

forecasts and snowmelt volume forecasts using SSDP formulation. Côté et al. (2011), introduced in

SSDP a new hydrological state variable given as a linear combination of snow water equivalent and

soil moisture. More recently, Desreumaux et al. (2014) presented the effect of using various hydro-

logical variables on SDP-derived policies of the Kemano hydropower system in British Columbia.

Anghileri et al. (2016) presented a forecast-based adaptive management framework for water supply

reservoirs and evaluate the contribution of long-term inflow forecasts to reservoir operations.

However, most of the studies described above are limited to small-scale problems, meaning that a

trade-off must be found between the complexity of system to be studied and the complexity of the hy-

drologic processes that can be captured. When analyzing a large-scale system, operators must decide

whether to use a simplification of the system (e.g. aggregating storage capacity), or use fewer hy-

9

drologic state variables to describe the hydrologic process, leading somehow to a loss of information,

and to a loss of spatiotemporal synergies that can be captured when analyzing the whole system. This

trade-off can largely be removed by using stochastic dual DP (SDDP).

SDDP, first introduced by Pereira and Pinto (1991), is one of the few available algorithms to opti-

mize the operating policies of large-scale hydropower systems. The solution approach is based on

the approximation of the expected future benefit function (FBF) of SDP by piecewise linear functions

(Figure 1.7). With SDDP, there is no need to evaluate the FBF over a dense grid as the function

can now be derived from extrapolation (and not interpolation). The accuracy of the approximation is

increased by adding new linear segments through a two-phase iterative algorithm. The set of linear

segments can be interpreted as Benders cuts in a stochastic multistage decomposition algorithm, and

its determination relies on the primal and dual information of the optimal solution of each subproblem.

To implement the efficient decomposition scheme, each nonlinear SDP subproblem must be formu-

lated as a convex problem, such as a linear program (LP). This constitutes the main drawback of the

technique since all the relations associated with the problem, objective function, and constraints, must

be linear.

End-of-period storage

Ben

efit

s $

Stochastic variable

End-of-period storage

Ben

efit

s $

Stochastic variable

Sampling Point 2

Sampling Point 1

Piecewise linear approximation

True function

Hyperplane 1

Hyperplane 2

Figure 1.7: SDP and SDDP principles

SDDP has largely been used in hydropower systems, such as Norway (Rotting and Gjelsvik, 1992;

Mo et al., 2001; Gjelsvik et al., 2010), South and Central America (Pereira, 1989; Homen-de Mello

et al., 2011; Shapiro et al., 2013), New Zealand (Kristiansen, 2004) and Turkey (Tilmant and Kelman,

2007). Some improvements to deal with the nonlinear water head effects have been developed by

using a convex hull approximation of the hydropower function and can be found in Goor et al. (2011)

and Cerisola et al. (2012). The SDDP algorithm constitutes the core of generic hydro-economic

models that have been used to analyze a variety of policy issues in the Euphrates-Tigris River basin

(Tilmant et al., 2008), the Nile River basin (Goor et al., 2011), the Zambezi River basin (Tilmant and

Kinzelbach, 2012) or in Spain (Pereira-Cardenal et al., 2016; Macian-Sorribes et al., 2016).

10

Infl

ow

s

Time

Hydrologic uncertainty

MPAR(p)

SDDP

FBF approximations

Hyd

ro-c

lima

tic

info

rma

tio

n

Time

Hydrologic uncertaintyMPARX(p,b)

SDDPX

Figure 1.8: SDDP and exogenous variables principles

In SDDP, the hydrologic uncertainty is captured through a multi-site periodic autoregressive model.

This model is capable of representing seasonality, serial and spatial streamflow correlations within

a river basin and among different basins. Furthermore, it is required to analytically derived the ap-

proximation of the FBF, and to synthetically generate the different scenarios for the simulation phase.

Recent works reveal a particular interest in improving the built-in hydrologic model. For example,

Lohmann et al. (2015) presented a new approach to include spatial information. Pritchard (2015)

modeled inflows as a continuous process with a discrete random innovation, Poorsepahy-Samian et al.

(2016) proposed a methodology to estimate the cuts parameters when a Box-Cox transformation is

used to normalize inflows, and more recently, Raso et al. (2017) present a streamflow model with a

multiplicative stochastic component and a non-uniform time step.

This PhD thesis fits into this trend and seeks to assess the value of different hydro-climatic information

when operating large-scale water resource systems. To achieve this, additional exogenous hydrologic

state variables are incorporated into the SDDP algorithm. Since the modeling of the hydrologic un-

certainty in SDDP is restricted to linear additive models (Infanger and Morton, 1996; De Queiroz

and Morton, 2013), the natural extension to include climatic variability into the autoregressive model

is the MPAR model with exogenous variables (MPARX) (Figure 1.8). The reader should refer to

Ljung (1999), Ltkepohl (2007) and Hannan and Deistler (2012) for detailed presentations of MPARX

models.

11

Chapter 2

Methods

2.1 Outline

This chapter explains how the exogenous hydrologic variables are incorporated into the SDDP al-

gorithm. It starts with overview of the optimization techniques available for solving the reservoir

operation problem with a particular attention given to Stochastic Dynamic Programming (SDP). Al-

ternative SDP formulations, each employing different hydrologic information, are presented, their

strengths and weaknesses discussed. Finally, the Stochastic Dual Dynamic Programming (SDDP)

algorithm is described.

2.2 Optimization problem

Recall that the reservoir operation problem is a multistage decision-making problem (Figure 2.1).

When it is formulated as an optimization problem, the goal is to determine a sequence of optimal

allocation decisions xt (e.g. reservoir release and spillage, water withdrawals, etc) that maximizes

the expected sum of benefits from system operation Z, over a planning period time T , while meeting

operational and/or institutional constraints. The mathematical formulation of the multistage decision-

making problem can be written as:

Z = max

E

[T

∑t=1

αtbt(St ,qt ,xt)+αT+1ν(ST+1)

] (2.1)

subject to:

12

St+1 = ft(St ,qt ,xt) (2.2)

gt+1(St+1)≤ 0 ∀t (2.3)

at+1(xt+1)≤ 0 ∀t (2.4)

Stage

t

Stage

T

Stage

1 State transition

Allocation decision

State Returns

𝑞𝑡 𝑥𝑡 𝑞𝑇 𝑥𝑇𝑞1 𝑥1

𝑠𝑡 𝑠𝑡+1 𝑠𝑇 𝑠𝑇+1𝑠1

𝑏1(𝑆1, 𝑞1, 𝑟1)

𝑠2

𝑏𝑡(𝑆𝑡, 𝑞𝑡, 𝑟𝑡) 𝑏𝑇(𝑆𝑇 , 𝑞𝑇 , 𝑟𝑇)

Figure 2.1: Multistage decision problem scheme

where bt(·) is the immediate benefit function, ν(·) is the terminal value function, αt is the discount

factor at stage t and E[·] is the expectation operator. In many reservoir operation problems the vector

of the state variables St includes the beginning-of-period storage st and any hydrological variable

ht . For a hydropower-dominated system of J reservoirs and D demand sites for off-stream uses (e.g.

municipal and industrial uses, irrigated agriculture), the immediate benefit function includes the net

benefits from hydropower HPt [$], the benefits from off-stream uses NBt [$], and penalties for not

meeting target water demands and/or violating operating constraints:

bt(·) = HPt +NBt −ξᵀt zt (2.5)

where zt (J× 1) is the vector of slack variables with the violations of operational constraints (e.g.

energy deficit, environmental flows, etc.) which are penalized in the objective function by the vector

ξt (J×1) of penalties [$/unit].

2.3 Linear Programming

Linear Programming (LP) is one of the most popular optimization techniques applied in water re-

sources management. Its attractiveness lies in ability to handle large scale problems, to converge to a

global solution, to allow for a sensitivity analysis from the duality characteristics of linear program-

ming, and the availability of generic softwares for solving LP problems. The main disadvantage is

13

the fact that the LP requires that all the relations associated with the problem, objective function and

constraints, to be linear or linearizable. Moreover, if the stochastic formulation of LP is rarely im-

plemented because of the computational burden, the two-stage stochastic linear programming with

recourse, first introduced independently by Dantzig (1955) and Beale (1955), can be used for solving

the problem deterministically for each of the the several scenarios of future inflows. This formulation

leads to an extremely large-scale linear programming problem, which can be reduced by utilizing Ben-

ders decomposition. The basic idea is to express the expected value of the second stage by a scalar and

to replace the second-stage constraints sequentially by cuts, which are necessary conditions expressed

only in terms of the first stage variables (Infanger, 1993). Another decomposition strategy for solving

large-scale stochastic programs is progressive hedging (PH) (Rockafellar and Wets, 1991) which is a

scenario-based decomposition technique. PH has been used as an effective heuristic technique for ob-

taining approximate solutions to multistage stochastic programs (Hart et al., 2012; Carpentier et al.,

2013)

2.4 Stochastic Dynamic Programming

Next to linear programming, Dynamic programming (DP) has been the most popular optimization

technique applied to water resources planning and management. The method was first introduced by

Bellman (1957) and solves the problem (Equations 2.1 to 2.4) by breaking the multistage problem

into simpler subproblems over each stage, which are solved recursively. DP can handle non-linear

relationships and discontinuous functions. DP performs an optimization on all discrete combinations

of the state variables. These optimal solutions are generalized for other points of the state variables

by a continuous function, using an interpolation approach (e.g. linear, cubic spline) (Johnson et al.,

1993; Tejada-Guibert et al., 1993; Kitanidis et al., 1987). Stochastic DP (SDP), often referred to a

Markov decision process, solves the problem by discretizing stochastic variables, as well as the system

status, to obtain an optimal policy for each discrete value of the reservoir system (Rani and Moreira,

2009). If the vector of state variables St includes the beginning-of-period storage st and any choice

of hydrologic state variable ht , the recursive SDP equation can be written as (Tejada-Guibert et al.,

1995):

Ft(st ,ht) = Eqt |ht

[maxxt{αtbt(·)+ E

ht+1|ht ,qt

[αt+1Ft+1(st+1,ht+1)]}] (2.6)

where E[·] is the expectation operator to observe hydrological condition ht+1 given the hydrological

state ht , and it is obtained from the conditional probabilities P(ht+1 | ht). SDP directly incorporates

both the probability distributions of random variables and the temporal persistence between successive

flows through the use of flow transition probabilities. Since the expectation operator acts on the max-

imization, the release decisions are made after the hydrologic variable ht is known (Tejada-Guibert

et al., 1995).

14

When the temporal persistence is not modeled, the hydrologic variable is not included in the SDP and

the equation 2.6 is reduced to the deterministic DP equation 2.7. The Pseudo-code description of the

DP procedure is presented in Algorithm 1.

Ft(st) = maxxt{αtbt(·)+αt+1Ft+1(st+1)} (2.7)

Initialize the FBF of the last stage;FT (sT+1) = 0;

for t = T,T −1, ...,1 dofor each storage level st = s1

t , ...,sKnt do

solve the one-stage problem Equation 2.7Ft(st) = max

xt{αtbt(·)+αt+1Ft+1(st+1)}

s.t operational constraints equations 2.2 to 2.4endCreate a complete FBF, Ft(st), for the previous stage by interpolating the values{

Ft(sknt ),kn = 1, ...,Kn

}end

Algorithme 1 : Pseudo-code description of the DP procedure

The most common choices for hydrological state variable have been the current flow qt and the pre-

vious flow qt−1. When the current flow is used the expectation Eqt |ht can be omitted and equation 2.6

becomes:

Ft(st ,qt) = maxxt{αtbt(·)+ E

qt+1|qt

[αt+1Ft+1(st+1,qt+1)]} (2.8)

Likewise, if the state hydrological variable is the previous flow equation 2.6 becomes:

Ft(st ,qt−1) = Eqt |qt−1

[maxxt{αtbt(·)+αt+1Ft+1(st+1,qt)}] (2.9)

and the Pseudo-code description of the SDP procedure is presented in algorithm 2.

2.5 Hydrologic information and large scale systems

Among several concerns, the representation of streamflow persistence and hydrologic forecasting in-

formation in the decision process is a critical issue when applying SDP (Labadie, 2004; Kelman et al.,

1990). Thus, better hydrologic information, incorporated as state variables, has the potential for en-

hancing SDP-derived policies. For example, Bras et al. (1983) presented the introduction of real time

forecast with an adaptive control technique where flow transition probabilities and system objective

are continuously updated over finite transient periods before achieving steady state conditions. Ste-

dinger et al. (1984) developed a SDP model which employs the best inflow forecast of the current

15

Initialize the FBF of the last stage;FT (sT+1,qT ) = 0;

for t = T,T −1, ...,1 dofor each storage level st = s1

t , ...,skmt , ...,sKm

t dofor each past inflow qt−1 = q1

t−1, ...,qkkt−1, ..., ,q

Kkt−1 do

for each inflow scenario of stage t conditioned to the past inflow scenarioqt = q1

t , ...,qk jt , ...,qK j

t dosolve the one-stage problem Equation 2.9 considering an initial storage skm

t and theinflow scenario qk j

t

Fk jt (skm

t ,qkkt−1) = max

xt{αtbt(·)+αt+1Ft+1(st+1,q

k jt )}

s.t operational constraints equations 2.2 to 2.4endcalculate the expected value of the benefits obtained across the conditioned inflowscenarios

Ft(skmt ,qkk

t−1) = ∑k j

(P

k j|kk

Fk jt (skm

t ,qkkt−1)

)endCreate a complete FBF, Ft(st ,qt−1), for the previous stage by interpolating the values{

Ft(skmt ,qkk

t−1),km = 1, ...,Km;kk = 1, ...,Kk

}end

endAlgorithme 2 : Pseudo-code description of the SDP procedure

period to define the policy. Kelman et al. (1990) proposed a sampling SDP (SSDP) to better capture

the complex temporal and spatial structure of the streamflow process. SSDP uses a large number

of streamflows scenarios, instead of assuming that inflow stochasticity in SDP follows a probability

density function. SSDP implementation relies on two different models: i) a decision model which

chooses an optimal release that maximizes the future and current benefit for each stage, state, and

scenario taking into account the streamflow uncertainty, and ii) a simulation model which uses the

optimal releases for each scenario to update the future value maintaining a realistic description of

streamflow series.

Karamouz and Vasiliadis (1992) developed the Bayesian SDP (BSDP) which includes inflow, storage,

and forecast as state variables. BSDP describes streamflows with a discrete lag-1 Markov process, and

uses Bayesian decision theory to incorporate new information by updating the transition probabilities.

In BSDP, natural and forecast uncertainties are both captured in the model. When the current inflow

qt and the seasonal or monthly flow forecast ft+1 are used as hydrologic state variables equation 2.6

becomes:

Ft(st ,qt , ft+1) = Eqt |qt , ft+1

[maxxt{αtbt(·)+ E

qt+1 ft+2|qt , ft+1,qt

[αt+1Ft+1(st+1,qt+1, ft+2)]}] (2.10)

Depending on the selection of the current hydrologic state variables ht and the next state hydrological

16

variables ht+1, the conditional expectations in Equation 2.10 can be determined for four different cases

(further details see Kim and Palmer (1997)). Therein, the potential advantage of using the seasonal

flow forecast and other hydrologic information is illustrated by comparing the performance of the

BSDP formulation and three alternative SDP models.

As mentioned earlier, since the optimization is performed conditionally on all discrete combinations

of the state variables, the problem gives rise to the curse of dimensionality: assuming k state variables,

discretized into N values the computational effort required to solve equation 2.6 increases exponen-

tially with the number of reservoirs J as(

NkJ)

. Then, if a large-scale system is studied using SDP,

a balance must be found between the complexity of the system (e.g number of reservoirs) and the

complexity of the hydrologic process that can be captured. Although strategies such as aggregating

the storage capacity or using fewer hydrologic variables can be implemented, they can somehow lead

to a loss of information, and to a loss of spatiotemporal synergies that can be captured when analyzing

the whole system. This trade-off between system and hydrologic complexities can largely be removed

by using stochastic dual DP (SDDP).

2.6 Stochastic Dual Dynamic Programming

Stochastic dual DP (SDDP) was first proposed by Pereira and Pinto in 1991. SDDP is not affected by

the curse of dimensionality and can therefore be used to optimize the operating policies of large-scale

hydropower systems. The solution approach is based on the approximation of the FBF functions of

SDP by piecewise linear functions (Figure 2.2). There is no need to evaluate Ft+1 over a dense grid as

the function can now be derived from extrapolation (and not interpolation).

Sampling Point 2

Sampling Point 1

Piecewise linear approximation

True function

Hyperplane 1

Hyperplane 2

𝑠𝑡+1

𝑞𝑡

𝐹𝑡+1

Figure 2.2: Piecewise linear approximation of FBF function Ft+1

17

With SDDP the multistage optimization problem (Equations 2.1 to 2.4) can be broken into a series of

one-stage linear programming (LP) problems which are solved recursively. At stage t, describing the

state of the system with the storage st and using as hydrologic state variable the previous inflow qt−1,

the recursive equation can be written as:

Ft(st ,qt−1) = max{αtbt(·)+αt+1Ft+1} (2.11)

The problem is bounded by L Bender’s cuts which are represented by the inequality constraints:

Ft+1−ϕ

lt+1

ᵀst+1 ≤ γ

lt+1

ᵀqt +β

lt+1

...

Ft+1−ϕLt+1

ᵀst+1 ≤ γ

Lt+1

ᵀqt +β

Lt+1

(2.12)

ϕt+1 (J×1), γt+1 (J×1), βt+1 are the linear parameters of the approximated FBF Ft+1. Likewise, the

stage to stage transformation function (i.e. the mass balance equation) is expressed as:

st+1−CMR(rt + lt)+ et(st ,st+1) = st +qt (2.13)

where st+1 is the vector (J× 1) of storage at the end of the period, rt is the vector (J× 1) of the

turbined flows, lt and et are the vectors (J× 1) of spillage and evaporation losses respectively, CMR

is the reservoir system connectivity matrix, CMRj,k= 1(-1) when reservoir j receives (releases) water

from (to) reservoir k. Figure 2.3 displays an example of a reservoir system configuration and the

connectivity matrix.

The linear segments Ft+1 are obtained from the dual solutions of the optimization problem at each

stage and can be interpreted as Benders cuts in a stochastic, multistage decomposition algorithm.

SDDP uses an iterative optimization/simulation strategy to increase the accuracy of the solution by

adding new cuts. To implement the decomposition scheme, the one-stage optimization problem must

be formulated as a convex problem, such as a linear program. Nonetheless, the power generation

function depends on the product of the turbined outflow and the net head on the turbine:

Pt = η(st ,st+1,rt) ·ρ ·g · rt ·ht(st ,st+1,rt) (2.14)

where Pt[W] is the power produced in the plant, η is the overall efficiency of the power plant, ρ

[kg/m3] is the density of water, g [m/s2] is the acceleration due to gravity, rt [m3/s] is the release

through the turbines and ht [m] is the net head which is a non linear function depending on the storage

levels at the begening, st and at the end st+1 of the period and the head losses.

To deal with the head effects on the hydropower production function Pt , a convex hull approximation

is stored in the constraints set (2.15). The linear parameters ψ , ω and δ are determined using the

procedure described in Goor et al. (2011).

18

1

2

3

4

Reservoir

Power Plant

𝐶𝑀𝑅 =

−1 0 0 00 −1 0 00 +1 −1 0+1 0 +1 −1

𝑞1

𝑞2

𝑞3

𝑞4

𝑟1

𝑟2

𝑟3

𝑟4

Figure 2.3: Example reservoir system configuration and connectivity matrix (adapted from Labadie(2004))

Pt −ψ

1st+1/2−ω1rt ≤ ψ

1st/2+δ1

...

Pt −ψHst+1/2−ω

Hrt ≤ ψHst/2+δ

H

(2.15)

Then, the immediate benefits of equation 2.11 are calculated as:

bt(st ,qt ,st+1,rt) =J

∑j=1

(πh( j)−θh( j))Pt( j)τt −ξ

ᵀt zt (2.16)

where τt is the number of hours in period t, π is the energy price [$/Wh] and θ is the operation and

maintenance cost [$/Wh]. As defined in Equation 2.5, zt is the vector of slack variables, penalized in

the objective function by the vector ξt of penalties [$/unit].

The decision variables such as storage st+1, releases rt and spillage losses lt are limited by lower and

upper boundaries:

st+1 ≤ st+1 ≤ st+1

rt ≤ rt ≤ rt

lt ≤ lt ≤ lt (2.17)

19

Figure 2.4: Normalized Convex-Hull approximation at Passes-Dangereuses power station- Rio Tintosystem, Quebec, Canada

2.6.1 Future Benefit Function Approximation

In SDDP, the hydrologic uncertainty is typically captured through a multi-site periodic autoregres-

sive model (MPAR). This model is capable of representing seasonality, serial and spatial stream-flow

correlations within a river basin and among different basins. It is also needed to analytically derive

the FBF approximations, and to produce synthetic streamflows scenarios for the simulation phase of

the iterative procedure. Furthermore, the convexity requirement of SDDP is guaranteed because the

MPAR is linear (Infanger and Morton, 1996; De Queiroz and Morton, 2013).

Autoregressive Model

Autoregressive (AR) models have been extensively used in hydrology and water resources since

1960’s. Its popularity and attractiveness rely on the simplest formulation and its intuitive type of time

dependence: variables at time t are dependent on the preceding ones. First introduced by Thomas

and Fiering (1962) and later by Box and Jenkins (1970), AR models can be represented as models

with constant parameters, parameters varying with time and combination of both. The models with

constant parameters are often implemented for modeling annual time series. Models with periodic

parameters are usually used with time series of intervals that are fraction of the year (e.g. seasons,

months, weeks, etc.) Salas et al (1980). The latter models are referred to as periodic AR (PAR)

20

models, and the periodicity may be in the mean, variance and/or autoregressive parameters. Since the

reservoir operation analysis involves time series at various geographic locations, spatial correlation

is also required. Then, at each site j the hydrologic process can be modeled by a multi-site periodic

autoregressive MPAR model of order p represented by:

(qt( j)−µqt ( j)

σqt ( j)

)=

p

∑i=1

φi,t( j)

(qt−i( j)−µqt−i( j)

σqt−i( j)

)+ εt( j) (2.18)

where qt is the time dependent variable for year v and time t, with t=1,2,...,52 weeks. µqt and σqt are

the periodic mean and standard deviation of qt , respectively, φi,t( j) are the autoregressive parameters

of the p order periodic model, and εt( j) is a time independent-spatially correlated stochastic noise.

Assuming that the noise εt( j) follows a 3-parameters (µv( j), σv( j) and κt( j)) log normal distribution:

fεt( j) =1

(εt( j)−κt( j))√

2πσv( j)e−0.5

(log(εt ( j)−κt ( j))−µv( j)

σv( j)

)2

(2.19)

with mean µεt , variance σ2ε,t

µεt ( j) = κt( j)+ e

(µv( j)+ σ2

v ( j)2

)

σ2ε,t( j) = e2(µv( j)+σ2

v ( j))+ e(2µv( j)+σ2v ( j)) (2.20)

the lower bound κt( j) which ensure non-negative inflows qt( j) > 0, is defined from equation 2.18 is

defined as:

εt( j)>−µqt ( j)σqt ( j)

−p

∑i=1

φi,t( j)

(qt−i( j)−µqt−i( j)

σqt−i( j)

)= κt( j) (2.21)

and the parameters µv( j) and σv( j) determined as:

µv( j) = 0.5Log σ2ε,t

Λ( j)(Λ( j)−1) (2.22)

σv( j) =√

log(Λ( j)) (2.23)

Λt( j) = 1+ σ2ε,t

κ2t ( j) (2.24)

the standardized stochastic noise Vt is estimated as:

Vt( j) =log(εt( j)−κt( j))−µv( j)

σv( j)(2.25)

21

The spatial statistical dependence of reservoir inflows is introduced by the lag0-covariances and cross

variances of the standardized stochastic noise. Then for the collection of independent standardized

noises Vt of each node j of the system Vt = [Vt(1), . . .Vt(J)], the spatial model can be written as:

Vt = AtWt (2.26)

where Wt is a column vector of J independent elements consisting of white noises, normally distributed

with zero mean and variance equal to 1. The estimation of matrix A can be obtained from the Cholesky

factorization of the covariance matrix of standardized noise at each node j:

At Aᵀt =Cov(Vt) (2.27)

SDDP and MPAR(1)

To derive the mathematical formulation of the FBF approximation let us assume an autoregressive

model of order p = 1, then equation 2.18 can be written as:

qt( j)−µqt ( j)σqt ( j)

= φt( j)

(qt−1( j)−µqt−1( j)

σqt−1( j)

)+ εt( j) (2.28)

SDDP uses a two phases strategy to increase the accuracy of the solution by adding new cuts: a

backward optimization and a forward simulation. Both phases require different sets of inflows. In the

backward phase, K inflows scenarios (backward openings) at each node of the system are generated

by the using the MPAR model. These scenarios are needed to analytically calculate the hyperplanes’

parameters, and ultimately to derive the upper bound to the true expected FBF. In the forward phase,

the MPAR model generates M synthetic reservoir inflows sequences to simulate the system behavior

over the planning period.

The calculation of the linear parameters ϕ lt , γ l

t and β lt of the approximated Ft+1 (equation 2.12) relies

on the primal and dual information available at the optimal solution. Let us say that at stage t, s◦t and

q◦t−1 are sampled and, in order to include the stochasticity of the problem, the K vectors of inflows

qKt are generated. The one-stage SDDP subproblem 2.11 to 2.17 is solved for K reservoir inflow

branches qkt . The expected FBF Ft+1, stored in the form of cuts, is the expected value of the K FBF

Fkt+1 calculated for each inflow branch (Figure 2.5) (Tilmant and Kelman, 2007).

According to the Kuhn-Tucker conditions for optimality the derivative of the objective function with

respect to the state variables S is given by:

∂F∂Si

= ∑λi∂gi

∂Si(2.29)

22

being λi the dual information of the optimization problem and gi the linear constraints 2.13 to 2.15.

Then, the slopes ϕl,kt and γ

l,kt of the functions Fk

t

Fkt ≤∑

Jϕ

l,kt ( j)s◦t ( j)+∑

Jγ

l,kt ( j)q◦t−1( j)+β

l,kt (2.30)

which will be added to the expected cost-to-go function at stage (t−1), can be calculated as:

∂Fkt

∂ st( j) = ϕ

l,kt ( j) = λ

kw,t( j)+

H

∑h=1

λh,khp,t( j)ψh

t+1( j)/2 (2.31)

∂FKt

∂qt−1( j) = γ

l,kt ( j) =

∂FKt

∂qt

∂qt

∂qt−1

=

(λ

l,kw,t ( j)+

L

∑l=1

λl,kc,t ( j)γ l

t+1( j)

)∂qt

∂qt−1(2.32)

The partial derivative of the current inflow respect to the previous inflow, from equation 2.28 is ex-

pressed as:

∂qt

∂qt−1= φt( j)

(σt( j)

σt−1( j)

)(2.33)

and λ kw,t (J×1), λ

h,khp,t (J×H×1) and λ

l,kc,t [L×1] are the dual information associated to water balance

(equation 2.13), the L cuts of the FBF (equation 2.12) and the H linear segments of the power functions

(equation 2.15), respectively.

Taking the expectation over the K artificially generated flows, the slope vectors ϕ lt and γ l

t,1 can be

determined:

ϕlt ( j) =

1K

K

∑k=1

ϕl,kt ( j) (2.34)

γlt ( j) =

1K

K

∑k=1

γl,kt ( j) (2.35)

Finally, the constant term is given by:

βlt =

1K

K

∑k=1

Fkt −∑

Jϕ

lt ( j)s◦t ( j)−∑

Jγ

lt,1( j)q◦t−1( j) (2.36)

As mention earlier, the backward optimization generates an outer approximation of the FBF Ft+1

(Figure 2.2). The accuracy of the approximation is evaluated at the end of the forward simulation. This

23

Sampled storage(period t, iteration 1)

Storage(period t+1, branch 1)

st + 1

Cut 1,1

Storage(period t+1, branch k)

Cut 1,k

Storage(period t+1, branch K)

Cut 1,K

Inflowbranch k

Ft + 1

AggregatedCut 1

Ft + 1

Ft + 1

Ft + 1

Sampled storage(period t, iteration 2)

Storage(period t+1, branch 1)

Cut 2,1

Storage(period t+1, branch k)

Cut 2,k

Storage(period t+1, branch K)

Cut 2,K

Inflowbranch k

Ft + 1

AggregatedCut 2

Ft + 1

Ft + 1

Ft + 1

Ft + 1

ApproximatedCut

Inflowbranch 1

Inflowbranch 1

Inflowbranch K

Inflowbranch K

st + 1

st + 1

st + 1

st + 1

st + 1

st + 1

st + 1

Figure 2.5: Backwards openings and the approximation of the FBF (adapted from Tilmant and Kelman(2007))

phase yields all the successive states and decision for each of M historical or synthetically generated

hydrologic sequences. Then the expected lower bound on the optimal solution is defined as:

Z =1M

T

∑t=1

bmt (st ,qm

t ,st+1,rt) =Zm

M(2.37)

where bmt is the immediate benefit at stage t for the hydrologic sequence m ∈ [1,2, . . . ,M]. This

forward simulation phase provides a lower bound with a 95% confidence intervals which allow us to

determine whether the upper bound is a good a approximation or not. If the upper bound does not fall

inside the confidence interval of the lower bound, the approximation is statistically not acceptable and

a new backward recursion is implemented with a new set of hyperplanes build on the storage volumes

that were visited during the last simulation phase.

24

The 95% confidence interval around the estimated lower bound Z is calculated as:[µZ−1.96

σZ√M,µZ +1.96

σZ√M

](2.38)

2.7 Climatological information as exogenous variable

Incorporating exogenous hydrologic variables into the state-space vector of SDDP offers the po-

tential to improve the performance of SDDP-derived release policies. The natural extension to in-

clude climatic variability into the autoregressive model is the MPAR model with exogenous variables

(MPARX). Using p previous inflows qt and b past exogenous variables Xt , the incremental flow at

node j, qt( j), can be derived from a multisite periodic autoregressive model with exogenous variables

MPARX(p,b):

qt( j)−µqt ( j)σqt ( j)

=p

∑i=1

φi,t( j)

(qt−i( j)−µqt−i( j)

σqt−i( j)

)+

b

∑κ=ι

ϑκ,t( j)

(Xt−κ( j)−µXt−κ

( j)σXt−κ

( j)

)+ εt( j)

(2.39)

where µXt and σXt are respectively the vectors of the periodic mean and the standard deviation of the

exogenous variables, and ϑκ,t the vector of the exogenous regressors. As indicated in Equation (2.39)

the exogenous variables may cover a different range of past input values, from ι to b, not necessarily

starting from t−1. This is significant in time-delay systems where the effect of an input may become

active after a certain time period (Marmarelis and Mitsis, 2014).

MPARX(p,b)

Exogenousvariables

Endogenousvariables.

𝑞𝑡−𝑝 𝑞𝑡

𝑋𝑡−𝜅

Figure 2.6: The ARX model structure (adapted from Ljung (1999))

2.8 Hydropower scheduling and contract management

Since the deregulation of the electricity market, in both developed and developing countries during

the eighties and nineties (Boubakri and Cosset, 1998), a variety of tools and methods have been

developed to jointly analyze operation scheduling and contract management (Mo et al., 2001; Gjelsvik

et al., 2010; Mo and Gjelsvik, 2002; Kristiansen, 2004; Flach et al., 2010). The typical hydro-based

producers face different types of risk such as price risk and quantity risk caused by both inflow and

25

demand uncertainty. In order to reduce the risk exposure, producers usually trade a variety of contracts

dealing with physical (power plants) and financial assets (energy contracts). As examples of contracts,

we can find future and forward contracts, option contracts, and load factor contract. The latter, usually

named as flexible contract is a physical or financial contract between two parts where price, energy and

maximum power (load factor) is predetermined, but its use is flexible, meaning that buyer determines

the amount of energy to be bought (Mo et al., 2001).

The mathematical description of a flexible contract follows the stage to stage energy balance, ac-

counting for the maximum possible withdrawal of energy ut = Pcτt [MWh], and the amount of energy

remaining of the contract wt [MWh]:

wt+1 = wt −ut (2.40)

with the initial energy amount of the flexible contract w0 [GWh]

wt = w0 (2.41)

and

Pwτt ≤ ut ≤ Pwτt (2.42)

where Pw [MW] is the instant power that can be withdrawn and τ the number of hours in period t.

Moreover, the demanded load Dt [MWh] must be met with the energy produced by the system Ptτ

and the sale/purchases through the contracts:

Ptτt +up,t −us,t = Dt (2.43)

These energy purchases up,t[MWh], and energy sales us,t[MWh] can be introduced in the immediate

benefit function (Equation 2.16) through the vector of slack variables zt , and consequently the prices

would be included in the vector ξt .

2.9 SDDPX formulation

A variant of the SDDP algorithm capable of incorporating various hydrologic information in the

decision-making process is developed. This new formulation, called SDDPX, incorporates exoge-

nous variables Xt , such as snow water equivalent and/or sea surface temperature in the state space

vector together with the previous inflows qt . Using p previous inflows qt and b past exogenous vari-

ables Xt , the vector of hydrologic state variables ht becomes [qt−1, qt−2,..., qt−p,Xt−κ ,..., Xt−b]. This

hydrologic information is encapsulated in SDDPX through a built-in MPARX model.

26

𝑠𝑡

𝑞𝑡 𝑟𝑡

𝑠𝑡+1Staget

𝑞𝑡+1 𝑟𝑡+1

𝑠𝑡+2Stage t+1

𝑠𝑇

𝑞𝑇 𝑟𝑇

𝑠𝑇+1StageT

𝑠1

𝑞1 𝑟1

𝑃1𝜏1

𝑠2Stage1

𝑤0 𝑤𝑡+1 𝑤𝑡+2 𝑤𝑇𝑤1 𝑤2

𝐷𝑡 𝐷𝑡+1 𝐷𝑇𝐷1

𝑢𝑝,1

𝑢𝑠,1

𝑃𝑡𝜏𝑡

𝑢𝑝,𝑡

𝑢𝑠,𝑡

𝑃𝑇𝜏𝑇

𝑢𝑝,𝑇

𝑢𝑠,𝑇

𝑃𝑡+1𝜏𝑡+1

𝑢𝑝,𝑡+1

𝑢𝑠,𝑡+1

𝑤𝑇+1

Figure 2.7: Schematization of energy trade

If at stage t, the system status is described by the storage st , the hydrological variable, and the amount

left of energy wt in C energy contracts, the one-stage SDDPX optimization problem can be written as:

Ft(st ,qt−1, ...,qt−p,Xt−κ , ...,Xt−b,wt) = max{bt(st ,qt ,rt ,st+1,ut)+Ft+1} (2.44)

subject to: Ft+1−ϕ

lt+1

ᵀst+1−χ

lt+1

ᵀwt+1 ≤ Γ

lt+1

ᵀht+1 +β

lt+1

...

Ft+1−ϕLt+1

ᵀst+1−χ

Lt+1

ᵀwt+1 ≤ Γ

Lt+1

ᵀht+1 +β

Lt+1

(2.45)

Γt+1ᵀht+1 = γt+1,1

ᵀqt + γt+1,2ᵀqt−1+, ...,γt+1,p

ᵀq(t−p)+1

γt+1,p+κᵀX(t−κ)+1+, ...,+γt+1,p+b

ᵀX(t−b)+1

(2.46)

the mass balance equation (2.13), the approximation of the hydropower functions 2.15, the trade

energy balance (2.43), the energy balance in the purchase contracts (2.40), and the lower and upper

boundaries (2.17 and 2.42).

In the backward phase the main modification to the traditional SDDP formulation lies in the calcula-

tion of the hyperplanes’ parameters ϕt+1, χt+1, βt+1, γt+1,1, γt+1,2,..., γt+1,p, γt+1,p+κ ,..., and γt+1,p+b

(see Equations 2.45 and 2.46). Using the Kuhn-Tucker conditions for optimality, the change of the

one-stage objective function Ft respect to the state variables st , wt , qt−1, qt−2,...,qt−p,Xt−κ ,..., and

27

Xt−b, can be determined by:

∂Ft

∂ st= λw,t +

H

∑h=1

λhhp,tψ

ht+1/2 (2.47)

∂Ft

∂wt= λe,t (2.48)

∂Ft

∂qt−1= λw,t

∂qt

∂qt−1+

L

∑l=1

λlc,t

(γ

lt+1,1

∂qt

∂qt−1+ γ

lt+1,2

)(2.49)

∂Ft

∂qt−2= λw,t

∂qt

∂qt−2+

L

∑l=1

λlc,t

(γ

lt+1,1

∂qt

∂qt−2+ γ

lt+1,3

)(2.50)

...∂Ft

∂qt−p= λw,t

∂qt

∂qt−p+

L

∑l=1

λlc,tγ

lt+1,1

∂qt

∂qt−p(2.51)

∂Ft

∂Xt−κ

= λw,t∂qt

∂Xt−κ

+L

∑l=1

λlc,t

(γ

lt+1,1

∂qt

∂Xt−κ

+ γlt+1,p+κ+1

)(2.52)

...∂Ft

∂qt−b= λw,t

∂qt

∂qt−b+

L

∑l=1

λlc,tγ

lt+1,1

∂qt

∂Xt−b(2.53)

where λw,t , λ hhp,t , λe,t and λ l

c,t are respectively the vectors with the dual information associated to the

mass balance (2.13), the H linear segments of the power functions (2.15), the energy balance (2.40)

and the L cuts of the benefit-to-go function (2.45).

Now, at stage t, s◦t , w◦t , q◦t−1, q◦t−2,..., q◦t−p, X◦t−κ ,..., and X◦t−b are sampled and K vectors of inflows qKt

are generated using the MPARX(p,b) (Equation 2.39). Since Fkt , which will be added to the expected

FBF at stage (t−1), can be approximated by:

Fkt ≤ ϕ

l,kt s◦t +χ

l,kt w◦t + γ

l,kt,1q◦t−1 + ...+ γ

l,kt,pq◦t−p + γ

l,kt,p+κX◦t−κ , ...,γ

l,kt,p+bX◦t−b +β

l,kt (2.54)

The slopes ϕl,kt and χ

l,kt are directly determined for each hydrologic scenario k using equations (2.47)

and (2.48):

ϕl,kt ( j) = λ

kw,t( j)+

H

∑h=1

λh,khp,t( j)ψh

t+1( j)/2 (2.55)

χl,kt (c) = λ

ke,t(c) (2.56)

and by using Equation 2.39 to find the partial derivative of qt respect to the hydrologic state variables,

the slopes γl,kt,1 , ...,γ

l,kt,p,γ

l,kt,p+κ , ..., and γ

l,kt,p+b can be determined using Equations (2.49) to (2.53) as:

28

γl,kt,1( j) = ϒt( j)

σqt ( j)σqt−1( j)

φt,1( j)+L

∑l=1

λl,kc,t ( j)γ l

t+1,2( j)

γl,kt,2( j) = ϒt( j)

σqt ( j)σqt−2( j)

φt,2( j)+L

∑l=1

λl,kc,t+1( j)γ l

t+1,3( j)

...

γl,kt,p( j) = ϒt( j)

σqt ( j)σqt−p( j)

φt,p( j)

...

γl,kt,p+κ( j) = ϒt( j)

σXt ( j)σXt−κ

( j)ϑt,κ( j)+

L

∑l=1

λl,kc,t+1( j)γ l

t+1,p+κ+1( j)

γl,kt,p+b( j) = ϒt( j)

σXt ( j)σXt−p( j)

ϑt,b( j) (2.57)

where ϒ( j) as:

ϒt( j) = λl,kw,t ( j)+

L

∑l=1

λl,kc,t ( j)γ l

t+1,1( j) (2.58)

Taking the expectation over the K artificially generated flows, the vector of slopes ϕ lt , χ l

t,1, γ lt,1,

γ lt,2,...,γ l

t,p, γ lt,p+κ ,...,γ l

t,p+b can be determined:

ϕlt ( j) =

1K

K

∑k=1

ϕl,kt ( j) (2.59)

χlt (c) =

1K

K

∑k=1

χl,kt (c) (2.60)

γlt,arx( j) =

1K

K

∑k=1

γl,kt,arx( j), (2.61)

∀arx = 1,2, ..., p, p+κ, ..., p+b

Finally, the constant term is given by:

βlt =

1K

K

∑k=1

Fkt −∑

Jϕ

lt ( j)s◦t ( j)−∑

Cχ

lt (c)w

◦t (c) . . .

−∑J

γlt,1( j)q◦t−1( j)−∑

Jγ

lt,2( j)q◦t−2( j)−∑

Jγ

lt,p( j)q◦t−p( j) . . .

−∑J

γlt,p+κ( j)X◦t−κ( j)−·· ·−∑

Jγ

lt,p+b( j)X◦t−b( j) (2.62)

As it was explain earlier, the accuracy of the FBF approximations is evaluated in a forward simulation

phase. Two different options exist to generate the M sequences required to simulate the system: i)

29

one can use the MPARX(p,b) to generate synthetic streamflow sequences based on historical records

of both endogenous and exogenous hydrologic variables, ii) or one can rather rely on series generated

outside of SDDPX using any relevant hydrologic model.

2.9.1 Toolbox description

The SDDPX model is coded in MATLAB®and the optimization scheme uses the linear Solver Gurobi®.

The toolbox processes two types of input data under Excel type format: i) the system configuration

file containing different characteristics of the physical river network, reservoir and demand data, and

some economic aspects (i.e. prices, contracts, energy demands), and ii) the hydrologic information

data file containing incremental flow data in m3/s for a set period of time: the incremental flow is

the natural runoff added to the network at each node. An additional Excel file is required for each

exogenous variable included in the analysis. These files must preserved the same structure of the

incremental flow data.

SDDPX Toolbox

Hydrologic Function MPARX model

Backward Optimization• FBF approximation

Forward Simulation• FBF Lower bound

Model Parametrization

Backwards openings (generated)

Forward sequences (historical endogenous and exogenous variables)

Re-optmization

Convergence criteria

Yes

no

Optimization modelSDDPX

Inputs

•Reservoir configuration•Topology•Physical and financial constraints

• Inflows• Exogenous variables

System configuration

Hydrologic information

Figure 2.8: Schematic SDDPX toolbox

The built-in MPARX model processes the hydrologic and climatologic information, determines the

parameters of the model and generates the backward openings required to implement the backward

optimization phase. Additionally, the Matlab function stores the historical inflows and the exogenous

variables, in such a manner to be used in the forward simulation and in the re-optimization phase. The

optimization module executes a main loop containing the backward and simulation phases. Once the

convergence criteria is met, a re-optimization procedure, in which the SDDPX-derived FBF approxi-

mations are used to determine the releases decision for the entire inflow series, is executed

30

Chapter 3

Case study

3.1 Outline

Two different hydropower systems have been used to test the different SDDP formulations: The

Gatineau River Basin and the Saguenay-Lac-Saint-Jean River Basin. This chapter describes the main

characteristics of the hydropower systems, the hydrologic regime, and the different management ob-

jectives.

3.2 The Gatineau River Basin and hydropower system

The Gatineau River, located in southwestern Quebec, rises in lakes north of the Baskatong Reservoir

and flows south to join the Ottawa River. The main river channel is about 400 km and drains an area of

23,700 km2. The watershed is used mainly for hydropower production and the cascade of power sta-

tions is operated by Hydro-Québec. Hydro-Québec is the main power generator in the province with

62 hydroelectric generating stations located across the province (installed capacity of 34,490 MW).

Hydro-Québec Production generates power for the Quebec market and sells its surpluses on whole-

sale markets primarily in Ontario and in the US. It is also active in arbitraging and purchase/resale

transactions (Retrieved from http://www.hydroquebec.com/generation/).

The Gatineau hydro-power system consists of a cascade of four power stations and two large reservoirs

(Figure 3.1). The upstream reservoir, Cabonga, has a storage capacity of 1.6 km3 and drains an area

of 2,201 km2. The second reservoir, Baskatong, was formed following the construction of the Mercier

Dam; it drains an area of 12,540 km2, and has a power plant with an installed capacity of 50 MW.

The Paugan station is a run-of-river power plant (R-O-R) with total capacity of 219 MW and a small

reservoir of 30 km2. The last R-O-R power stations are Chelsea and Rapides Farmer, with a total

installed capacity of 149 and 95 MW, respectively.

The watershed is characterized by a continental climate: warm and humid during the summer, and

31

Cabonga

Baskatong

Paugan

Chelsea

Inflow

Inflow

Inflow

Inflow

R. Farmers

Pembroke

Rapides Farmers Chelsea

Paugan

Baskatong

Cabonga

Joliette

Laval

MontrealOttawa

Cornwall

Longue

Trois-R

Reservoir

Power Plant

N

EW

S

77° W 76° W 75° W 74° W 73° W

48° N

47° N

46° N

45° N

Figure 3.1: Gatineau River Basin and hydropower system

generally wet, cold and snow covered in the winter. The nival regime can be described by (i) a

very high discharge during spring-summer (the snowmelt season; weeks 12-24), (ii) large variability

during spring and autumn and (iii) low flows during winter. Figure 3.2 displays the weekly averages

and weekly standard deviations of the inflows at four different nodes of the multireservoir system.

Table 3.1: Gatineau hydropower system

Storage Installed CapacityNode Name Hm3 MW

1 Cabonga 1633 -2 Baskatong 3175 503 Paugan 93 2194 Chelsea ROR 1485 Rapides Farmers ROR 95

a ROR: Run of the river power plant

A secondary objective of the operation of the multireservoir system is to control flooding at Wakefield.

32

4 8 12 16 20 24 28 32 36 40 44 48 520

10

20

30

40

50

60Node 1: Cabonga

[Hm

3]

4 8 12 16 20 24 28 32 36 40 44 48 520

50

100

150

200Node 3: Paugan

[Hm

3]

weeks

4 8 12 16 20 24 28 32 36 40 44 48 520

100

200

300

400

500Node 2: Baskatong

[Hm

3]

4 8 12 16 20 24 28 32 36 40 44 48 520

5

10

15

20

25

30Node 4: Chelsea

[Hm

3]

weeks

µq

t

σq

t

µq

t

σq

t

µq

t

σq

t

µq

t

σq

t

Figure 3.2: Weekly average (µ) and standard deviation (σ ) of inflow series - Gatineau River Basin

The town of Wakefield is located 35 km downstream of the Paugan plant and 20 km upstream of the

Chelsea plant. Wakefield is at the upstream end of the Chelsea reservoir. A rapid exists just upstream

of Wakefield, followed by a flatter river section. To prevent flooding at Wakefield, the Chelsea dam

pond is lowered by as much as 1 m at certain times. However, the lowering of the Chelsea pond

might not prevent minor floods at Wakefield, then the peaking capability at the Paugan plant is limited

(Abdelnour and Limited, 2001).

3.3 The Saguenay-Lac-Saint Jean River Basin and Rio Tinto system

Rio Tinto (RT), a major Aluminum producer in Canada, owns and operates a 3150MW-hydro-power

system in Quebec. The hydroelectric system, located in the Saguenay-Lac-Saint Jean River Basin,

includes 4 reservoirs and 5 hydropower plants: three on the Péribonca River and two on the Sague-

nay River. The drainage area is about 78,000 km2. In the northern part of the basin, there are two

large reservoirs: Manouane and Passes-Dangereuses with a storage capacity of 2.7 and 5.2 km3, re-

spectively. The downstream reservoir system, which includes the Lac-Saint-Jean Reservoir, drains

the Basins of the Péribonka, Ashuapmushuan, Mistassini and Mistassibi rivers. Figure 3.3 provides a

33

schematic representation of the RT hydroelectric network and Table 3.2 lists the main characteristics

of the system.

RLM

RPDCCP

CCS

SH

CIM SaguenayAlmaRLSJ

Riviere du Loup

Mis

tass

ibi

Riv

er

CCD

St Lawrence RivLEGEND

Power stationRiver basin

N

74⁰W 73⁰W

52⁰N

51⁰N

72⁰W 71⁰W 70⁰W

50⁰N

49⁰N

48⁰N

R. Lac-Manuane

R. Passes-Dangereuses

CCP

CCD

CCS

SH

CIM

Inflow

Inflow

Inflow

Inflow

Inflow

R. Chute-du-Diable

R. Lac-Saint-Jean

Figure 3.3: Rio Tinto hydropower system

Figure 3.4 displays the weekly averages and the standard deviations of inflows at 4 nodes of the

multireservoir system. We can see that the hydrologic regime is characterized by a high discharge

during spring-summer (weeks 14 to 24) and very low flows during winter (weeks 48 to 13); high

variability during spring influenced by the snowmelt season, and moderate variability during summer-

autumn influenced by the precipitation on the river basin. The total annual inflow at Lac-Manouane,

Passes-Dangereuses, Chute-du-Diable and Lac-Saint-Jean reservoirs are respectively 3.6; 7.3; 5.7 and

27.1 Km3. The total annual inflow at Chute-à-la-Savane is 0.924 Km3.

The water stored in Lac-Saint-Jean Reservoir includes direct rainfall onto the water body, natural

runoff, and water coming in from the Péribonka River, which is regulated by the Passes-Dangereuses

and Chute-du-Diable reservoirs. Lac-Saint Jean Reservoir is regulated by the Isle-Maligne (CIM)

34

Table 3.2: Rio Tinto hydro-power system characteristics

Storage Installed CapacityNode Id Name Hm3 MW

1 RLM Reservoir Lac-Manouane 2657 -2 RPD Reservoir Passes-Dangereuses 5228 7503 RCD Reservoir Chute-du-Diable 396 2054 CCS Chute-à-la-Savane ROR 2105 RLSJ Reservoir Lac-St-Jean 5083 4026 SH Shipshaw ROR 1587

a ROR: Run of the river power plant

4 8 12 16 20 24 28 32 36 40 44 48 520

50

100

150

200

250node:RLM

weeks

[Hm

3]

µqt

σqt

4 8 12 16 20 24 28 32 36 40 44 48 520

100

200

300

400

500

600node:RPD

weeks

[Hm

3]

µqt

σqt

4 8 12 16 20 24 28 32 36 40 44 48 520

100

200

300

400

500node:RCD

weeks

[Hm

3]

µqt

σqt

4 8 12 16 20 24 28 32 36 40 44 48 520

500

1000

1500

2000node:RLSJ

weeks

[Hm

3]

µqt

σqt

Figure 3.4: Weekly average and standard deviation of inflow series - Saguenay-Lac-St-Jean RiverBasin

35

power plant and the spillways on the Grande Décharge and Petite Décharge rivers, before flowing

towards the Saguenay River. During the spring flood period (i.e. April 1 to June 30), the volume of

water entering the Reservoir is over five times its storage capacity: spillways on the Petite Décharge

and the Grande Décharge tend to be open. During the summer, fluctuations in the reservoir levels are

influenced by precipitation, natural inputs, regulated inputs from the Péribonka River and hydroelec-

tric production at CIM (Retrieved from http://energie.riotinto.com, July 18 2017).

3.3.1 Recreational and fishing level constraints at LSJR

Reservoir Lac-St-jean has important recreational and sport-fishing industries which highly constrain

the storage levels during summer and autumn seasons. An agreement between the Government of

Québec and Rio Tinto, signed in 1986, established a maximum storage elevation of 101.84 m a.s.l.

However, since 1991 RT has voluntary established a maximum elevation of 101.54 m a.s.l. Fur-

thermore, it has been set that from July 24 to September 1, the minimum elevations is 100.78 m

a.s.l., and the maximum elevation must no exceed 101.39 m a.s.l. (Retrieved from http://energie.

riotinto.com, July 18 2017). Figure 3.5 displays a schematic diagram of the maximum and mini-

mum elevations at Lac-Saint-Jean Reservoir agreed between RT and the Government of Québec.

4 8 12 16 20 24 28 32 36 40 44 48 52Weeks

Ele

vation (

m a

.s.l.)

Min Level

Max Level

Historical Level

101.54

100.78

101.39

Figure 3.5: Schematic of minimum and maximum weekly levels at Lac-Saint Jean

3.3.2 Portfolio of energy contracts

RT hydro-power system has the capacity to produce more than 17 TWh/year, which constitutes

roughly 90% of the electricity required for the production of aluminum, forcing RT to buy energy

to fully ensure the production of the mineral. RT has two (2) energy contracts with Hydro-Quebec

limited in instant power and in total energy: P-Annual contract, available the entire year, and an 11-

week P-Pre-Freshet contract available after winter. Likewise, as part of the agreement, hydro-Quebec

purchases the RT power surpluses.

36

3.3.3 Climatological variables

Climate data, including precipitation, maximum and minimum temperature as well as snow water

equivalent (SWE), is collected by RTA’s private network of 22 weather stations and 11 SWE transect

field measurement sites. Figure 3.6 displays the weekly average SWE and weekly precipitation for

each node of the system.

4 8 12 16 20 24 28 32 36 40 44 48 520

150

300node:RLM

7SWE

<SWE

4 8 12 16 20 24 28 32 36 40 44 48 520

20

40node:RLM

4 8 12 16 20 24 28 32 36 40 44 48 520

150

300node:RPD

4 8 12 16 20 24 28 32 36 40 44 48 520

20

40node:RPD

4 8 12 16 20 24 28 32 36 40 44 48 52

Sno

w W

ater

Equ

ival

ent [

mm

]

0

150

300node:RCD

4 8 12 16 20 24 28 32 36 40 44 48 52Pre

cipi

tatio

n [m

m]

0

20

40node:RCD

4 8 12 16 20 24 28 32 36 40 44 48 520

75

150node:CS

4 8 12 16 20 24 28 32 36 40 44 48 520

20

40node:CS

weeks4 8 12 16 20 24 28 32 36 40 44 48 52

0

150

300node:RLSJ

weeks4 8 12 16 20 24 28 32 36 40 44 48 52

0

20

40node:RLSJ

7p

<p

Figure 3.6: Weekly average and standard deviation of Snow Water Equivalent and precipitation -Saguenay-Lac-St-Jean River Basin

37

Chapter 4

Overview of Results

4.1 Outline

As outlined in the introduction, this research focuses on the analysis of multireservoir system when

different hydrologic and climatic information are incorporated into the SDDP hydro-economic model.

Following the methodological activities presented therein, this chapter summarizes the main findings

of this research. More details are provided in the scientific Papers I and II.

4.2 Incorporation of multi-lag autocorrelation

In the first activity, an existing SDDP model with a built-in lag-1 hydrologic model, is modified to

take into account multiple lags through a MPAR(p) model. The hydrologic model developed during

this activity is capable of analyzing different spatially distributed hydrologic series, estimating the

parameters of the model, selecting the order p of the periodic model, and generating the two sets

of hydrologic information required to implement the SDDP algorithm. Annexe C presents the main

characteristics of the MPAR(p) model with an illustrative example of parameters estimation and order

selection.

The mathematical formulation to couple the higher order MPAR(p) hydrologic model and SDDP can

be found in Paper I. In that paper, we present an assessment of climate change impact on the operating

policies of the hydroelectric system in the Gatineau River Basin.

4.2.1 Results

Before exploring Paper I, this subsection presents the analysis of including the multi-lag autocorre-

lation in the decision making problem. A comparison between the lag-1 (SDDP(1)) and multi-lag

(SDDP(p)) is performed. Both SDDP models are implemented under the same conditions: K=25

backwards openings and M= 25 simulation sequences for a planing period of T = 260 weeks (5

years). The analysis is performed on the simulation results achieved after a re-optimization procedure

38

Figure 4.1: Draw-down refill cycle Baskatong and Cabonga Reservoirs. Lag-1 SDDP (Left panel) andmulti-lag SDDP(p) (right panel)

(Tejada-Guibert et al., 1993). The SDP-derived future benefit function approximations (Ft+1) from

the third year (t = 105,...,157), are used to determine the release decisions for the entire inflow series.

Figure 4.1 displays the draw-down refill cycle of Cabonga and Baskatong Reservoirs for the lag-1 and

the multi-lag SDDP models. Generally speaking, the annual cycles follow the same behavior: a draw-

down phase during the first weeks of the year (winter season) and a refill phase during the snow-melt

season (weeks 12-24, see hydrologic regime) . However, the storage levels reached using the multi-lag

model are much more variable than those achieved with the lag-1 SDDP. The information that can be

captured with the extended formulation better anticipates the incoming inflows and therefore yields

policies that better exploit the storage capacity of the reservoirs.

Figure 4.2 displays the statistical distribution of the annual spillage losses (panel (a)) and energy

production (panel (c)) for both formulations, SDDP(1) and SDDP(p). As we can see, regardless of the

non-exceedance probability the spillage losses are greater with the SDDP(1) model; panel (b) shows

the horizontal difference for the same probabilities. On average, the reduction spillage losses is around

824 Hm3/yr. Similarly, in panel (c) and (d), we can observe that 95% of the time the annual energy

production increases with the SDDP(p) model : an average annual increase of 78 GWh is identified

for the Gatineau hydro-power system.

39

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Spillage losses (Km3)

No

n-E

xce

ed

an

ce

pro

ba

bili

ty [-]

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2

Deviation (Km3)

(b)

2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Energy (TWh)

(c)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.2 0 0.2Deviation (TWh)

(d)

SDDP(1)

SDDP(p)

SDDP(1)

SDDP(p)

Figure 4.2: (a) Statistical distribution of annual spillage losses (b) spillage deviation respect SDDP(1)(c) Statistical distribution of annual energy production (d) energy production deviation with respectSDDP(1)

4.2.2 Operating policies facing climate change

In Paper I we develop and test a horizontal approach to assess climate change impact on water re-

sources systems, where hydro-climatic projections are not rapidly available. The typical approach

to assess climate change is based on a vertical integration/coupling of models. General Circulation

Models (GCM) provide future climatic projections, which are downscaled and used as inputs to hy-

drologic models. Afterwards, water systems models process these hydrologic projections to adapt the

operation policies to climate change. From the decision-making point of view this is challenging due

to the intensive computational and resource requirements, which are often increased when analyzing

more sources of uncertainty. Since hydro-climatic projections are becoming more often available for

different water systems, the purpose of the study is to analyze whether the computationally intensive

vertical approach is necessary in the first place, or the analysis could rely on projections available

in neighboring systems. To do so, we present a comparative assessment of the performance of the

Gatineau hydroelectric system under future climate conditions using the vertical approach and the

horizontal approach in which available projections are extrapolated from a neighboring system to the

system of interest.

Historical and projected climate information is provided by Hydro-Quebec, the Centre d’Expertise

Hydrique du Québec (CEHQ) and Ouranos. Climatic projections are obtained from the third gen-

eration coupled global climate model (CGCM3) of the Canadian Centre for Climate Modeling and

Analysis. Future projections are dynamically downscaled using the Canadian regional climate model

40

78° W 77° W

48° N

47° N

46° N

45° N

76° W 75° W 74° W

Cabonga

Baskatong

Paugan

Rapides Farmers ChelseaReservoir

Power Plant

Gatineau Basin

Du Lièvre Basin

G A T I N E A U

UD L I È V R E

N

EW

S

Ottawa

Cornwall

Pembroke

M

L

Figure 4.3: Gatineau and Du Lievre river basins

(CRCM version 4.2.3). Even though the uncertainty regarding the intensity of future consumption of

fossil fuels is carried out for various emission scenarios, our methodology is illustrated for scenario

A2 only.

Hydrologic projections are generated by CEHQ using the hydrologic model Hydrotel. To account for

the structural (hydrologic) uncertainty, inherent to the Gatineau River Basin, a multimodel approach

based on 20 different lumped hydrologic models is also used. Eventually a total of 21 time series

are made available for the stochastic multireservoir optimization model SDDP(p). Implementing

SDDP(p) on each time series yields an empirical distribution that reflects the impact of the structural

uncertainty of the hydrological processes on the energy generated by the system. This statistical

distribution constitutes the benchmark to assess the quality of the extrapolation techniques.

To implement the horizontal approach, the Du-Lievre River Basin which flows south and parallel to

the Gatineau River is selected as source river basin (Figure 4.3). The reference and future inflows

from DuLievre River Basin, as generated by the CEHQ using the Hydrotel model, are used to test four

different extrapolation techniques to derive the river discharges in the Gatineau River Basin. Table 4.1

summarizes the assumptions of the four extrapolation techniques tested. Further details in Paper I.

41

Table 4.1: Extrapolation techniques

Extrapolation technique AssumptionDrainage area ratio Both basins are hydrologically similar.Change in the mean Changes in the average weekly discharges in the

basin are proportional to those observed in theneighboring system.

Reference weekly regression Spatial correlation between river inflows remainsunchanged in the future. The current linear relation-ships will not be affected by climate change.

Autoregressive model The changes in mean and standard deviation areidentical to those observed in the source basin. Spa-tial correlation of the noise is preserved.

2 2.5 3 3.5 4Energy (TWh)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Non

-exc

eeda

nce

prob

abili

ty (

-)

Empirical CDF

Structural uncertaintyBenchmarkArea RatioChange in meanLinear RegressionAutoregressive Model

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-20 0 20Relative difference (%)

Change in mean

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-20 0 20

Area Ratio

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-20 0 20

Linear Regression

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-20 0 20

AutoregressiveModel

(a) (b)

Figure 4.4: Annual energy generation (a) cumulative distribution functions (b) relative differencesbetween the distribution functions

The SDDP(p) model of the Gatineau System is run 21 times, each time exploiting a different time

series of river discharges obtained from the 21 hydrologic models. The process was then repeated for

each extrapolation technique. As part of the results, Figure 4.4(a) presents the statistical distribution

of the annual energy generation using both the vertical and the horizontal approaches. The structural

uncertainty is represented in gray and its expected value (black CDF) is the benchmark against which

the extrapolation techniques are compared. This is the mean of the 21 different hydrologic models

that are part of the vertical approach directly implemented on the Gatineau River Basin.

Figure 4.4(b) displays the relative horizontal differences between the CDFs corresponding to the dif-

ferent extrapolation techniques and the benchmark CDF. These subplots show to what extent a par-

42

ticular extrapolation technique performs across contrasting hydrologic conditions. We can see that

the extrapolation based on the weekly linear regressions performs better than the others because the

relative differences are smaller over a large spectrum of probabilities. In other words, regardless of

the hydrologic conditions, the energy outputs obtained after extrapolating the flows with linear regres-

sions almost match those of the benchmark (with the exception of the driest year - low non-exceedance

probabilities).

The analysis reveals that the statistical distribution of energy generation can already be well ap-

proached using the extrapolation technique based on linear regressions. Further discussion on the

weekly basis can be found in Paper I. Generally speaking, annual and weekly productions are within

their respective structural uncertainty ranges, indicating that the extrapolation provides a reasonable

approximation of the amount of energy that can be expected.

For illustrative purposes, Figure 4.5 shows the expected draw-down refill cycle at Baskatong Reservoir

for the current hydrologic conditions and the hydrologic projections derived from the linear regres-

sion approach. We can identify that for the selected emission scenario, the refill phase starts earlier

following the earlier spring freshet. Since for this emission scenario, weekly discharges during the

snowmelt season are larger than under current conditions, the reservoir must be depleted earlier and a

further to free enough storage capacity, avoiding spillage losses during the snowmelt period.

0 4 8 12 16 20 24 28 32 36 40 44 48 52

500

1000

1500

2000

2500

3000

Sto

rage [hm

3]

Current Conditions

4 8 12 16 20 24 28 32 36 40 44 48 52

500

1000

1500

2000

2500

3000

Sto

rage [hm

3]

Expected storage value

Weeks

Current Condition

Climate change Horizontal app

Min Storage

Max Storge

0 4 8 12 16 20 24 28 32 36 40 44 48 52

500

1000

1500

2000

2500

3000

Sto

rage [hm

3]

Climate change - Horizontal approach

Figure 4.5: Baskatong drawdown-refill cycle - Climate change and current conditions

4.3 Incorporation of exogenous variables

Paper II presents SDDPX, an extension of SDDP capable of analyzing large scale water resources

allocation problems taking into account various hydro-climatic information. In practice, reservoir

inflows can be affected by other observable variables, which can help better capture the hydrologic

processes (e.g. snow water equivalent and/or sea surface temperature). Such variables, called exoge-

nous variables, are incorporated into the SDDP algorithm through a multi-site periodic autoregressive

model with exogenous variables (MPARX). With this extension the allocation policies are function of

the storage, the p previous inflows and relevant external variables. Paper II presents the new mathe-

matical formulation to derive the cuts approximating the future benefit functions based on the MPARX

43

model. The Rio Tinto (RT) system is used as case study to illustrate the potential gain associated with

the use of exogenous variables when operating a multireservoir system.

Four different formulation are tested (Table 4.2). The lag-1 SDDP(1), the multi-lag SDDP(p) and

two formulations based on the MPARX(p,b), differing in the selected exogenous variables. the snow

water equivalent SWEt and the combination of the SWEt and the accumulated winter precipitation Pwt

(i.e. accumulated precipitation from October until time t (Kim and Palmer, 1997)). A re-optimization

procedure is implemented in which SDDPX-derived future benefit functions are employed within a

simulation carried out over the historical record of both the endogenous and exogenous hydrologic

variables.

Table 4.2: SDDP formulations

MPARX(p,b)Formulation p b XSDDP(1) 1 - -SDDP(p) p - -SDDPX(p,SWE) p 1 SWESDDPX(p,SWE,Pw) p 1 SWE,Pw

4.3.1 Results

For this activity, the configuration is a simplified, hypothetical, system with a single objective: the

maximization of hydropower generation. Figure 4.6 illustrates the expected value of the aggregated

storage of the two largest reservoirs in the system: Passes-Dangereuses and Lac-Saint-Jean Reservoirs.

We can observe that the policies achieved using SDDPX show a more aggressive depletion of the

reservoirs during the draw-down phase (lower levels), indicating that the incorporation of exogenous

hydrologic variables can better anticipate the snow-melt runoff during spring. These lower levels

impact the levels beyond the summer therefore reducing the unproductive spills. In contrast, the

SDDP(1) and SDDP(p) tend to hedge more against hydrologic uncertainty, reaching higher levels

during the depletion phase in order to avoid potential water shortage in the future.

Table 4.3 summarizes a comparison of the different formulations with respect to SDDP(1). It shows

the mean spillage reduction, the mean release increase and the annual gain of energy. Basically, we can

see that increasing the number of lags from 1 to p already increases the amount of energy produced by

54 GWh. Incorporating exogenous variables further increases the production of energy by 46 GWh

and 51 GWh depending on whether SWE or the combination of SWE and winter precipitation are

used. Assuming an average market price of 45 US$/MWh (NYISO, 2017), these gains correspond to

a 2.45 to 4.70 million US$ increase in annual energy value. These energy gains are made possible by

improved operating policies that better exploit the storage capacity of the system.

Figure 4.7 presents the statistical distribution of the annual spillage losses (left panel) and the marginal

value of water (right panel) at each power plant for the SDDP(1) and SDDPX(p,SWE,Pw) formula-

tions. Generally speaking, results reveal that, regardless of the hydrologic condition, the spills are

44

4 8 12 16 20 24 28 32 36 40 44 48 522000

3000

4000

5000

6000

7000

8000

9000

10000

11000

Weeks

Sto

rag

e H

m3

SDDP(1)

SDDP(p)

SDDPX(p,SWE)

SDDPX(p,SWE,Pw)

Figure 4.6: Accumulated drawdown-refill cycle Passes Dangereuses and Lac-Saint-Jean reservoirs

Table 4.3: Average annual results - Differences with respect to the SDDP(1) model

Spillage Release Annual gainreduction increase of energy

Model (m3/s) (m3/s) (GWh)SDDP(p) 15 16 54SDDPX(p,SWE) 35 36 101SDDPX(p,SWE,Pw) 33 34 105

reduced when SDDPX is used to determine the policies; the horizontal gap between the lines repre-

sents the reduction of spillage at each power plant with respect to SDDP(1) formulation. Likewise,

the right panel shows the impact associated with the incorporation of more hydrological information

on the marginal value of water (i.e. Lagrange multiplier associated with the mass balance equation).

It reveals that at any given plant and regardless of the hydrologic condition the operator would be

willing to pay more for the same unit of water because the unproductive spills are reduced.

4.4 Joint optimization of physical and financial assets

In Paper II, a joint optimization of the physical infrastructure (i.e. Reservoirs and power plants) and

financial assets (i.e. energy contracts) is presented. As mentioned earlier, Rio Tinto’s hydropower

system is capable to produce 90% of the energy required to satisfy the production of aluminum,

45

35 40 45 50 550

0.5

1

Marginal water values $/1000 m3

RPD

18 20 22 24 260

0.5

1

Marginal water values $/1000 m3

RCD

14 16 18 200

0.5

1

Marginal water values $/1000 m3

Non-E

xceeda

nce pr

obabili

ty [-]

CS

10 11 12 13 140

0.5

1

Marginal water values $/1000 m3

RLSJ

7.5 8 8.5 9 9.50

0.5

1

Marginal water values $/1000 m3

SH

0 10 20 30 400

0.5

1

Spillage losses m3/s

RPD

0 20 40 600

0.5

1

Spillage losses m3/s

RCD

0 20 40 60 800

0.5

1

Spillage losses m3/sNon-E

xceeda

nce pr

obabili

ty [-]

CS

0 200 400 6000

0.5

1

Spillage losses m3/s

RLSJ

0 100 200 3000

0.5

1

Spillage losses m3/s

SH

SDDP(1) SDDPX(p,SWE,Pw)

Figure 4.7: Statistical distribution of the annual spillage losses (left panel) and the marginal value ofwater (right panel)

46

Table 4.4: Average annual results - Differences with respect to the SDDP(1) model

Annual gain Net Purchasesof energy reduction

Model (GWh) (%)SDDP(p) 30 4.0SDDPX(p,SWE) 50 6.6SDDPX(p,SWE,Pw) 51 6.7

forcing RT to buy energy to fully ensure the production of the mineral. RT has with Hydro-Québec

two (2) energy purchases contracts limited to a maximum annual amount of energy and maximum

instant power: P-Annual, available the entire year and P-Pre-Freshet only available 11 weeks during

the pre-freshet period. Besides, hydro-Québec has a power agreement with RT for the purchase of the

RT power surpluses. A fourth dummy and expensive contract (price 20 times higher than the regular

purchase price) is included in the configuration of the system to make sure that the load is met under

the worst conditions.

4.4.1 Results

This joint optimization also considers the secondary operating objectives such as recreation and flood

control at Lac-Saint-Jean. Results reveal that the expected gain in energy production using SDDP(p)

and SDDPX formulations is reduced. It means that when a more restrictive configuration is analyzed,

the operating policies tend to hedge more against the hydrologic risk, therefore accepting energy

deficits in the current period to reduce the probability of greater energy shortage in the future. This re-

duction in power output is the opportunity cost of hedging, and it is compensated by energy purchases.

Compared to SDDP(1) the net energy purchases are reduced by 6.6% and 6.7% when the exogenous

variables are included (Table 4.4). We can also see that the energy gains/purchases reduction seem

to follow the principle of diminishing marginal returns: the incorporation of the second exogenous

variable produce a marginal improvement.

Figure 4.8 displays the statistical distribution of the purchases made from the portfolio of contracts

using two different formulations: the simplest SDDP(1) and the most complex SDDPX formulation.

The use of contract "P-Pre-Freshet", which is available 20% of the year, is comparable for both for-

mulations. However, the SDDP(1) operating policy uses less energy from this ephemeral contract and

instead purchases more energy from the P-Annual and/or P-Extras contracts. The statistical distribu-

tion for the P-Extras contract reveals that regardless of the non-exceedance probability the amount

of energy bought is higher when the SDDP(1) formulation is used. These differences are marginal

if they are compared with those found in the P-annual contract statistical distribution. While with

the SDDP(1) formulation the maximum instant power is equaled 38% of the time (non-exceedance

probability > 0.62), with the SDDPX-derived policies this amount is equaled only 22% of the time.

Let us now analyze the annual power efficiency of the system (i.e. ratio between the power generated

47

0 25 50 75 1000

0.2

0.4

0.6

0.8

1

No

n-E

xce

ed

an

ce

pro

ba

bili

ty [

-]P-Pre-Freshet

0 25 50 75 1000

0.2

0.4

0.6

0.8

1

Weekly energy purchase [-]

P-Annual

0 25 50 75 1000

0.2

0.4

0.6

0.8

1P-Extras

SDDP(1)

SDDPX(p,SWE,Pw)

SDDP(1)

SDDPX(p,SWE,Pw)

SDDP(1)

SDDPX(p,SWE,Pw)

Figure 4.8: Statistical weekly distribution of energy purchases from the portfolio of contracts

-4 -2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

% Difference in Power Efficiency

No

n-E

xce

ed

an

ce

pro

ba

bili

ty [

-]

SDDP(1)

SDDPX(p,SWE,Pw) 1st config

SDDPX(p,SWE,Pw) 2nd config

Figure 4.9: % of difference in the power efficiency respect SDDP(1) formulation for both configura-tion

48

by the system and the total outflow). In general terms, the re-optimization results reveal a marginal im-

provement in the average power efficiency of the system when more hydrologic variables are included

in the state-vector: going from 1,399.7 to 1,401.4 W/m3/s when increasing the number of lags (from

1 to p), up to 1,403.3 and 1,403.4 W/m3/s when respectively (SWE) and (SWE,Pw) are incorporated

as exogenous variables. In order to compare the results obtained from the two different configuration

(i.e. maximization of hydropower generation and joint optimization with secondary operating ob-

jectives), Figure 4.9 displays the statistical distribution of the annual differences in power efficiency

between SDDPX(p,SWE,Pw) and SDDP(1) formulation. Two main findings can be observed. First,

for both configurations, the efficiency of the system is enhanced more than 65 % of the time (i.e. non-

exceedance probability greater than 0.35) when the exogenous hydrologic variables are incorporated

in the analysis. Secondly, the variability of the efficiency gains/losses is less pronounced for the sec-

ond configuration. In that case, both the upside and the downside of the hydrological risk associated

with imperfect forecasts are partly suppressed. As energy shortages are increasingly costly, i.e. com-

pensated by purchases through increasingly more expensive contracts, release policies for the second

configuration act as a hedging mechanism thereby yielding more conservative release decisions over

the entire spectrum of the hydrologic uncertainty.

49

Conclusion

As the availability of various hydro-climatic information keeps increasing due to new technologies,

such as high resolution satellite imagery, there is a need to incorporate such information into the

decision making processes that we use to manage our water resources. Up until recently, the main

bottleneck was to be found in the optimization algorithms which were unable to handle either a large

quantity of hydro-climatic information or large water resource systems. This trade-off is now largely

removed by new algorithmic strategies which seek to identify approximate solutions to the complex

decision-making problem. One of such algorithms is Stochastic Dual Dynamic Programming (SDDP).

This PhD thesis fits within this context and aims at assessing the economic value of various hydro-

logic and climatic information when managing a hydropower system consisting of physical (reser-

voirs and power stations) and financial assets (contracts). To achieve this, the methodology relies on

a gradual improvement of an available SDDP model that uses the past inflows as the only hydrologic

state variable. The extended formulation jointly optimizes the production of hydroelectricity and the

sale/purchase of energy through multiple contracts taking into account various hydro-climatic data.

The original SDDP model was first upgraded to consider the multi-lag autocorrelation in the stream-

flow series. The built-in MPAR(1) was substituted by a MPAR(p) model and the SDDP formulation

was adjusted accordingly. The resulting SDDP(p) was then tested on a hydroelectric system in the

Gatineau River Basin in Quebec. The simulation results showed that the SDDP(p) -derived policies

perform better than SDDP(1), especially during the snowmelt season. It turns out that the inclusion of

multiple lags improves the forecast of the incoming flows, therefore avoiding unproductive spills at the

end of the refill phase. This SDDP(p) formulation was also used to determine the optimal operating

strategies of the Gatineau hydropower system under climate change.

The second extension, denoted SDDPX, incorporates various hydro-climatic data such as snowpack,

precipitation or sea surface temperature, through the use of exogenous hydrologic state variables.

Here, at each time step, a built-in MPAX(p,b) model processes both past inflows (lag p) and past

exogenous data (lag b). The parameters of the MPARX model are then used to approximate the

objective function to be maximized by the SDDPX algorithm. The implementation of SDDPX is

illustrated with the hydro-power system of Rio Tinto (RT) located in the Saguenay-Lac-Saint-Jean

River Basin. A hypothetical system configuration, which aimed maximizing hydropower generation

only, revealed the extent of the energy gain that can be expected when more hydrologic information

50

is incorporated into SDDP. Since the hydrologic processes are better captured and the uncertainty

reduced, SDDPX-derived policies are even more wiling to take risk than SDDP(p) policies. The

joint optimization of physical and financial assets showed that all four SDDP-derived polices hedge

more against potential costly shortages, inducing a reduction in power output. Nevertheless, SDDPX-

derived policies still revealed better performances in terms of net purchases reductions, meaning that

the opportunity cost of hedging is lower since the hydrologic risks is reduced when more hydrologic

information is included in the analysis. However, the gains associated with the incorporation of this

information seems to follow the principle of diminishing marginal returns. Therefore, an input variable

selection assessment would be necessary to determine the number of variables to be incorporated.

Although the optimization model was tested for two hydropower systems characterized by a snow-

driven hydrology, SDDPX is capable of analyzing large-scale water systems for different flow regimes,

and consequently process relevant exogenous variables. The current SDDPX forward phase is carried

out based on historical records of both, exogenous and endogenous hydrologic variables, which might

become a limitation if limited information is available.

51

Future work

The optimization model SDDPX developed in this research should be used as departing point for

future research. The future research may be focused on, but not limited, the following specific recom-

mendations.

• Evaluate the economic value of using the SDDPX-derived FBF approximations as terminal

value function in the short-term optimization. Author suggests the incorporation of ensemble

streamflow predictions ESP to determine the SDDPX terminal value functions, and assess their

potential to enhance the short-term operating policies.

• Test the SDDPX model in water systems with a different hydrologic regime, other than the

snow-melt regime analyzed in this project. Since macro-climatic phenomenon signals such El

Niño Southern Oscillation (ENSO) and Pacific Decadal Oscillation PDO are now well recog-

nized for forecasting streamflow in some regions, author suggests the implementation of SD-

DPX model in large-scale system where these climatic signals have demonstrated the potential

of enhancing the streamflow foresight.

• Implement SDDPX on water systems with other water use sectors, such irrigated agriculture,

municipal water supply, navigation, environmental flows, etc, and assess the economic impact

of the climatic variables on the water allocation in these different sectors.

• Incorporate the stochasticity of the exogenous variables in the SDDPX-FBF approximations,

using MIMO-type hydrologic models (i.e. Multiple Input Multiple Output). While the back-

ward optimization is solved for k synthetic openings, the variability on the exogenous variables

was omitted, as they are not required in the mass balance.

• Further analysis in the sampling strategies of the hydrologic state are required. Preliminary

analysis have shown an important variability in the performance of the system.

52

Bibliography

Abdelnour, R., and F. T. Limited (2001), Ice Booms in Rivers ; Lessons Learned and the Development

of Reliable Solutions Ice Booms Main Characteristics The Wakefield Ice Boom, 11th Work. River

Ice, pp. 1–10.

Ahmad, A., A. El-Shafie, S. F. Mohd Razali, and Z. S. Mohamad (2014), Reservoir optimization in

water resources: A review, doi:10.1007/s11269-014-0700-5.

Anghileri, D., N. Voisin, A. Castelletti, F. Pianosi, B. Nijssen, and D. P. Lettenmaier (2016), Value

of long-term streamflow forecasts to reservoir operations for water supply in snow-dominated river

catchments, Water Resour. Res., 52(6), 4209–4225, doi:10.1002/2015WR017864.

Archibald, T., K. McKinnon, and I. Thomas (1997), An aggregate stochastic dynamic programming

model of multireservoir systems, Water Resour. Res., 33(2), 333–340.

Archibald, T., K. McKinnon, and I. Thomas (2006), Modeling the operation of multireservoir systems

using decomposition and stochastic dynamic programming, Nav. Res. Logist., 53(3), 217–225, doi:

10.1002/nav.

Beale, E. (1955), On Minimizing a convex function subject to Linear Inequalities, J. R. Stat. Soc. Ser.

B, 17, 173–184.

Bellman, R. (1957), Dynamic Programming, 1 ed., Princeton University Press, Princeton, NJ, USA.

Bellman, R., and S. Dreyfus (1962), Applied dynamic programming, Tech. rep., United States Air

force project RAND, Princeton University Press.

Boubakri, N., and J.-C. Cosset (1998), The financial and operating performance of newly privatized

firms: Evidence from developing countries, J. Finance, 53(3), 1081–1110, doi:10.1111/0022-1082.

00044.

Box, G.E.P. and Jenkins, G.M. (1970). Time Series Analysis Forecasting and Control. Holden Day,

San Francisco.

Bras, R. L., R. Buchanan, and K. C. Curry (1983), Real time adaptive closed loop control of

reservoirs with the High Aswan Dam as a case study, Water Resour. Res., 19(1), 33–52, doi:

10.1029/WR019i001p00033.

53

Carpentier, P. L., M. Gendreau, and F. Bastin (2013), Long-term management of a hydroelectric mul-

tireservoir system under uncertainty using the progressive hedging algorithm, Water Resour. Res.,

49(5), 2812–2827, doi:10.1002/wrcr.20254.

Cerisola, S., J. M. Latorre, and A. Ramos (2012), Stochastic dual dynamic programming applied to

nonconvex hydrothermal models, Eur. J. Oper. Res., 218(3), 687–697, doi:10.1016/j.ejor.2011.11.

040.

Côté, P., D. Haguma, R. Leconte, and S. Krau (2011), Stochastic optimisation of Hydro Quebec

hydropower installations: a statistical comparison between SDP and SSDP methods, Can. J. Civ.

Eng., 1434(April), 1427–1434, doi:10.1139/l11-101.

Dantzig, G. (1955), Linear programming under Uncertainty, Manag. Sci. 1, pp. 197–206.

Day, G. N. (1985), Extended Streamflow Forecasting Using NWSRFS, J. Water Resour. Plan. Manag.,

111(2), 157–170, doi:10.1061/(ASCE)0733-9496(1985)111:2(157).

De Queiroz, A. R., and D. P. Morton (2013), Sharing cuts under aggregated forecasts when decom-

posing multi-stage stochastic programs, Oper. Res. Lett., 41(3), 311–316, doi:10.1016/j.orl.2013.

03.003.

Desreumaux, Q., P. Côté, and R. Leconte (2014), Role of hydrologic information in stochastic dy-

namic programming: a case study of the Kemano hydropower system in British Columbia, Can. J.

Civ. Eng., 41(9), 839–844, doi:10.1139/cjce-2013-0370.

Faber, B., and J. Stedinger (2001), Reservoir optimization using sampling SDP with ensemble stream-

flow prediction (ESP) forecasts, J. Hydrol., 249, 1113–133, doi:10.1016/S0022-1694(01)00419-X.

Flach, B., L. Barroso, and M. Pereira (2010), Long-term optimal allocation of hydro generation for a

price-maker company in a competitive market: latest developments and a stochastic dual dynamic

programming approach, IET Gener. Transm. Distrib., 4(2), 299, doi:10.1049/iet-gtd.2009.0107.

Garen, D. C. (1992), Improved Techniques in Regression-Based Streamflow Volume Forecasting, J.

Water Resour. Plan. Manag., 118(6), 654–670, doi:10.1061/(ASCE)0733-9496(1992)118:6(654).

Gelati, E., H. Madsen, and D. Rosbjerg (2010), Markov-switching model for nonstationary runoff con-

ditioned on El Niño information, Water Resour. Res., 46, W02,517, doi:10.1029/2009WR007736.

Gelati, E., H. Madsen, and D. Rosbjerg (2014), Reservoir operation using El Niño forecasts case study

of Daule Peripa and Baba, Ecuador, Hydrol. Sci. J., 59(8), 1559–1581, doi:10.1080/02626667.2013.

831978.

Georgakakos, A. (1989), The Value of Streamflow Forecasting in Reservoir Operation, Water Resour.

Bull., 25(4).

54

Gjelsvik, A., B. Mo, and A. Haugstad (2010), Long and Medium term Operations Planning and

Stochastic Modelling in Hydro-dominated Power Systems Based on Stochastic Dual Dynamic Pro-

gramming, Hanb. power Syst. I, Energy Syst., pp. 33–55, doi:10.1007/978-3-642-02493-1.

Goor, Q., R. Kelman, and A. Tilmant (2011), Optimal multipurpose multireservoir operation model

with variable productivity of hydropower plants, J. water Resour. Plan. Manag., 137(3), 258–267,

doi:10.1061/(ASCE)WR.1943-5452.

Hannan, E. J., and M. Deistler (2012), The Statistical Theory of Linear Systems, Society for Industrial

and Applied Mathematics, doi:10.1137/1.9781611972191.

Hart, W. E., C. Laird, J.-P. Watson, and D. L. Woodruff (2012), Pyomo – Optimization Modeling in

Python, vol. 67, 13–28 pp., doi:10.1007/978-1-4614-3226-5.

Homen-de Mello, T., V. L. de Matos, and E. C. Finardi (2011), Sampling strategies and stopping cri-

teria for stochastic dual dynamic programming: a case study in long term hydrothermal scheduling,

Energy Syst., 2(1), 1–31, doi:10.1007/s12667-011-0024-y.

Infanger, G. (1993), Planning under Uncertainty: Solving large scale stochastic linear programs,

Boyd and Fraser publishing company, Danvers, Massachusetts.

Infanger, G., and D. P. Morton (1996), Cut sharing for multistage stochastic linear programs with

interstage dependency, Math. Program., 75(2), 241–256, doi:10.1007/BF02592154.

Johnson, S. A., J. R. Stedinger, C. A. Shoemaker, Y. Li, and J. Tejada-Guibert (1993), Numerical

solution of continous state dynamic programs using linear and spline interpolation, Oper. Res.,

41(3), 484–500.

Karamouz, M., and H. V. Vasiliadis (1992), Bayesian stochastic optimization of reservoir operation

using uncertain forecasts, Water Resour. Res., 28(5), 1221–1232, doi:10.1029/92WR00103.

Karamouz, M., F. Szidarovszky, and B. Zahraie (2003), Water Resources Systems Analysis with Em-

phasis on Conflict Resolution, 608 pp.

Karunanithi, N., W. J. Grenney, D. Whitley, and K. Bovee (1994), Neural Networks for River Flow

Prediction, J. Comput. Civ. Eng., 8(2), 201–220, doi:10.1061/(ASCE)0887-3801(1994)8:2(201).

Kelman, J., J. Stedinger, L. Cooper, E. Hsu, and S. Yuan (1990), Sampling stochastic dynamic

programming applied to reservoir operation, Water Resour. Res., 26(3), 447–454, doi:10.1029/

WR026i003p00447.

Kim, Y., and R. Palmer (1997), Value of seasonal flow forecasts in Bayesian stochastic programming,

J. water Resour. Plan. Manag., 123(6), 327–335, doi:10.1061/(ASCE)0733-9496(1997)123:6(327).

55

Kim, Y.-o., H. Eum, E. Lee, and I. Ko (2007), Optimizing operational policies of a Korean multireser-

voir system using sampling stochastic dynamic programming with ensemble streamflow prediction,

J. water Resour. Plan. Manag., 133(1), 4–14.

Kitanidis, P. K., E. F. I. Foufoula-georgiou, and S. Anthony (1987), Error analysis of conventional

discrete and gradient dynamic programming, Water Resour. Res., 23(5), 845–858.

Kristiansen, T. (2004), Financial risk management in the electric power industry using stochastic

optimization, Adv. Model. Optim., 6(2), 17–24.

Marmarelis, V., and G. Mitsis (Eds.) (2014), Data-driven Modeling for Diabetes, Lecture Notes in

Bioengineering, Springer Berlin Heidelberg, Berlin, Heidelberg, doi:10.1007/978-3-642-54464-4.

Labadie, J. W. (2004), Optimal Operation of Multireservoir Systems: State-of-the-Art Review, J.

Water Resour. Plan. Manag., 130(2), 93–111, doi:10.1061/(ASCE)0733-9496(2004)130:2(93).

Ljung, L. (1999), System Identification: Theory for the User, in Wiley Encycl. Electr. Electron. Eng.,

John Wiley & Sons, Inc., Hoboken, NJ, USA, doi:10.1002/047134608X.W1046.

Ltkepohl, H. (2007), New Introduction to Multiple Time Series Analysis, Springer Publishing Com-

pany, Incorporated.

Lohmann, T., A. S. Hering, and S. Rebennack (2015), Spatio-temporal hydro forecasting of mul-

tireservoir inflows for hydro-thermal scheduling, Eur. J. Oper. Res., 255(1), 243–258, doi:10.1016/

j.ejor.2016.05.011.

Macian-Sorribes, H., A. Tilmant, and M. Pulido-Velazquez (2016), Improving operating policies of

large-scale surface-groundwater systems through stochastic programming, Water Resour. Res., doi:

10.1002/2016WR019573.

Mo, B., A. Gjelsvik, and A. Grundt (2001), Integrated risk management of hydro power scheduling

and contract management, Power Syst. IEEE Trans. , 16(2), 216–221, doi:10.1109/59.918289.

Mo, B., and A. Gjelsvik (2002), Simultaneous optimisation of withdrawal from a flexible contract and

financial hedging, 14th Power Syst. Comput. Conf., (June), 24–28.

NYISO (2017), New York Independent System Operator.

Oliveira, R., and D. P. Loucks (1997), Operating rules for multireservoir systems, Water Resour. Res.,

33(4), 839–852, doi:10.1029/96WR03745.

Pagano, T. C., D. Garen, and S. Sorooshian (2004), Evaluation of Official Western U. S. Sea-

sonal Water Supply Outlooks , 1922 – 2002, J. Hydrometeorol., 5(5), 896–909, doi:10.1175/

1525-7541(2004)005<0896:EOOWUS>2.0.CO;2.

56

Pereira, M. (1989), Optimal stochastic operations scheduling of large hydroelectric systems, Int. J.

Electr. Power Energy Syst., 11(3), 161–169, doi:10.1016/0142-0615(89)90025-2.

Pereira, M., N. Campodónico, and R. Kelman (1998), Long-term Hydro Scheduling based on Stochas-

tic Models, Int. Conf. Electr. Power Syst. Oper. Manag., pp. 1–22.

Pereira, M. V. F., and L. M. V. G. Pinto (1991), Multi-stage stochastic optimization applied to energy

planning, Math. Program., 52(1-3), 359–375, doi:10.1007/BF01582895.

Pereira-Cardenal, S. J., B. Mo, A. Gjelsvik, N. D. Riegels, K. Arnbjerg-Nielsen, and P. Bauer-

Gottwein (2016), Joint optimization of regional water-power systems, Adv. Water Resour., 92, 200–

207, doi:10.1016/j.advwatres.2016.04.004.

Pina, J., A. Tilmant, and F. Anctil (2016), Horizontal Approach to Assess the Impact of Climate

Change on Water Resources Systems, J. Water Resour. Plan. Manag., p. 04016081, doi:10.1061/

(ASCE)WR.1943-5452.0000737.

Pina, J., A. Tilmant, and P. Côté (2017), Optimizing Multireservoir System Operating Policies Using

Exogenous Hydrologic Variables, Water Resour. Res., (1990), 1–20, doi:10.1002/2017WR021701.

Poorsepahy-Samian, H., V. Espanmanesh, and B. Zahraie (2016), Improved Inflow Modeling in

Stochastic Dual Dynamic Programming, J. Water Resour. Plan. Manag., p. 04016065, doi:

10.1061/(ASCE)WR.1943-5452.0000713.

Pritchard, G. (2015), Stochastic inflow modeling for hydropower scheduling problems, Eur. J. Oper.

Res., 246, 496–504, doi:10.1016/j.ejor.2015.05.022.

Rani, D., and M. M. Moreira (2009), Simulation Optimization Modeling: A Survey and Potential

Application in Reservoir Systems Operation, Water Resour. Manag., 24(6), 1107–1138, doi:10.

1007/s11269-009-9488-0.

Raso, L., P. O. Malaterre, and J. C. Bader (2017), Effective Streamflow Process Modeling for Optimal

Reservoir Operation Using Stochastic Dual Dynamic Programming, J. Water Resour. Plan. Manag.,

p. 04017003, doi:10.1061/(ASCE)WR.1943-5452.0000746.

Rockafellar, R. T., and R. J. B. Wets (1991), Scenarios and Policy Agregation in Optimization under

Uncertainty, Math. Oper. Res., 16(1), 119–147, doi:10.1287/moor.16.1.119.

Rotting, T., and A. Gjelsvik (1992), Stochastic dual dynamic programming for seasonal scheduling in

the Norwegian power system, IEEE Trans. Power Syst., 7(1), 273–279, doi:10.1109/59.141714.

Saad, M., P. Bigras, a. Turgeon, and R. Duquette (1996), Fuzzy Learning Decomposition for the

Scheduling of Hydroelectric Power Systems, Water Resour. Res., 32(1), 179–186, doi:10.1029/

95WR02971.

57

Salas, J., Delleur, J., Yevjevich, V., and Lane, W. (1980). Applied Modeling of Hydrologic Time Series.

Water Resources Publications, Littleton, Colorado, USA.

Sene, K. (2010), Hydrometeorology, 1–427 pp., Springer Netherlands, Dordrecht, doi:10.1007/

978-90-481-3403-8.

Séguin, S., S.-E. Fleten, P. Côté, A. Pichler, and C. Audet (2017), Stochastic short-term hydropower

planning with inflow scenario trees, Eur. J. Oper. Res., 259(3), 1156–1168, doi:10.1016/j.ejor.2016.

11.028.

Shapiro, A., W. Tekaya, J. P. da Costa, and M. P. Soares (2013), Risk neutral and risk averse Stochastic

Dual Dynamic Programming method, Eur. J. Oper. Res., 224(2), 375–391, doi:10.1016/j.ejor.2012.

08.022.

Stedinger, J. R., B. F. Sule, and D. P. Loucks (1984), Stochastic dynamic programming mod-

els for reservoir operation optimization, Water Resour. Res., 20(11), 1499–1505, doi:10.1029/

WR020i011p01499.

Tejada-Guibert, J., S. Johnson, and J. Stedinger (1993), Comparison of two approaches for imple-

menting multireservoir operating policies derived using stochastic dynamic programming, Water

Resour. Res., 29(12), 3969–3980.

Tejada-Guibert, J. A., S. A. Johnson, and J. R. Stedinger (1995), The Value of Hydrologic Informa-

tion in Stochastic Dynamic Programming Models of a Multireservoir System, Water Resour. Res.,

31(10), 2571–2579, doi:10.1029/95WR02172.

Thomas, H.A. and Fiering, M.B. (1962). Mathematical synthesis of streamflow sequences for the

analysis of river basins by simulation. In Design of Water Resource Systems, Edited by Mass et al.,

Harvard University Press, Cambridge, 1962.

Tilmant, A., and R. Kelman (2007), A stochastic approach to analyze trade-offs and risks as-

sociated with large-scale water resources systems, Water Resour. Res., 43(6), W06,425, doi:

10.1029/2006WR005094.

Tilmant, A., and W. Kinzelbach (2012), The cost of noncooperation in international river basins, Water

Resour. Res., 48(1), W01,503, doi:10.1029/2011WR011034.

Tilmant, A., D. Pinte, and Q. Goor (2008), Assessing marginal water values in multipurpose

multireservoir systems via stochastic programming, Water Resour. Res., 44(12), W12,431, doi:

10.1029/2008WR007024.

Todini, E. (1988), Rainfall-runoff modeling - Past, present and future, J. Hydrol., 100(1-3), 341–352,

doi:10.1016/0022-1694(88)90191-6.

58

Turgeon, A., and R. Charbonneau (1998), An aggregation disaggregation approach to long term reser-

voir management, Water Resour. Res., 34(12), 3585–3594.

Yeh, W. W.-G. (1985), Reservoir Management and Operations Models: A State-of-the-Art Review,

Water Resour. Res., 21(12), 1797–1818, doi:10.1029/WR021i012p01797.

Zahraie, B., and M. Karamouz (2004), Hydropower Reservoirs Operation : A Time Decomposition

Approach, Sci. Iran., 11(April), 92–103.

59

Appendix A

Paper I: Horizontal Approach to assesthe Impact of Climate Change on WaterResources Systems

Résumé

L’approche généralement utilisée pour évaluer les impacts du changement climatique sur des sys-

tèmes de ressources en eau est fondée sur une intégration et/ou un couplage vertical de modèles. Étant

donné que la gamme d’incertitude qui peut être explorée avec le GCM est limitée, les chercheurs

préfèrent s’appuyer sur des ensembles pour élargir la propagation, ce qui rend l’approche par modéli-

sation encore plus exigeante en termes de temps de calcul et de ressources. Lorsqu’un système d’eau

particulier doit être analysé, la question est de savoir si cette approche verticale à forte intensité de

calcul est nécessaire en premier lieu, ou s’il est possible de compter sur des projections disponibles

dans les systèmes voisins ? L’étude proposée aborde cette question en comparant les performances

d’un système de ressources en eau dans des conditions climatiques futures, en utilisant les approches

verticales et horizontales. La méthodologie est illustrée par le système hydroélectrique du bassin de la

rivière Gatineau au Québec (Canada). L’analyse des résultats de la simulation, obtenus à partir des ap-

proches verticales et horizontales, révèle que les productions énergétiques annuelles et hebdomadaires

du système sont similaires.

http://ascelibrary.org/doi/abs/10.1061/(ASCE)WR.1943-5452.0000737.

60

Horizontal Approach to Assess the Impact of ClimateChange on Water Resources Systems

Jasson Pina1; Amaury Tilmant, Ph.D.2; and Francois Anctil, Ph.D.3

Abstract: The typical approach to assess climate change impacts on water resources systems is based on a vertical integration/coupling of

models. Since the range of uncertainty that can be explored with GCM is limited, researchers rely on ensembles to enlarge the spread, making

the modeling approach even more demanding in terms of computation time and resource. When a particular water system must be analyzed,

the question is to know whether this computationally intensive vertical approach is necessary in the first place or if we could rely on pro-

jections available in neighboring systems? The proposed study addresses this question by comparing the performance of a water resource

system under future climate conditions using the vertical and the horizontal approaches. The methodology is illustrated with the hydropower

system of the Gatineau River Basin in Quebec, Canada. The analysis of the simulation results reveals that the annual and weekly energy

productions of the system derived from both the vertical and the horizontal approaches are similar. DOI: 10.1061/(ASCE)WR.1943-

5452.0000737. © 2016 American Society of Civil Engineers.

Author keywords: Water resources systems; Climate change; Hydropower generation; Stochastic programming; Multimodel hydrology.

Introduction

The profligate use of fossil fuel energy that was initiated by the

industrial revolution in the nineteenth century has had profound

impacts on societies and on the environment worldwide. The mas-

sive release of carbon dioxide in the atmosphere by ever increasing

human activities is expected to change the composition of the

atmosphere and, hence, the components of the hydrological cycle

such as precipitation and temperature (IPCC 2013). Significant ef-

forts have been devoted over the past 30 years to assess the impacts

of climate change on water resources and the effectiveness of

adaptation measures to mitigate those impacts.There are essentially two categories of adaptation measures to

climate change in the water sector: structural and nonstructaural.

The first one includes traditional engineering-based solutions such

as investments in hydraulic infrastructure like levees and dams

(e.g., Jeuland and Whittington 2014). In the second category, alter-

native solutions like insurance mechanisms (e.g., Botzen and van

der Bergh 2008), demand management (e.g., Iglesias and Garrote

2015), and changes in operating policies (e.g., Vicuna et al. 2009)

have received a lot of attention over the last decade.The typical approach to assess climate change impacts on water

resources systems is based on a vertical integration/coupling of

models: GCM models are run to project future precipitations

and temperatures, which are then downscaled and used as inputsto hydrologic models whose outputs are processed by watersystems models (Yao and Georgakakos 2001; Carpenter andGeorgakakos 2001; Harrison and Whittington 2002). From a deci-sion-making point of view, this top-down vertical approachpresents some challenges. Since the range of uncertainty that canbe explored with GCM is limited, researchers are relying on ensem-bles to enlarge the spread, making the modeling approach evenmore demanding in terms of computation time and resource. Forexample, in an attempt to cover all sources of uncertainty, Chenet al. (2011) combine various emission scenarios, initial GCM con-ditions, GCMs, downscaling techniques, and hydrological models.

Another challenge associated with this vertical, top-down ap-proach lies in the difficulty of attaching probabilities to the scenar-ios and hence to carry out a risk analysis. Yet, designing a portfolioof adaptation measures requires that the effectiveness of these mea-sures, whether they are structural or nonstructural, be properly as-sessed. Decision analysis is the theoretical framework that hastraditionally been implemented to deal with this planning issue.When rooted in a traditional benefit-cost analysis framework, de-cision analysis requires that key uncertainties be well characterized,which is often not the case with climate change scenarios (Dessaiand Hulme 2004; New et al. 2007).

To address this issue, Wilby and Dessai (2010), Prudhommeet al. (2010), and Brown et al. (2012) propose an alternative ap-proach for climate risk assessment that links bottom-up, stochasticvulnerability analysis with the traditional top-down use of GCMprojections. This alternative approach, coined Decision Scaling,has been used to manage climate risks in the Upper Great Lakesby Brown et al. (2011). Decision scaling is a robust decision mak-ing approach, a set of decision analytic methods to help decisionmakers identify solutions that satisfactorily perform across a widerange of potential climate changes (Lempert and Schlesinger 2000).

These two approaches (top-down versus bottom-up) illustratethe ongoing debate among scientists on how best to tackle the chal-lenges associated with climate risk assessment and management.On the one hand, proponents of the top-down approach argue theneed to improve the predictive capability of climate models in orderto reduce the uncertainty attached to climate projections. On the

1Ph.D. Student, Dept. of Civil Engineering and Water Engineering,Université Laval, Av. de la Médecine 1065 Pavillon Adrien-Pouliot,2983, QC, Canada G1V 0A6. E-mail: jasson.pina-fulano.1@ulaval.ca

2Professor, Dept. of Civil Engineering and Water Engineering, Univer-sité Laval, Av. de la Médecine 1065 Pavillon Adrien-Pouliot, 2978, QC,Canada G1V 0A6 (corresponding author). E-mail: amaury.tilmant@gci.ulaval.ca

3Professor, Dept. of Civil Engineering and Water Engineering, Univer-sité Laval, Av. de la Médecine 1065 Pavillon Adrien-Pouliot, 2988, QC,Canada G1V 0A6. E-mail: Francois.Anctil@gci.ulaval.ca

Note. This manuscript was submitted on October 31, 2015; approved onSeptember 8, 2016; published online on November 10, 2016. Discussionperiod open until April 10, 2017; separate discussions must be submittedfor individual papers. This paper is part of the Journal of Water Resources

Planning and Management, © ASCE, ISSN 0733-9496.

© ASCE 04016081-1 J. Water Resour. Plann. Manage.

J. Water Resour. Plann. Manage., 04016081

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TE

LA

VA

L o

n 11

/14/

16. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

61

other hand, proponents of the bottom-up approach rather considerthat, in light of the deep uncertainties involved, more efforts mustbe devoted to understanding the climate-related vulnerabilities(Weaver et al. 2013).

For managers of large hydropower systems, this scientific debateis important, but so are more practical (pressing) issues. One of themis how to rapidly appraise the impacts of a changing flow regime onthe energy produced by a particular system. Because the verticalapproach is computationally intensive, and with the increasingavailability of projections in neighboring systems, the question isto know whether we could obtain a decent approximation of thechanges in the energy generated by a particular system by extrapo-lating the hydro-climatic projections already available in a broaderregion encompassing that system. This study addresses this questionby comparing the performance of a water resource system underfuture climate conditions using the vertical approach and the hori-zontal approach in which available projections are extrapolated tothe system of interest. The methodology is illustrated with a cascadeof power stations on the Gatineau River in Quebec, Canada.

The paper is organized as follows: section “Material andMethod” presents the modelling of the climate and the hydrologicalprocesses, the multireservoir operation model, and then the selectedextrapolation methods. Section “Analysis of Simulation Results”describes the case study, which is followed by a presentationand a discussion of simulated results (energy generation). Finally,conclusions are given in section “Conclusion.”

Material and Method

This section starts with an overview of the methodology, which willbe explain in more detail in the following subsections. This meth-odology, which is illustrated on Fig. 1, is implemented on the hy-dropower system of the Gatineau River Basin in Quebec, Canada.This system consists of a cascade of four power stations and twolarge reservoirs located in the Laurentian region north of Ottawa.The top-down approach implemented in many river basins in theregion relies on the projections of the Canadian GCM, dynamicallydownscaled, and processed by a hydrological model to projectfuture discharges. Hydrologic projections are then used as inputsto a stochastic multireservoir optimization model (stochastic dualdynamic programming, SDDP) to derive energy outputs from thesystem. To assess the horizontal approach, we need a benchmark:the hydrologic projections derived from an extended vertical ap-proach implemented in the Gatineau River Basin (Fig. 1). Theextension is needed to account for the structural (hydrologic)uncertainty inherent to that basin. To achieve this, the vertical ap-proach in the Gatineau also relies on another set of 20 hydrologicmodels, each having a different structure. In a neighboring riverbasin (Du Lievre), the vertically obtained projections are extrapo-lated to the Gatineau River Basin using four different extrapolationtechniques. A reservoir operation model then processes all hydro-logic projections. Energy simulation results, derived either directlyfrom the vertical approach or indirectly by extrapolating the pro-jections available in Du Lievre River Basin, are compared.

To summarize, comparing the performance of the system underthe vertical and the horizontal approaches requires• H hydrologic models to capture the structural uncertainty in the

target river basin (here the Gatineau). These models are part ofthe vertical approach implemented in that basin. The mean ofthe projections derived from these models is used as a bench-mark to compare the accuracy of the extrapolation techniques.Here, H ¼ 21.

• K hydrologic sequences representative of the natural uncertaintyin the target river basin. These sequences are required by the

SDDP algorithm to derive the optimal reservoir operatingpolicies. Here, K ¼ 23.

• E extrapolation techniques are used to derive river dischargesin the target river basin (Gatineau) based on the projectionsavailable in the source river basin (Du Lievre). Here, fourextrapolation techniques (E ¼ 4) are tested and compared interms of energy generation and reservoir trajectories.The following subsections describe the methodology in more

detail.

Gatineau River Basin and Hydropower System

The Gatineau River, located in southwestern Quebec, rises in lakesnorth of the Baskatong Reservoir and flows south to join theOttawa River. The main river channel is about 400 km and drainsan area of 23,700 km2. The hydroelectric system consists of a cas-cade of four power stations and two large reservoirs (Fig. 2). Theupstream reservoir, Cabonga, has a storage capacity of 1.633 km2

and drains an area of 2,201 km2. The second reservoir, Baskatong,was formed following the construction of the Mercier Damdrains an area of 12,540 km2, and has a power plant with an in-stalled capacity of 55 MW. The Paugan station is a run-of-riverpower plant (R-O-R) with total capacity of 219 MW and a smallreservoir of 30 km2. The last R-O-R power stations are Chelseaand Rapides Farmer, with a total installed capacity of 149 and95 MW, respectively.

Climate Modeling and Projections

Historical data were provided by Hydro-Québec and the Centred’Expertise Hydrique du Québec (CEHQ), and climatic scenarios

Fig. 1. Schematics of the methodology

© ASCE 04016081-2 J. Water Resour. Plann. Manage.

J. Water Resour. Plann. Manage., 04016081

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TE

LA

VA

L o

n 11

/14/

16. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

62

by Ouranos. Climatic projections were obtained from the third-generation coupled global climate model (CGCM3) of theCanadian Centre for Climate Modelling and Analysis, whichhas a surface grid with a roughly spatial resolution of 3.75°latitude/longitude; daily data for several 2D variables are availablefor years 2001–2100. Future projections were dynamically down-scaled to match the temporal and spatial scales of the hydrologicalmodels, using the Canadian regional climate model (CRCMversion 4.2.3), time slice simulation for 2041–2070 over the NorthAmerican domain. The CRCM horizontal grid is uniform in a polarstereographic projection, with a typical 45-km grid mesh (true at60°N), and its vertical resolution is variable using a Gal-Chenscaled height terrain-following coordinate (Music and Caya 2007).Local Intensity scaling method were implemented by Ouranos tocorrect biases between observations and simulations for the refer-ence period (1961–1990).

To account for the uncertainty regarding the intensity of futureconsumption of fossil fuels, the analysis should be carried out forvarious emission scenarios. However, the proposed methodologywill be illustrated for scenario A2 only.

Hydrologic Modeling and Projections

Hydrologic projections were generated by CEHQ using the hydro-logical model Hydrotel, a semi-physically based distributed modellargely used in Quebec for dam management (Fortin et al. 2001;Turcotte et al. 2004). To account for the structural (hydrologic)

uncertainty inherent to the Gatineau River Basin, a multimodel ap-proach based on 20 different lumped hydrologic models was alsoimplemented. The model selection was initially carried out byPerrin (2000) and revised by Seiller et al. (2012) for hydrologicalprojection purposes. It is mainly based on known performance andstructural diversity: from 4 to 10 free parameters, and 2 to 7 storageunits. The structural diversity ensures that the simple average of themodels’ output is much more robust to temporal transposabilitythan the output of the models taken individually (Seiller et al.2012). This arises from the fact that the models are somehow un-correlated and tend to cancel out each other’s errors. The gain thuslies in the combination of models and not in their individual per-formance. Indeed, Thiboult and Anctil (2015) compared Hydrotelto the above multimodel in validation and for short-term forecastingover 38 basins spread across the province of Quebec. Results invalidation showed that Hydrotel clearly stands among the bestmodels for all catchments, but is outperformed by the simple aver-age of the 20 lumped (much simpler) models. As for Thiboult andAnctil (2015), all models were applied here in exactly the sameconditions: they were run on a daily time step and fed with identicalprecipitations and potential evapotranspirations (computed fromOudin et al. (2005) temperature-based formulation). Snow accumu-lation and melt are simulated with the CemaNeige snow accountingmodule (Valéry et al. 2014).

Split sampling test (SST) (Klemeš 1986) was implemented forthe calibration and validation procedures. The principle is to cal-ibrate the models on a time series with selected characteristics and

Cabonga

Baskatong

Paugan

Chelsea

Inflow

Inflow

Inflow

Inflow

R. Farmers

Pembroke

Rapides Farmers Chelsea

Paugan

Baskatong

Cabonga

Joliette

Laval

MontrealOttawa

Cornwall

Longue

Trois-R

Reservoir

Power Plant

N

EW

S

77° W 76° W 75° W 74° W 73° W

48° N

47° N

46° N

45° N

Fig. 2. Gatineau River Basin [data from CEHQ 2012; Esri, DeLorme, TomTom, Intermap, Increment P Corp, GEBCO, USGS, FAO, NPS,

NRCAN, GeoBase, IGN, Kadaster NL, Ordnance Survey, Esri Japan, METI, Esri China (HongKong), Swisstopo, MapmyIndia, © OpenStreetMap

Contributors, and the GIS User Community (2015); basemaps ArcGIS]

© ASCE 04016081-3 J. Water Resour. Plann. Manage.

J. Water Resour. Plann. Manage., 04016081

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TE

LA

VA

L o

n 11

/14/

16. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

63

to validate it on a contrasted time series. Model calibration wasperformed for the period 1969–1988, and the validation phase from1988 to 2005. The shuffled complex evolution (SCE-UA) (Duanet al. 1993) was used to calibrate the model parameters. The ob-jective function was the root-mean square error applied to theroot-squared transformed streamflow (RMSEsqrt). The evaluationof model performance in validation relied on the Nash-Sutcliffe efficiency criterion (Nash and Sutcliffe 1970) calculatedon the root-squared transformed streamflows.

Once all hydrological models were calibrated, climate projec-tions were used to obtain time series of future inflows at each nodeof the system (reservoirs and ROR power plants) as pictured onFig. 2. A total of 21 time series were made available for the sto-chastic multireservoir optimization model: one set from the distrib-uted hydrologic model (Hydrotel) and 20 sets from the lumpedhydrologic models. The 21 time series capture the structural uncer-tainty associated with the hydrologic processes in the river system(here the Gatineau). This uncertainty is inherent to the system andwould be present regardless of the emission scenario considered. Inother words, for a given set of meteorological inputs and at a givenpoint in space and time, an empirical distribution of river flows canbe traced out from the simulations of the 21 hydrologic models.Implementing the multireservoir optimization model on each timeseries yields an empirical distribution of the energy generated atthat point, i.e., at a particular power plant and time of the year. Thisstatistical distribution reflects the impact of the structural uncer-tainty of the hydrological processes on the energy generated bythe system. It will be used to assess the quality of the extrapolationtechniques discussed in the next section.

Extrapolating Future River Discharges

All daily hydrologic times were aggregated on a weekly timestep in order to be compatible with the reservoir operation model.Different extrapolation techniques were used to derive the riverdischarges in the Gatineau from the weekly projections availablein neighbouring systems (river basins). The reference and futureinflows from Du Lievre River Basin, as generated by the CEHQusing Hydrotel, were selected to test different extrapolation tech-niques. Du Lievre River Basin drains an area of 9,946 km2, flowssouth and parallel to the Gatineau River, and empties into theOttawa River (Fig. 3). Four different techniques were tested1. Drainage area ratio: The first, most simple, extrapolation tech-

nique relies on the ratios between the drainage areas. Future pro-jection at Du Lievre River Basin were directly extrapolated tothe Gatineau River Basin by using Eq. (1) and the drainagearea ratios presented in Table 1. This technique makes the as-sumption that both river basins, Gatineau and Du Lievre, arehydrologically similar

QFutðjÞ ¼ QFutð6ÞAðjÞAð6Þ ð1Þ

where j ¼ 1, : : : ,5 = subbasins in the Gatineau River System;j ¼ 6 = Du Lievre River Basin. QfutðjÞ are the future inflowprojections and and AðjÞ is the (sub)basin area.

2. Change in mean: The second technique extrapolates flows in theGatineau Basin from the ratios between future and current aver-age weekly flows in Du Lievre River Basin. Here, we assumethat changes in the average weekly discharges in one basin areproportional to those observed in the neighbouring system.Based on this assumption, the inflows in the Gatineau sub-basins are given by

QFut;tðjÞ ¼ QRef;tðjÞμFut;tð6ÞμRef;tð6Þ

ð2Þ

where μRef;t and μFut;t = average weekly discharges for thereference series and the future projections, respectively.

3. Reference weekly regression: The reference weekly regressionapproach exploits the linear relationship among the referenceinflows in both rivers basins, QrefðjÞ

QRef;tðjÞ ¼ atðjÞt · QRef;tð6Þ þ ctðjÞ ð3ÞParameters aðjÞ and cðjÞ are used to determine the future

values in the Gatineau River sub-basins using the hydrologicprojections in the Du Lievre River QFut;tð6Þ

QFut;tðjÞ ¼ atðjÞ · QFut;tð6Þ þ ctðjÞ ð4ÞThis third extrapolation technique assumes that current linear

relationships between weekly river discharges in both basinswill not be affected by climate change. In other words, for any

78° W 77° W

48° N

47° N

46° N

45° N

76° W 75° W 74° W

Cabonga

Baskatong

Paugan

Rapides Farmers ChelseaReservoir

Power Plant

Gatineau Basin

Du Lièvre Basin

G A T I N E A U

UD L I È V R E

N

EW

S

Ottawa

Cornwall

Pembroke

M

L

Fig. 3. Gatineau and Du Lievre River Basins [data from CEHQ 2012;

Esri, DeLorme, TomTom, Intermap, Increment P Corp, GEBCO,

USGS, FAO, NPS, NRCAN, GeoBase, IGN, Kadaster NL, Ordnance

Survey, Esri Japan, METI, Esri China (HongKong), Swisstopo, Map-

myIndia, © OpenStreetMap Contributors, and the GIS User Commu-

nity (2015); basemaps ArcGIS]

Table 1. Drainage Area Ratios

(j) Basin Area (km2) Ratio

1 Cabonga 2,201 0.222 Baskatong 12,540 1.263 Paugan 7,740 0.784 Chelsea 1,189 0.125 Rapides Farmers — —

6 Du Lievre 9,946 1

© ASCE 04016081-4 J. Water Resour. Plann. Manage.

J. Water Resour. Plann. Manage., 04016081

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TE

LA

VA

L o

n 11

/14/

16. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

64

given week, the spatial correlation between river flows in bothbasins remains unchanged in the future regardless of the extentof the changes in river flows.

4. Correlation approach: The last technique takes into account thechanges in the weekly means and standard deviations fromDu Lievre River while assuming that the spatial correlationamong the inflows in the reference period remains identical.This extrapolation technique was chosen because it has thesame structure as the built-in hydrological model found in themutireservoir stochastic optimization model described inthe next section.The keep the notation simple, let us assume that the inflows

can be described by a lag-1 multisite autoregressive model (seeAppendix for a general formulation to p lags)

�

QtðjÞ − μtðjÞσtðjÞ

�

Ref

¼ ϕt

�

Qt−1ðjÞ − μt−1ðjÞσt−1ðjÞ

�

Ref

þ ϵtðjÞ ð5Þ

where σt = weekly standard deviation; and ϵtðjÞ = time independent(but spatially correlated) stochastic noise with mean zero andweekly variance σ2

��ε;t. ϕt is the lag-1 autoregressive parameterand is determined by using the method of the moments as thecorrelation ρi;t. Likewise, the variance is determined as σ2

��ε;t ¼1 − ϕ2

t . Once the parameters for the reference series are determined,the changes in the mean, standard deviation and in the autoregres-sive parameters from the reference inflows to the future projectionscalculated in Du Lievre River Basin

Δμt ¼μFut;tð6ÞμRef;tð6Þ

; Δσt ¼σFut;tð6ÞσRef;tð6Þ

; Δϕt ¼ϕFut;tð6ÞϕRef;tð6Þ

ð6Þ

are used to project the parameters for the future period on theGatineau River

μFut;tðjÞ ¼ Δμt · μRef;tðjÞσFut;tðjÞ ¼ Δσt · σRef;tðjÞϕFut;tðjÞ ¼ Δϕt · ϕRef;tðjÞ ð7Þ

Assuming that the noise ϵt follows a three-parameters lognormal distribution with μv, σv and lower bound ψt

fϵt ¼1

ðϵt − ψtÞffiffiffiffiffiffiffiffiffiffi

2πσv

p e−0.5f½logðϵt−ψtÞ−μv�=σvg2 ð8Þ

with mean μ��εt and variance σ2��ε;t

μ��εtψt þ eðμvþσ2v=2Þ σ2

��ε;te2ðμvþσ2vÞ þ eð2μvþσ2vÞ ð9Þ

Then the weekly future projections in the Gatineau sub-basinsare synthetically generated from time t0 to period T (number ofyears times number of weeks). The starting points of the generationprocedure Q0

t−1 are retrieved form the reference period and pro-jected directly using Eq. (13)

ψtðjÞ ¼ −μtðjÞσtðjÞ

− ϕt

�

Q0t−1ðjÞ − μt−1ðjÞ

σt−1ðjÞ

�

Fut

ð10Þ

λtðjÞ ¼ 1þ σ2ϵ;t

ψ2t ðjÞ

μvðjÞ ¼ 0.5Logσ2ϵ;t

λðjÞ½λðjÞ − 1� ð11Þ

σvðjÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

log½λðjÞ�p

ð12Þ

QFut;tðjÞ ¼ σFut;t · eVtðjÞ·σvðjÞþμvðjÞ ð13Þ

where Vt = cross-correlated white noise.

Stochastic Multireservoir Optimization Model—SDDP

Reservoir operation problems are multistage, stochastic and non-linear optimization problem. The goal is to determine a sequenceof optimal decisions xt that maximize the expected sum of benefitsfrom system operation Z, over a planning period time T, whilemeeting operational and/or institutional constraints (Fig. 4).

The mathematical formulation of the multistage decision-making problem can be written as

maxxt

fZg ¼ maxxt

�

E

�

X

T

t¼1

αtbtðst; qt; rtÞ þ αTþ1νðsTþ1; qTÞ��

ð14Þ

where btð·Þ = immediate benefit function at stage t; νð·Þ = terminalvalue function; αt = discount factor at stage t; E½·� = expectationoperator; st = vector of storages at the beginning of stage t;qt = vector of inflows; and rt = vector of releases. The immediatebenefit function will include the net benefits from system genera-tion and penalties for not meeting target demands and/or violatingconstraints.

The immediate benefit function, at time t, can be written as

btðst; qt; rtÞ ¼ τ tðπ − θÞ 0Pt − ξ 0t zt ð15Þ

where τ t = number of hours in period t; Pt (MW) = vector of powergenerated; π = vector of energy price ($=MWh); θ = vector of theoperation and maintenance cost ($=MWh); and zt = vector ofdeficits or surpluses (unit deficit or surplus) penalized by the vectorξ 0t of penalties ($=unit deficit or surplus).

The stage to stage transformation function corresponds to themass balance equation

stþ1 − CRðrt þ ltÞ ¼ st þ qt − et ð16Þ

where stþ1 = vector of storage at the end of the period; lt = vector ofspills; CR = reservoir system connectivity matrix, CR

j;k ¼ 1ð−1Þwhen reservoir j receives (releases) water from (to) reservoirk; and et = vector of evaporation losses.

This problem can be solved with discrete stochastic dynamicprogramming (SDP), which recursively constructs a benefit-to-go function Ftþ1 at the grid points defining the state space. Withtwo reservoir storages and a minimum of four hydrologic state var-iables, the reservoir operation problem in the Gatineau cannot besolved with SDP as the computational effort increases exponen-tially with the number of state variables. To deal with the dimen-sionality issue, we use stochastic dual dynamic programming, an

Fig. 4. Multistage decision-making problem

© ASCE 04016081-5 J. Water Resour. Plann. Manage.

J. Water Resour. Plann. Manage., 04016081

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TE

LA

VA

L o

n 11

/14/

16. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

65

extension of SDP that can handle a much larger state space (Tilmantet al. 2008).

The solution approach is based on the approximation of the ex-pected cost-to-go functions of SDP by piecewise linear functions(hyperplanes), meaning there is no need to evaluate Ftþ1 over adense grid. SDDP uses an iterative procedure to increase the accu-racy of the solution by adding new hyperplanes in the regions ofthe state-space where they are most needed, i.e., in the regions ex-plored by the system.

To further save computation time, SDDP relies on an analyticalrepresentation of the natural hydrologic uncertainty through amulti-site periodic autoregressive model MPAR(p). More detailson the MPAR(p) model are provided in Appendix. The natural un-certainty must not be confused with the structural uncertainty dis-cussed earlier. The natural uncertainty originates from thestochasticity of the hydrologic processes generating runoff. Thestructural uncertainty comes from our inability to perfectly modelthese processes. To describe the natural uncertainty, several realiza-tions of the stochastic processes are considered through, forexample, multiyear time series of river flows, while the structuraluncertainty is captured through the use of multiple hydrologicmodels, each having a different structure.

As mentioned earlier, SDDP is an iterative algorithm organizedaround two phases: a backward optimization and a forward simu-lation. In the backward phase, the MPAR model is used to generateK inflows scenarios (backward openings) at each node of thesystem. These scenarios are needed to analytically calculate the hy-perplanes’ parameters, and ultimately to derive the upper bound tothe true expected benefit-to-go function. The accuracy of thisapproximation is evaluated at the end of a forward simulation phasebased on M hydrological sequences representative of the naturaluncertainty. If the upper bound is inside the confidence interval ofthe lower bound, the approximation is statistically acceptable andthe algorithm stops, otherwise a new backward recursion is imple-mented with a new set of hyperplanes build on the storage volumesthat were visited during the last simulation phase.

Analysis of Simulation Results

The SDDP model of the Gatineau System was run 21 times, eachtime exploiting a different time series of river discharges obtainedby one of the 21 hydrologic models. The process was then repeatedfor each extrapolation technique discussed above. For each run, theoptimal operating policies were simulated over 23 historical hydro-logic sequences of 260 weeks (M ¼ 23 and T ¼ 260 weeks). Twas chosen so as to be sufficiently long to avoid the effectsof the boundary conditions on reservoir operating policies on in-termediate years (Goor et al. 2011). Here, results are analyzedfor year three only as the first and last two years are influencedby the boundary conditions (initial storages at time t ¼ 1 as wellas FTþ1 ¼ 0).

Table 2 lists the average and standard deviation (Std) of annualenergy generation in the system after extrapolating the hydrologicprojections in Du Lievre River Basin using the four techniques de-scribed above. The standard deviation reflects the natural uncer-tainty of the hydrologic processes in the basin. For example, forthe extrapolation technique based on the drainage areas, the annualhydroelectric production could vary from 2.5 to 3.8 TWh depend-ing on whether we have a (naturally) dry or a wet year. On average,depending on the extrapolation technique, the annual productionswould vary from 3.13 to 3.29 TWh. This 5% difference is signifi-cant, yet within the range of the structural uncertainty representedby the grey area on Fig. 5.

The mean of the multimodel (black CDF on Fig. 5) is the bench-mark against which the extrapolation techniques must be com-pared. Recall that this is the mean of the 21 different hydrologicmodels that a part of the vertical approach directly implementedon the Gatineau River Basin. We are interested in the horizontaldifferences between the CDFs derived after extrapolating the pro-jections and the benchmark; the smaller the differences across awide range of hydrologic conditions (nonexceedance probabilities),the better is the extrapolation technique. Fig. 6 displays the hori-zontal differences between the CDFs corresponding to the differentextrapolation techniques and the mean of the CDFs correspondingto all 21 hydrologic models (the benchmark). This figure showsto what extent a particular extrapolation technique performsacross contrasting hydrologic conditions, from very dry years(low nonexceedance probabilities) to very wet years (high nonex-ceedance probabilities). The examination of Fig. 6 reveals that theextrapolation based on the weekly linear regressions performsbetter than the others because the relative differences are smallerover a large spectrum of probabilities. In other words, regardlessof the hydrologic conditions, the energy outputs obtained afterextrapolating the flows with linear regressions almost match thoseof the benchmark (with the exception of the driest year).Table 3 shows the root-mean-square differences (TWh) betweenthe energy outputs obtained after extrapolating the flows and thebenchmark for various probability ranges (100, 90, 80, and50%) centered on the median. As expected from the visual exami-nation of Fig. 6, we can see that the extrapolation approach basedon weekly linear regressions generates the smallest differences withthe benchmark, especially for close to a normal hydrologic condi-tions (50% probability range). However, the predictive capability ofthe linear regression degrades as more extreme hydrologic years(wet or dry) are included in the analysis. Note that the latter is truefor all extrapolation techniques.

Table 2. Annual Energy Generation: Mean and Standard Deviation

EnergyArearatio

Changein mean

Linearregression AR model

Mean (TWh) 3.29 3.13 3.22 3.17Std (TWh) 0.3190 0.3545 0.3007 3.1916

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Energy (TWh)

Non

exce

eden

ce p

roba

bilit

y (

)

Empirical CDF

Structural uncertainty

Benchmark

Area Ratio

Change in mean

Linear Regression

Autoregressive Model

Fig. 5. Annual energy generation—cumulative distribution functions

© ASCE 04016081-6 J. Water Resour. Plann. Manage.

J. Water Resour. Plann. Manage., 04016081

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TE

LA

VA

L o

n 11

/14/

16. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

66

As the technique based on the weekly linear regression performsbetter for this case study, it seems that climate change will not affectthe spatial correlation between weekly river discharges in bothbasins. This observation would probably hold for other basins inthe region but cannot be generalized to a larger scale.

The extrapolation technique based on the weekly linear regres-

sion performing well on an annual basis, let us examine the pro-

duction in the Gatineau System on a weekly time step. Fig. 7 shows

the weekly means while Fig. 8 displays the weekly standard devia-

tions of the energy generated with the horizontal and the vertical

approaches. Recall that the vertical approach relies on 21 hydro-

logic models, meaning that both the average weekly values and

standard deviations are given by an interval. If the production pat-

terns are similar, we can see that the extrapolation of Du Lievre

River discharges tend to overestimate the average weekly produc-

tion during the summer season: the average weekly productions are

close to the upper bound of the structural uncertainty. During the

rest of the year, the opposite behavior is observed. Since the mean

values are most of the time within the structural uncertainty range,

we can conclude that the extent of the weekly productions derived

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120

10

0

10

20Area Ratio

Rel

ativ

e di

ffere

nce

(%)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120

10

0

10

20Linear Regression

Rel

ativ

e di

ffere

nce

(%)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120

10

0

10

20Change in mean

Rel

ativ

e di

ffere

nce

(%)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120

10

0

10

20Autoregressive Model

Rel

ativ

e di

ffere

nce

(%)

Non exceedance probability [ ]

Fig. 6. Annual energy generation—relative differences between the distribution functions

Table 3. Root-Mean-Square Deviations (TWh) between the FourExtrapolation Techniques and the Benchmark

Probabilityrange (%)

Arearatio

Changein mean

Linearregression AR model

100 0.81 1.09 0.61 0.7190 0.65 1.01 0.31 0.6780 0.59 0.85 0.22 0.5850 0.62 1.18 0.29 0.46

© ASCE 04016081-7 J. Water Resour. Plann. Manage.

J. Water Resour. Plann. Manage., 04016081

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TE

LA

VA

L o

n 11

/14/

16. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

67

with a horizontal approach are consistent with that obtained with a

diversity of hydrologic models.The erratic rainfalls that characterize the beginning of the winter

season result in an increased variability of the production as indi-

cated on Fig. 8. This also reflects in the structural uncertainty,

which is larger during the second half of the year. As we can also

see from the examination of Fig. 8, the standard deviation associ-

ated with the horizontal approach lies most of the time within the

range defined by the structural uncertainty. It means that the

horizontal approach yields weekly energy productions whose

variability is consistent with that obtained with a diversity of

hydrologic models.Fig. 9 shows the average drawdown-refill cycle of the

Baskatong Reservoir when the inflows are derived from the linear

regression. As one can see, the drawdown starts in the summer and

goes on until weeks 12–13 when the snow melt season starts and

the reservoir is quickly refilled at the beginning of the next summer.

For illustrative purposes, the drawdown refill cycle corresponding

to current hydrometeorological conditions are also displayed on thesame figure. We can see that for the selected emission scenario, therefill phase starts two weeks earlier following earlier snowmelt.Under that scenario, weekly river discharges during the first halfof the year (winter) are larger than under the current climate, mean-ing that the reservoir must be depleted at little bit further before thespring freshet. In the summer and the fall, lower rainfalls leadto a steady decline in the storage levels. The reapportion of thisreservoir is an example of nonstructural adaptation measure tomitigate the impacts of a changing flow regime (for a givenemission scenario).

Conclusion

This paper examines the issue of assessing the impact of climatechange in a river basin (target river basin) for which hydrologicprojections have not yet been calculated but are available in neigh-bouring river basins (source river basins). A methodology ispresented to determine whether or not the extrapolation of the pro-jections already available can be used as a substitute to the verticalapproach to climate change assessment. The methodology com-pares the performance of a hydropower system with the streamflowseries derived either directly from a vertical approach or indirectlyby extrapolating the projections available outside the system of in-terest. To account for the structural hydrologic uncertainty inherentto the system, a multimodel approach is implemented within thevertical chain of hydro-climate models. To account of the naturaluncertainty, a stochastic water resource model is developed todetermine the optimal allocation policies using either the vertically-obtained or the extrapolated river discharges. Different extrapola-tion techniques are tested and the performance of the systemcompared to the benchmark, i.e., the statistical distribution ofenergy generation obtained with the vertical chain of models. Ahydropower system in Quebec, Canada, is used as a case studyand the analysis is done for a particular emission scenario.

The analysis of simulation results reveals that the statistical dis-tribution of energy generation can already be well approachedusing an extrapolation technique based on linear regressions.Assuming that current linear relationships between weekly riverdischarges in both basins will not be affected by climate change,simple linear regressions can be used to derive future discharges inthe target river basin from the projected discharges in the sourceriver basin. Annual and weekly productions are within their respec-tive structural uncertainty ranges, indicating that the extrapolationprovides a reasonable approximation of the amount of energy thatcan be expected from the system. This is probably particular to thecase study and cannot be generalized to the rest of Quebec (andbeyond). However, similar tests should be carried out to identifywhat extrapolation technique (assumption) is best for each hydro-logic region in Quebec. Then, the results of the vertical approach

10 20 30 40 500

20

40

60

80

Week

Ene

rgy

(GW

h)

Structural uncertainty

Horizontal

Fig. 7. Energy generation—weekly averages

5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

Week

Ene

rgy

(GW

h)

Structural uncertaintyHorizontal

Fig. 8. Energy generation—weekly standard deviations

4 8 12 16 20 24 28 32 36 40 44 48 52

500

1000

1500

2000

2500

3000

Sto

rage

[hm

3 ]

Expected storage value

Weeks

Current Condition

Climate change Horizontal app

Min Storage

Max Storge

Fig. 9. Baskatong drawdown-refill cycle—horizontal approach versus current conditions

© ASCE 04016081-8 J. Water Resour. Plann. Manage.

J. Water Resour. Plann. Manage., 04016081

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TE

LA

VA

L o

n 11

/14/

16. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

68

implemented on a particular system could be extrapolated to therest of the hydrologic region encompassing that system.

Appendix. DDP and MPAR(p)

The built-in hydrologic model found is SDDP is a multisite, multi-period autoregressive model with cross-correlated residuals

�

qtðjÞ − μtðjÞσtðjÞ

�

¼X

Pm

i¼1

ϕi;tðjÞ�

qt−iðjÞ − μt−iðjÞσt−iðjÞ

�

þ ϵtðjÞ ð17Þ

where j = site’s index (j ¼ 1; 2; : : : ; J); qt = river discharge at timet (t ¼ 1,2; : : : ; 52 weeks); μt and σt = periodic mean and standarddeviation; ϕi;tðjÞ = lag-i autoregressive parameter at site j and timet; and ξt = time independent stochastic noise with mean zero andperiodic variance σ2

ϵ;tðjÞ. Eq. (17) is compatible with the Benders’decomposition scheme on which SDDP relies to mitigate the curseof dimensionality associate with DP formulations.

The set of parameters to be determined in the model are μtðjÞ,σtðjÞ, ϕi;tðjÞ, and σ2

ϵ;tðjÞ. Given N years of historical (reference) orprojected (future) incremental inflow data qtðjÞ available at site j,the parameters μtðjÞ, σtðjÞ can be estimated by the first and secondmoment as

μqðjÞ ¼1

N

X

N

v¼1

qv;tðjÞσ2t ðjÞ ¼

1

N − 1

X

N

v¼1

½qv;tðjÞ − μtðjÞ�2 ð18Þ

The temporal Lag-k autocorrelation ρtðkÞ between qt andqt−k for each season t is obtained by multiplying Eq. (17) byf½qt−kðjÞ − μt−kðjÞ�=σt−kðjÞg and taking the expectation termby term

ρtðkÞ ¼ ϕ1;tðjÞρt−1jk − 1j þ ϕ2;tðjÞρt−2jk − 2jþ ϕPm;t

ðjÞρt−Pmjk−Pmj ð19Þ

Eq. (19) constitutes a set of Pm linear equations, fork ¼ 1; 2; : : : ;Pm

ρtð1Þ¼ϕ1;tρt−1ð0Þþϕ2;tρt−2ð1Þþ : : :þϕPm;tρt−Pm

ð1−PmÞρtð2Þϕ1;tρt−1ð1Þþϕ2;tρt−2ð0Þþ : : :þϕPm;t

ρt−Pmð2−PmÞ

..

.

ρtðpmÞ¼ϕt;1ρt−1ðPm−1Þþϕt;2ρt−2ðPm−2Þþ : : :þϕt;Pmρt−Pm

ð0Þð20Þ

which may be written as

Pq;tðjÞ · ϕi;tðjÞ ¼ ρi;tðjÞ ð21Þ

where Pq;tðjÞ = periodic covariance matrix given as

2

6

6

6

6

4

1 ρt−1ð1Þ : : : ρt−1ðPm − 1Þρt−1ð1Þ 1 : : : ρt−2ðPm − 2Þ

..

. ... ..

.

ρt−1ðPm − 1Þ ρt−2ðPm − 2Þ : : : 1

3

7

7

7

7

5

and ϕi;tðjÞ and ρi;tðjÞ, respectively, are the vectors of autoregressivecoefficients of the MPAR(P) and autocorrelation coefficients.

ϕi;tðjÞ ¼

2

6

6

6

6

6

6

6

6

4

ϕ1;t

ϕ2;t

ϕ3;t

..

.

ϕPm;t

3

7

7

7

7

7

7

7

7

5

ρi;tðjÞ ¼

2

6

6

6

6

6

6

6

6

4

ρtð1Þρtð2Þρtð3Þ...

ρtðPmÞ

3

7

7

7

7

7

7

7

7

5

According to Salas et al. (1980) the residual variance for eacht, σ2

ϵ;tðjÞ can be estimated as a function of the periodic autore-gressive coefficients ϕi;tðjÞ and the periodic autocorrelationcoefficients ρi;tðjÞ

σ2ϵ;tðjÞ ¼ 1 −

X

Pm

j¼1

ϕj;tðjÞρj;tðjÞ ð22Þ

When the hydrological state variable in SDDP is denoted bythe previous p inflows the one-stage optimization problem canbe written as

Ftðst; qt−1; qt−2; :::; qt−PmÞ ¼ maxfαtbtðst; qt; rtÞ þ Ftþ1g ð23Þ

subject to

stþ1 þ rt þ lt ¼ st þ qt ð24Þ

Ftþ1 − φltþ1stþ1 ≤ γltþ1,1qt þ γltþ1,2qt−1 þ : : : þ γltþ1;Pm

qtþ1−Pmþ βl

tþ1

..

.

Ftþ1 − φLtþ1stþ1 ≤ γLtþ1,1qt þ γLtþ1,2qt−1 þ : : : þ γLtþ1;Pm

qtþ1−Pmþ βL

tþ1 ð25Þ

The parameters φtþ1; γtþ1,1; γtþ1,2; : : : ; γtþ1;Pmand βtþ1 of the

linear segments of Ftþ1 can be calculated from the primal and the

dual information available at the optimal solution of the one stage

optimization problem. According to the Kuhn-Tucker conditions

for optimality the derivative of the objective function with respectto the state variables x is given by

∂F

∂xi¼

X

λi

∂gi

∂xið26Þ

© ASCE 04016081-9 J. Water Resour. Plann. Manage.

J. Water Resour. Plann. Manage., 04016081

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TE

LA

VA

L o

n 11

/14/

16. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

69

where λi = dual information of the optimization problem; and gi =linear constraintsin Eqs. (24) and (25).

∂Ft

∂xi¼ λw;t

∂

∂xiðstþ1 þ rt þ lt − st − qtÞ

þX

L

l¼1

λlc;t

∂

∂xiðFtþ1 − φl

tþ1stþ1 − γltþ1,1qkt

− γltþ1,2qt−1− γltþ1,3qt−2 − : : : − γltþ1;Pmqtþ1−Pm

− βltþ1Þð27Þ

By taking the partial derivative with respect to the state variablesst; qt−1; qt−2; : : : ; qt−Pm

, the change of the original function Ft

respect the state variable associated to the constraints, isdetermined as

∂Ft

∂st¼ λw;t ð28Þ

∂Ft

∂qt−1¼ λw;t

∂qt

∂qt−1þX

L

l¼1

λlc;t

�

γltþ1,1∂qt

∂qt−1þ γltþ1,2

∂qt−1

∂qt−1

�

∂Ft

∂qt−2¼ λw;t

∂qt

∂qt−2þX

L

l¼1

λlc;t

�

γltþ1,1∂qt

∂qt−2þ γltþ1,3

∂qt−2

∂qt−2

�

..

.

∂Ft

∂qt−Pm

¼ λw;t∂qt

∂qt−Pm

þX

L

l¼1

λlc;t

�

γltþ1,1∂qt

∂qt−Pm

�

ð29Þ

Now, let us say that at stage t; s°t; q°t−1; q

°t−2; : : : ; q

°t−Pm

aresampled and, in order to include the stochasticity of the problem,K vector of inflows qKt are generated by using the periodic autor-egressive model Eq. (17).

Having into account that Fkt , which will be added to the ex-

pected cost-to-go function at stage ðt − 1Þ, can be approximated by

Fkt ≤ φl;k

t s°t þ γl;kt;1q°t−1 þ γl;kt;2q

°t−2 þ : : : þ γl;kt;Pm

q°t−Pmþ βl;k

t

ð30Þ

The slopes φl;kt ; γl;kt;1; γ

l;kt;2; :::; γ

l;kt;Pm

are determined using the dualinformation of the optimal solution at stage t Eqs. (28) and (29), foreach k scenario as

∂Fkt

∂st¼ φl;k

t ðjÞ ¼ λkw;tðjÞ ð31Þ

∂FKt

∂qt−1¼ γl;kt;1ðjÞ ¼

�

λl;kw;tðjÞ þX

L

l¼1

λl;kc;tγltþ1,1ðjÞ

�

∂qt

∂qt−1þX

L

l¼1

λl;kc;tγltþ1,2ðjÞ

∂FKt

∂qt−2¼ γl;kt;2ðjÞ ¼

�

λl;kw;tðjÞ þX

L

l¼1

λl;kc;tγltþ1,1ðjÞ

�

∂qt

∂qt−2þX

L

l¼1

λl;kc;tγltþ1,3ðjÞ

..

.

∂FKt

∂qt−Pm

¼ γl;kt;PmðjÞ ¼

�

λl;kw;t þ

X

L

l¼1

λl;kc;tγ

ltþ1,1

�

∂qt

∂qt−Pm

ð32Þ

Defining Γ as

ΓtðjÞ ¼ λl;kw;tðjÞ þ

X

L

l¼1

λl;kc;tγltþ1,1ðjÞ ð33Þ

and by using Eq. (17) to find the derivatives of qt respect the hydro-logic variables, the set of Eq. (31) can be rewritten as

γl;kt;1ðjÞ ¼ ΓtðjÞσtðjÞσt−1ðjÞ

ϕt;1ðjÞ þX

L

l¼1

λl;kc;tγltþ1,2ðjÞ ð34Þ

γl;kt;2ðjÞ ¼ ΓtðjÞσtðjÞσt−2ðÞj

ϕt;2ðjÞ þX

L

l¼1

λl;kc;tþ1γ

ltþ1,3ðjÞ ð35Þ

γl;kt;PmðjÞ ¼ ΓtðjÞ

σtðjÞσt−Pm

ðjÞϕt;PmðjÞ ð36Þ

Taking the expectation over the K artificially generated flows,the slope vectors φl

t; γlt;1; γ

lt;2; : : : ; γ

lt;Pm

can be determined

φltðjÞ ¼

1

K

X

K

k¼1

λkw;tðjÞ ð37Þ

γlt;pðjÞ ¼1

K

X

K

k¼1

γl;kt;pðjÞ; p ¼ 1,2, : : : ,Pm ð38Þ

The jth element of the vector constant term is given by

βltðjÞ ¼

1

K

X

K

k¼1

Fkt − φl

tðjÞs°tðjÞ − γt;1ðjÞlq°t−1ðjÞ

− γlt;2ðjÞq°t−2ðjÞ− · · · −γlt;PmðjÞq°t−Pm

ðjÞ ð39Þ

© ASCE 04016081-10 J. Water Resour. Plann. Manage.

J. Water Resour. Plann. Manage., 04016081

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TE

LA

VA

L o

n 11

/14/

16. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

70

Acknowledgments

The authors acknowledge NSERC, Ouranos, Hydro-Quebec, andMDDELCC. We also thank the reviewers for their constructivecomments and suggestions. All SDDP-derived simulation resultscan be accessed through the U. Laval webpage of the second author(https://www.gci.ulaval.ca/departement-et-professeurs/professeurs-et-personnel/professeurs/fiche/show/tilmant-amaury/models-and-data-518/). A READ-ME.txt file describes the Matlab structurecontaining the results.

References

Botzen, W., and van der Bergh, J. (2008). “Insurance against climatechange and flooding in the Netherlands: Present, future, and compari-son with other countries.” Risk Anal., 28(2), 413–426.

Brown, C., Ghile, Y., Laverty, M., and Li, K. (2012). “Decision scaling:Linking bottom-up vulnerability analysis with climate projections inthe water sector.” Water Resour. Res., 48(9), W09537.

Brown, C., Werick, W., Leger, W., and Fay, D. (2011). “A decision analyticapproach to managing climate risks: Application to the upper greatlakes.” J. Am. Water Resour. Assoc., 47(3), 524–534.

Carpenter, T. M., and Georgakakos, K. P. (2001). “Assessment of FolsomLake response to historical and potential future climate scenarios. 1:Forecasting.” J. Hydrol., 249(1–4), 148–175.

CEHQ (Centre d’Expertise Hydrique du Québec). (2012). “Délimitationdes bassins versants correspondant aux stations hydrométriquesouvertes et fermées.” ⟨ftp://ftp.mddep.gouv.qc.ca/⟩ (Jun. 18, 2015).

Chen, J., Brissette, F. P., Poulin, A., and Leconte, R. (2011). “Overalluncertainty study of the hydrological impacts of climate change fora Canadian Watershed.” Water Resour. Res., 47(12), W12509.

Dessai, S., and Hulme, M. (2004). “Does climate adaptation policy needprobabilities?” Clim. Policy, 4(2), 107–128.

Duan, Q., Gupta, V., and Sorooshian, S. (1993). “Shuffled complexevolution approach for effective and efficient global minimization.”J. Optim. Theory Appl., 76(3), 501–521.

Fortin, J., Turcotte, R., and Massicotte, R. (2001). “Distributed watershedmodel compatible with remote sensing and GIS data. I: Description ofmodel.” J. Hydrol. Eng., 10.1061/(ASCE)1084-0699(2001)6:2(91),91–99.

Goor, Q., Kelman, R., and Tilmant, A. (2011). “Optimal multipurpose mul-tireservoir operation model with variable productivity of hydropowerplants.” J. Water Resour. Plann. Manage., 10.1061/(ASCE)WR.1943-5452.0000117, 258–267.

Harrison, G. P., and Whittington, H. W. (2002). “Susceptibility of theBatoka Gorge hydroelectric scheme to climate change.” J. Hydrol.,264(1–4), 230–241.

Iglesias, A., and Garrote, L. (2015). “Adaptation strategies for agriculturalwater management under climate change in Europe.” Agric. Water

Manage., 155, 113–124.IPCC (Intergovernmental Panel on Climate Change). (2013). “Summary

for policymakers. Climate change 2013: The physical science basis.”Contribution of Working Group I to the 5th Assessment Rep. of the

Intergovernmental Panel on Climate Change, T. F. Stocker, et al.,eds., Cambridge University Press, Cambridge, U.K.

Jeuland, M., and Whittington, D. (2014). “Water resources planning underclimate change: Assessing the robustness of real options for the blueNile.” Water Resour. Res., 50(3), 2086–2107.

Klemeš, V. (1986). “Operational testing of hydrological simulation

models.” Hydrol. Sci. J., 31(1), 13–24.

Lempert, R., and Schlesinger, M. (2000). “Robust strategies for abating

climate change.” Clim. Change, 45(3–4), 387–401.

Music, B., and Caya, D. (2007). “Evaluation of the hydrological cycle over

the Mississippi River Basin as simulated by the Canadian regional

climate model (CRCM).” J. Hydrometeorol., 8(5), 969–988.

Nash, J., and Sutcliffe, J. (1970). “River flow forecasting through

conceptual models. Part I: A discussion of principles.” J. Hydrol.,

10(3), 282–290.

New, M., Lopez, A., Dessai, S., and Wilby, R. (2007). “Challenges in using

probabilistic climate change information for impact assessments: An

example from the water sector.” Philos. Trans. R. Soc. London A,

365(1857), 2117–2131.

Oudin, L., Hervieu, F., Michel, C., Perrin, C., Andréassian, V., Anctil, F.,

and Loumagne, C. (2005). “Which potential evapotranspiration input

for a lumped rainfallrunoff model? Part 2—Towards a simple and effi-

cient potential evapotranspiration model for rainfall-runoff modelling.”

J. Hydrol., 303(1–4), 290–306.

Perrin, C. (2000). “Vers une amélioration dun modèle global pluie-debit.”

Ph.D. thesis, Institut National Polytechnique de Grenoble-INPG,

Grenoble, France.

Prudhomme, C., Wilby, R., Crooks, S., Kay, A., and Reynard, N. (2010).

“Scenario-neutral approach to climate change impact studies: Applica-

tion to flood risk.” J. Hydrol., 390(3–4), 198–209.

Salas, J., Delleur, J., Yevjevich, V., and Lane, W. (1980). Applied modeling

of hydrologic time series, Water Resources Publications, Littleton, CO.

Seiller, G., Anctil, F., and Perrin, C. (2012). “Multimodel evaluation

of twenty lumped hydrological models under contrasted climate

conditions.” Hydrol. Earth Syst. Sci., 16(4), 1171–1189.

Thiboult, A., and Anctil, F. (2015). “Assessment of a multimodel ensemble

against an operational hydrological forecasting system.” Can. Water

Resour. J., 40(3), 272–284.

Tilmant, A., Pinte, D., and Goor, Q. (2008). “Assessing marginal water

values in multipurpose multireservoir systems via stochastic

programming.” Water Resour. Res., 44(12), W12431.

Turcotte, R., Lacombe, P., Dimnik, C., and Villeneuve, J.-P. (2004).

“Prévision hydrologique distribuée pour la gestion des barrages publics

du Québec.” Can. J. Civ. Eng., 31(2), 308–320.

Valéry, A., Andréassian, V., and Perrin, C. (2014). “Catchments, ‘As simple

as possible but not simpler’: What is useful in a temperature-based

snow-accounting routine? Part 2—Sensitivity analysis of the Cemaneige

snow accounting routine on 380.” J. Hydrol., 517, 1176–1187.

Vicuna, S., Dracup, J. A., Lund, J. R., Dale, L. L., and Maurer, E. P. (2009).

“Basin scale water system operations with uncertain future climate

conditions: Methodology and case studies.” Water Resour. Res.,

46(4), 1–19.

Weaver, C. P., Lempert, R. J., Brown, C., Hall, J. A., Revell, D., and

Sarewitz, D. (2013). “Improving the contribution of climate model

information to decision making: The value and demands of robust

decision frameworks.” Wiley Interdiscip. Rev. Clim. Change, 4(1),

39–60.

Wilby, R., and Dessai, S. (2010). “Robust adaptation to climate change.”

Weather, 65(7), 180–185.

Yao, H., and Georgakakos, A. (2001). “Assessment of Folsom Lake

response to historical and potential future climate scenarios. 2:

Reservoir management.” J. Hydrol., 249(1–4), 176–196.

© ASCE 04016081-11 J. Water Resour. Plann. Manage.

J. Water Resour. Plann. Manage., 04016081

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TE

LA

VA

L o

n 11

/14/

16. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

71

Appendix B

Paper II:Optimizing multireservoirsystem operating policies using exogenoushydrologic variables

Résumé

La programmation dynamique stochastique duale (SDDP) fait partie des quelques algorithmes disponibles

pour optimiser les politiques de gestion des systèmes hydroélectriques de grande taille. Ce manuscrit

présente une nouvelle extension, appelée SDDPX, dans laquelle des variables hydrologiques exogènes

telles que l’équivalent de neige en eau et/ou la température de la surface de la mer, sont incluses dans

le vecteur d’état avec les variables traditionnelles (endogènes), comme les débits des pas de temps

précédents. Avec cette extension, les décisions de relâche d’eau sont maintenant fonction des stocks,

des débits des pas de temps précédents et des variables exogènes pertinentes à même de représenter

des processus hydrologiques plus complexes que ceux des formulations SDDP traditionnelles. Pour

illustrer le gain potentiel associé à l’utilisation de variables exogènes pour la gestion d’un système à

réservoir multiples, le système hydroélectrique de Rio Tinto (RT) est utilisé comme cas d’étude. Il est

situé dans le bassin du fleuve Saguenay-Lac-St-Jean au Québec (Canada), et dispose d’une capacité

installée de 3137 MW. La performance du système est évaluée pour différentes combinaisons de vari-

ables d’état hydrologiques, allant du modèle autorégressif d’ordre 1 à une formulation plus complexe

impliquant les débits des pas de temps précédents, l’équivalent de neige en eau et les précipitations

hivernales. Une procédure de ré-optimisation est implémentée dans laquelle les fonctions de béné-

fices futurs dérivées de SDDPX sont utilisées au sein d’une simulation réalisée sur l’historique des

variables hydrologiques endogènes et exogènes.

http://onlinelibrary.wiley.com/doi/10.1002/2017WR021701/abstract.

72

RESEARCH ARTICLE10.1002/2017WR021701

Optimizing Multireservoir System Operating Policies UsingExogenous Hydrologic VariablesJasson Pina1, Amaury Tilmant1 , and Pascal Cot�e2

1Department of Civil Engineering and Water Engineering, Universit�e Laval, QC, Canada, 2Rio Tinto, Quebec PowerOperation, Jonqui�ere, QC, Canada

Abstract Stochastic dual dynamic programming (SDDP) is one of the few available algorithms tooptimize the operating policies of large-scale hydropower systems. This paper presents a variant, calledSDDPX, in which exogenous hydrologic variables, such as snow water equivalent and/or sea surfacetemperature, are included in the state space vector together with the traditional (endogenous) variables,i.e., past inflows. A reoptimization procedure is also proposed in which SDDPX-derived benefit-to-gofunctions are employed within a simulation carried out over the historical record of both the endogenousand exogenous hydrologic variables. In SDDPX, release policies are now a function of storages, past inflows,and relevant exogenous variables that potentially capture more complex hydrological processes than thosefound in traditional SDDP formulations. To illustrate the potential gain associated with the use of exogenousvariables when operating a multireservoir system, the 3,137 MW hydropower system of Rio Tinto (RT)located in the Saguenay-Lac-St-Jean River Basin in Quebec (Canada) is used as a case study. Theperformance of the system is assessed for various combinations of hydrologic state variables, ranging fromthe simple lag-one autoregressive model to more complex formulations involving past inflows, snow waterequivalent, and winter precipitation.

1. Introduction

The operation of a multireservoir system is a complex, multistage, stochastic decision-making probleminvolving, among others, (i) a trade-off between immediate and future consequences of a release decision,(ii) considerable risks and uncertainties, and (iii) multiple objectives and operational constraints (Oliveira &Loucks, 1997). For hydropower systems, the problem can also be nonlinear because the production ofhydroelectricity depends on the product of the outflow and the head (storage). This decision-making prob-lem has been studied for several decades and state-of-the-art reviews can be found in Yeh (1985), Labadie(2004), Rani and Moreira (2009)], and more recently in Ahmad et al. (2014).

Dynamic programming (DP), first introduced by Bellman (1957), has been one of the most popular optimiza-tion techniques to determine reservoir operating policies. The method solves the problem by decomposingthe multistage decision-making problem into simpler one-stage problems, which are then solved recur-sively. DP can be easily expanded to accommodate the stochasticity of the hydrologic input by adding ahydrologic variable to the state space vector. The resulting stochastic DP (SDP) formulation, often referredto as a Markov decision process, explicitly considers the streamflow lag-1 correlation found in the flowrecords. SDP solves the problem by discretizing stochastic variables, as well as the system status, to obtainan optimal solution for each discrete value of the state space that characterizes the system (Tejada-Guibertet al., 1995). Although conceptually attractive, SDP is however limited by the so-called curse of dimensional-ity, which limits its application to small systems involving no more than four state variables (storage andhydrologic).

Incorporating more hydrologic information in the state vector has the potential to enhance SDP-derivedpolicies and thus improve the performance of the system. Various approaches have been proposed the liter-ature to address this challenge. For example, Bras et al. (1983) combine real time forecasts with an adaptivecontrol technique in SDP to update flow transition probabilities. Stedinger et al. (1984) develop an SDPmodel which employs the best inflow forecast for the current period’s flow to define the policy. Kelmanet al. (1990) propose sampling SDP (SSDP) to better capture the complex temporal and spatial structures of

Key Points:� The SDDP algorithm can handle

multiple endogenous and exogenoushydrologic state variables� Gains in energy production can be

observed when more hydrologicvariables are included in the statespace vector� The marginal water values tend to

increase while spillage losses arereduced regardless of the hydrologicstatus of the system

Correspondence to:A. Tilmant,amaury.tilmant@gci.ulaval.ca

Citation:Pina, J., Tilmant, A., & Cot�e, P. (2017).Optimizing multireservoir systemoperating policies using exogenoushydrologic variables. Water ResourcesResearch, 53. https://doi.org/10.1002/2017WR021701

Received 19 AUG 2017

Accepted 10 NOV 2017

Accepted article online 15 NOV 2017

VC 2017. American Geophysical Union.

All Rights Reserved.

PINA ET AL. SDDPX EXOGENOUS VARIABLES 1

Water Resources Research

PUBLICATIONS

73

the streamflow process. Karamouz and Vasiliadis (1992) propose another alternative to SDP, coinedBayesian SDP (BSDP), in which prior flow transition probabilities are regularly updated using the Bayes’theorem. Tejada-Guibert et al. (1995) determine the value of hydrological information for several SDPformulations each employing a different set of hydrologic state variables and, similarly, Kim and Palmer(1997) illustrate the potential advantage of using the seasonal flow forecasts and other hydrologic informa-tion by comparing the performance of the BSDP against alternative SDP formulations. Furthermore, Faberand Stedinger (2001) and Kim et al. (2007) examine the use Ensemble Streamflow Prediction (ESP) withinthe SSDP optimization framework. Cot�e et al. (2011) incorporate a new hydrologic state variable in SSDP asa linear combination of snow water equivalent and soil moisture, and more recently, Desreumaux et al.(2014) present the effect of using various hydrological variables on SDP-derived policies of the Kemanohydropower system in British Columbia. Another active research line focuses on the incorporation of climatevariables such as the El Nino Southern Oscillation (ENSO) and Pacific Decadal Oscillation (PDO) climate sig-nals into reservoir operation models (Gelati et al., 2013; Hamlet & Lettenmaier, 1999; Kwon et al., 2009).However, most of the above-mentioned studies are limited to small-scale problems, meaning that a trade-off must be found between the complexity of system to be studied (number of reservoirs) and the complex-ity of the hydrologic processes that can be captured.

This trade-off between system and hydrologic complexities can largely be removed using stochastic dualDP (SDDP), an extension of SDP that is not affected by the curse of dimensionality (Pereira & Pinto, 1991).To achieve this, SDDP builds a locally accurate approximation of the expected-benefit-to-go functionsthrough piecewise linear functions. With such an approximation, there is no need to evaluate the functionover a dense grid; the benefits can now be derived from extrapolation and not interpolation as in SDP. Aswe will see later, these piecewise linear functions are constructed from the primal and dual solutions of theone-stage optimization problem and can be interpreted as Benders cuts in a stochastic, multistage decom-position algorithm. To increase the accuracy of the approximation, SDDP uses an iterative procedure inwhich new cuts are added to the most interesting region of the state space until the approximation is statis-tically acceptable. As explained in Tilmant and Kelman (2007), to implement the decomposition scheme,the one-stage optimization problem must be formulated as a convex problem, such as a linear program.SDDP has largely been used in hydropower-dominated systems such as Norway (Gjelsvik et al., 2010; Moet al., 2001; Rotting & Gjelsvik, 1992), South and Central America (Homen-de Mello et al., 2011; Pereira, 1989;Shapiro et al., 2013), and New Zealand (Kristiansen, 2004). The SDDP algorithm constitutes the coreof generic hydro-economic models that have been used to analyze a variety of policy issues in theEuphrates-Tigris River basin (Tilmant et al., 2008), the Nile River basin (Goor et al., 2011), the Zambezi Riverbasin (Tilmant & Kinzelbach, 2012), or in Spain (Macian-Sorribes et al., 2016; Pereira-Cardenal et al., 2016).

In SDDP, the hydrologic uncertainty is typically captured through a multisite periodic autoregressive modelMPAR(p). This model is capable of representing seasonality, serial, and spatial streamflow correlations withina river basin and among different basins. It is also needed to analytically derive some of the parameters ofthe linear segments used to approximate the benefit-to-go functions, and to produce synthetic streamflowsscenarios for the simulation phase of this iterative algorithm. Furthermore, the convexity requirement ofSDDP is guaranteed because the MPAR(p) is linear. Recent works reveal an interest in improving the built-inhydrological model. In Lohmann et al. (2015), a new approach to include spatial information is presented.Pritchard (2015) models inflows as a continuous process with a discrete random innovation. Poorsepahy-Samian et al. (2016) propose a methodology to estimate the cuts parameters when a Box-Cox transforma-tion is used to normalize inflows, and more recently, Raso et al. (2017) present a streamflow model with amultiplicative stochastic component and a nonuniform time step.

This study focuses on the incorporation of new hydrologic information into the SDDP algorithm. This newinformation, encapsulated as additional, exogenous, hydrologic state variables through a built-in MPARXmodel, aims at better capturing the hydrologic processes responsible for reservoir inflows. By definition,reservoir inflows (the endogenous variables) are causal dependent on the exogenous hydrologic state varia-bles while the opposite is not true. For example, winter snow pack, precipitation, or sea surface temperatureare exogenous to river discharges. Incorporating exogenous hydrologic state variables into SDDP requiresthat some aspects of the iterative algorithm be modified. Particular attention is devoted to the new mathe-matical formulation of the cuts approximating the benefit-to-go functions, and how the correspondingparameters can be analytically derived from the primal and dual information that become available as the

Water Resources Research 10.1002/2017WR021701

PINA ET AL. SDDPX EXOGENOUS VARIABLES 2

74

algorithm progresses backward (backward optimization phase). In the forward simulation phase, due to thestructure of the built-in hydrologic model (MPARX), the number of hydrologic sequences is now limited tothe length of hydrological records/direct measurements.

Following the temporal decomposition approach Zahraie and Karamouz (2004), the proposed SDDPX for-mulation solves the midterm hydropower scheduling problem and therefore seek to properly capture themidterm to long-term hydrologic uncertainties. Since the nonlinear nonconvex hydropower function can-not be used as such in the SDDP algorithm, usually simplifications of the function are used. We can rely, forexample, on a production coefficient (Archibald et al., 1999), convex hull approximations (Goor et al., 2011),McCormick envelopes (Cerisola et al., 2012), or a concave approximation (Zhao et al., 2014). To deal withthe nonconvexity of the optimization problem, the hydropower production functions are approximated byconvex hulls. This new formulation is illustrated with the 3,137 MW hydropower system of Rio Tinto (RT)located in the Saguenay-Lac-St-Jean River Basin in Quebec (Canada). The five hydropower plants have thecapacity to produce approximately 90% of the electricity required for the production of aluminum, forcingRT to purchase energy to fully ensure the production of the mineral. Therefore, a joint optimization of physi-cal (power plants) and financial (contracts) assets is developed using SDDPX, and a comparative analysis ofthe performance of the system with various combinations of endogenous and exogenous hydrologic statevariables is performed.

The paper is organized as follows: section 2 starts with a presentation of the reservoir operation problem,which is then followed by a description of the SDDP algorithm and its variant SDDPX with exogenoushydrologic variables. This section ends with a presentation of the case study. Afterward, optimization resultsare discussed in section 3. Finally, concluding remarks are given in section 4.

2. Materials and Methods

2.1. The Hydroelectric Reservoir Operation ProblemThe operation of a multireservoir system is a multistage decision-making problem. When framed as an opti-mization problem, the goal is to determine a sequence of optimal decisions xt that maximizes the expectedsum of net benefits from system operation Z over a given planning period. Let T be the number of stages inthe planning period, btð�Þ be the net benefit function at stage t, mð�Þ be the terminal value function, E½�� bethe expectation operator, and St be a vector of state variables characterizing the system at the beginning ofstage t, the objective function can be written as:

Z5EXT

t51

btðSt; xtÞ1mðST11Þ" #

(1)

This objective function will be maximized to the extent made possible by operational and/or institutionalconstraints affecting the state and decision variables.

2.2. One-Stage SDDP ProblemSDDP solves the optimization problem (1) by decomposing it into a sequence of one-stage problems thatare solved recursively. Let us first adopt the same notation as in Tilmant et al. (2008) and say that the waterresources system is represented by a network with J nodes (e.g., reservoir, power plant). Imagine that theobjective is to maximize the net benefits associated to the production of hydroelectricity. The immediatebenefit function btð�Þ includes the net benefits from hydropower generation and penalties for not meetingtarget demands and/or violating constraints is expressed as

btðSt; xtÞ5XJ

j51

Psys;tðjÞstðptðjÞ2htðjÞÞ� �

�2n0t zt (2)

where st is the number of hours in period t, Psys;t is the vector of power generated (MW), p is the vector ofenergy price ($/MWh), h is the vector of the operation and maintenance cost ($/MWh), and zt is the vectorof deficits or surpluses (e.g., energy demand, environmental flows) penalized by the vector n0t of penalties($/unit). Sales and purchases are included in vector zt and the penalty is the price of energy specified in thecorresponding contract ($/MWh).

Water Resources Research 10.1002/2017WR021701

PINA ET AL. SDDPX EXOGENOUS VARIABLES 3

75

At stage t, the system status St is described by the vector of storage st , the hydrological state variable ht ,and the amount left of energy wt in C contracts. The one-stage SDDP optimization problem is expressedas:

Ftðst; ht;wtÞ5maxfbtðst; ht;wt; xtÞ1Ft11g (3)

In SDDP, the benefit-to-go function is represented by the scalar Ft11 which is bounded by a locally accuratelinear approximation. The stage to stage transformation function corresponds to the mass balance equationinvolving the vector of spillage losses lt , the vector of inflows qt , the vector of the turbined outflows rt , thevector of evaporation losses et , and the connectivity matrix CMR

st112CMRðrt1ltÞ5st1qt2etðstÞ (4)

CMRðj; kÞ5 1(21) when reservoir j receives(releases) water from(to) reservoir k.

As mentioned earlier, the benefit-to-go function Ft11, which is represented by a scalar in equation (3), isbounded from above by inequality constraints:

Ft112ult11st112vl

t11wt11 � Clt11ht111bl

t11

�

Ft112uLt11st112vL

t11wt11 � CLt11ht111bL

t11

8>><>>: (5)

where L is the number of cuts. The parameters ut11; vt11; bt11, and Ct11 must have been calculatedfrom the primal and the dual information available at the optimal solution of the one-stage optimiza-tion problem at the stage t 1 1 (Tilmant et al., 2008). In previous SDDP application to water resourcessystems, the hydrological variables are the natural inflows observed during the last p periods htðjÞ5½qt21ðjÞ; qt22ðjÞ; . . . ; qt2pðjÞ� and the current inflow is described by a multisite periodic autoregressivemodel MPAR(p):

qtðjÞ2lqtðjÞ

rqt ðjÞ5Xp

i51

/i;tðjÞqt2iðjÞ2lqt2i

ðjÞrqt2i ðjÞ

� �1�tðjÞ (6)

where lqtand rqt are, respectively, the vectors of periodic mean and standard deviation of qt at period t,

/i;t is the vector of autoregressive coefficients, and �t is the time dependent and spatially correlated sto-chastic noise of zero mean and variance r2

�;t .

The nonlinear hydropower production Psys;t (MW), defined as the product of the net head ht (m), the releasert (m3=s), the turbines/generators efficiency g, and the specific weight of water cw (MN/m3):

Psys;t5cw � gðst; st11; rtÞ � hðst; st11Þ � rt (7)

is approximated by convex hulls and stored in the constraints set (8):

Psys;t2w1st11=22x1rt � w1st=21d1

�

Psys;t2wHst11=22xHrt � wHst=21dH

8>><>>: (8)

where Psys;t is the approximated power, H is the number of linear segments, w, x, and d are the vectors ofparameters determined according to the procedure described in Goor et al. (2011).

The load Dt (MWh) must be met with the energy produced by the system Psys;t and the sales/purchasesthrough the contracts: X

J

Psys;tðjÞst1X

C

utðcÞ1Pp;tDt2Ps;tDt5Dt (9)

Energy transactions can be handled by (i) instant power contracts in which a given amount of power Pp;t

can be bought and surpluses be sold Ps;t during time period Dt, and (ii) purchase energy contracts wtðcÞ. Inthe latter, the amount of energy utðcÞ5Pw;tst that can be withdrawn from the contracts follows the energybalance equation:

Water Resources Research 10.1002/2017WR021701

PINA ET AL. SDDPX EXOGENOUS VARIABLES 4

76

wt11ðcÞ1utðcÞ5wtðcÞ (10)

and it is limited by the minimum and maximum instant power withdrawn Pw;t

Pminw st � ut � Pmax

w st (11)

2.3. Incorporating Exogenous Hydrologic Variables into SDDPIncorporating exogenous hydrologic variables into the state space vector of SDDP offers the potential toimprove the performance of SDDP-derived release policies. Using p previous inflows qt and b past exoge-nous variables Xt, the hydrological state variable ht becomes ht5½qt21; qt22,. . .,qt2p; Xt2j,. . ., Xt2b�. Withthese variables, the incremental flow at node j, qtðjÞ, can be derived from a multisite periodic autoregressivemodel with exogenous variables MPARX(p,b) (Ljung, 1999):

qtðjÞ2lqtðjÞ

rqt ðjÞ5Xp

i51

/i;tðjÞqt2iðjÞ2lqt2i

ðjÞrqt2i ðjÞ

� �1Xb

j5i

#j;tðjÞXt2jðjÞ2lXt2j

ðjÞrXt2jðjÞ

� �1�tðjÞ (12)

where lXtand rXt are, respectively, the vectors of the periodic mean and the standard deviation of the

exogenous variables and #j;t is the vector of the exogenous regressors. As indicated in equation (12), theexogenous variables may cover a different range of past input values, from i to b, not necessarily startingfrom t 2 1.

The main modification to the traditional SDDP formulation lies in the calculation of the hyperplanes’ param-eters ut11; vt11; bt11, and Ct11 (see equation (5)). In particular, Ct11 is the vector of linear parameters[ct11;1; ct11;2,. . ., ct11;p; ct11;p1j,. . ., ct11;p1b] associated to ht11 5 ½qt; qt21,. . .,qðt2pÞ11; Xðt2jÞ11,. . ., Xðt2bÞ11�.The procedure to analytically derive the hyperplanes’ parameters when exogenous hydrologic variables areadded to the state space vector is described in Appendix A. This procedure is implemented in the backwardoptimization phase of the SDDP algorithm. The accuracy of the piecewise linear approximation of Ft11 isthen evaluated in a forward simulation phase and if they fail to pass the test, a new backward optimizationphase is implemented (otherwise the algorithm stops). At each iteration, new hyperplanes are added to theconstraints set, refining the approximation of Ft11.

Both phases require different sets of inflows. In the backward phase, K inflows scenarios (backward open-ings) at each node of the system are generated using the MPARX(p,b). Actually, as explained in Appendix A,these scenarios are needed to analytically calculate the hyperplanes’ parameters, and ultimately to derivethe upper bound to the true expected benefit-to-go function. In the forward phase, two different optionsexist to generate the M hydrologic sequences required to simulate the system: (i) one can use theMPARX(p,b) to generate synthetic streamflow sequences based on historical records of both endogenousand exogenous hydrologic variables, (ii) or one can rather rely on series generated outside of SDDPX usingany relevant hydrologic model. Hence, in contrast to the MPAR model used in SDDP, the built-in hydrologicmodel (MPARX) can no longer be used to generate any number of streamflow sequences because the exog-enous variables are independent from river discharges. This might become a limitation if limited hydrologicdata are available.

2.4. Case StudyThe hydroelectric system of Rio Tinto located in the Saguenay-Lac-Saint Jean River Basin, Quebec (Canada)is used as a case study. It includes four reservoirs and five hydropower plants: three on the P�eribonca Riverand two on the Saguenay River. The drainage area is about 78,000 km2. In the northern part of the basin,there are two large reservoirs: Manouane and Passes-Dangereuses with a storage capacity of 2.7 and5.2 km3, respectively. The downstream reservoir system, in which the reservoir Lac-Saint-Jean is included,drains the Basins of the P�eribonka River, Ashuapmushuan River, Mistassini River, and Mistassibi River. Reser-voir Lac-St-jean has important recreational and sport-fishing industries which highly constrain the storagelevels during summer and autumn seasons. Figure 1 shows the reservoir system configuration of the RThydroelectric network and Table 1 lists the main characteristics of the system.

These installations can generate more than 17 TWh/yr, which is roughly 90% of the electricity required forthe production of aluminum, forcing RT to buy energy to fully ensure the production of the mineral. Thereare two contracts (C 5 2): one yearly energy contract and one contract available at the end of the winterseason.

Water Resources Research 10.1002/2017WR021701

PINA ET AL. SDDPX EXOGENOUS VARIABLES 5

77

The 53 year hydrologic and climatic series (at nodes 1–5) provided by the Quebec Power Operation Depart-ment of RT are used to parameterize the multisite periodic autoregressive model with exogenous variablesMPARX(p,b) (equation (12)), which residuals �t are fitted to a three-parameter log normal distribution. Forthis comparative analysis, four SDDP formulations, each having a specific hydrologic model, are imple-mented and their performance compared. The first formulation relies on a MPAR(1) model as presented inTilmant et al. (2008). The second formulation attempts at better capturing the temporal persistence of thehydrologic processes through a MPAR(p) model (Matos & Finardi, 2012; Pina et al., 2016). Note that theorder p of the autoregressive model varies in space (node) and time (week). The third and fourth formula-tions are based on the MPARX(p,b) described above, but differ in the selected exogenous variables, whichare, respectively, the snow water equivalent SWEt and the combination of the SWEt and the accumulatedwinter precipitation Pw

t , i.e., winter precipitation from October until time t (Kim & Palmer, 1997). Table 2 liststhe main characteristics of the alternative SDDP formulations.

Figure 1. Rio Tinto hydropower system.

Table 1Rio Tinto Hydropower System Characteristics

Node Id Name Storage (hm3) Capacity (MW)

1 RLM Reservoir Lac-Manouane 2,6572 RPD Reservoir Passes-Dangereuses 5,228 8443 RCD Reservoir Chute du Diable 396 2354 CCS Chute-�a-la-Savane RORa 2505 RLSJ Reservoir Lac-St-Jean 5,083 4546 SH Shipshaw ROR 1,354

aROR: run of the river power plant.

Water Resources Research 10.1002/2017WR021701

PINA ET AL. SDDPX EXOGENOUS VARIABLES 6

78

3. Analysis of Simulation Results

All SDDP formulations are implemented on exactly the same hydrocli-matic data: K 5 40 backward openings, M 5 40 hydrological and cli-matic sequences over a planning period of 260 weeks (T 5 260 or5 years). The results are analyzed after reoptimizing the polices alongthe 53 year historical hydrologic sequences. To achieve this, theweekly SDDP-derived benefit-to-go functions of the third year (Ft11;t 5 105, 106,. . ., 157) are used to determine the corresponding releasedecisions by maximizing current plus expected future benefits, sub-ject to system constraints. Retrieving the cuts of the third year is moti-

vated by the fact that the impacts of the boundary conditions (initial storages and zero terminal valuefunctions) on the benefit-to-go functions are minimal when a time period of 2 years is used as a buffer zone(years 1 and 2 for the initial conditions and years 4 and 5 for the terminal conditions). When dealing withsystems with larger carry-over storage capacity, it might be needed to increase the length of the bufferzone by increasing the length of the planning period (e.g., from 5 to 10 years).

To better perceive the advantage of incorporating more hydrologic state variables, the problem is solvedfor two different configurations based on the RT power system. The first system configuration is a simplified,hypothetical, system with a single objective: the maximization of hydropower generation. Important sec-ondary objectives like recreation and flood control as well as the energy load and the contracts are ignoredin order to dedicate all the flexibility offered by the reservoirs to the production of energy. These secondaryobjectives are included in the second configuration which is the actual model of the system. Here the opti-mization is performed on both the physical (power plants, reservoirs) and financial assets (portfolio ofcontracts).

Using SDDP(1) as a benchmark, Table 3 lists, for both configurations, the mean annual reduction in spillagelosses and the annual energy gains one can expect when more hydrologic state variables (endogenous andexogenous) are added to the state space vector. For the second configuration, Table 3 also shows thereduction in energy purchases.

3.1. Maximization of Hydropower GenerationAs we can see in Table 3, increasing the number of lags from 1 to p in the first configuration alreadyincreases the amount of energy produced by 54.41 GWh. Incorporating exogenous variables furtherincreases the production of energy by 46.3 and 50.9 GWh depending on whether SWE or the combinationof SWE and winter precipitation are used. Assuming an average market price of 45 US$/MWh (New YorkIndependent System Operator, 2017), these gains correspond to a 2.45–4.70 million US$ increase in annualenergy value. These energy gains are made possible by improved operating policies that better exploit thestorage capacity of the system.

Figure 2 presents the drawdown-refill cycle of the two largest reservoirs of the system, Passes-DangereusesReservoir (RPD) and Lac-St-Jean Reservoir (RLSJ). The simulated trajectories reveal how the incorporation ofthe exogenous hydrologic variables affects the operating policies during the winter, and the extent towhich the reservoirs must be depleted before the spring snowmelt. In this power system, a large portion ofthe energy comes from run-of-river power stations that are prone to spilling. Consequently, lowering the

Table 2SDDP Formulations

MPARX(p,b)

Formulation p b X

SDDP(1) 1SDDP(p) pSDDPX(p, SWE) p 1 SWESDDPX(p; SWE; Pw ) p 1 SWE; Pw

Table 3Average Annual Results-Differences With Respect to the SDDP(1) Model

First configuration Second configuration

Model

Spillagereduction

(m3=s)

Annual gainof energy

(GWh)

Spillagereduction

(m3=s)

Annual gainof energy

(GWh)

Net purchasesreduction

(%)

SDDP(p) 15 54 10 30 4.0SDDPX(p, SWE) 35 101 26 50 6.6SDDPX(p; SWE; Pw ) 33 105 28 51 6.7

Water Resources Research 10.1002/2017WR021701

PINA ET AL. SDDPX EXOGENOUS VARIABLES 7

79

storage levels in the upstream reservoirs tend to reduce the spillage losses throughout the system, there-fore increasing the total energy output and compensating for the reduced efficiency at the storage powerplants. With better forecasts throughout the system, the upstream reservoirs are operated at lower pool ele-vation to prevent excessive spills not only at the reservoir site, but also downstream.

Since the decisions taken during the drawdown phase have consequences beyond the summer, Figure 3also explores the statistical distribution of the storage decisions for weeks 14, 15, and 16, when storage lev-els are the lowest. In week 14, for example, when the simple SDDP(1) and SDDP(p) formulations are used,the volume in storage exceeds 2 km3 100% of the time. However, when the exogenous variables areincluded in the algorithm, storage volumes exceed this capacity less than 40% of the time, meaning that 6years out of 10, the storage levels in week 14 will be lower than 2 km3. These results clearly indicate thatthe incorporation of exogenous variables yields policies that better anticipates snowmelt runoff, thereforeavoiding massive spills.

The impact of these decisions on the amount of water spilled is further analyzed in Figure 4 where we cansee the probability distributions of annual spillages (left plots) at each power plant for two SDDP formula-tions: the simple SDDP(1) and the more sophisticated SDDPX(p; SWE; Pw). With SDDPX, the spills arereduced regardless of the hydrologic conditions, and the reduction is more pronounced in RLSJ, the down-stream reservoir supplying the two largest power stations accounting for about two-thirds of the installed

Figure 2. Weekly mean storage at reservoirs Passes-Dangereuses (RPD) and Lac-St-Jean (RLSJ). First configuration.

0 2000 4000 60000

0.2

0.4

0.6

0.8

1

week14

Non

-Exc

eede

nce

prob

abili

ty [-

]

Storage Hm30 2000 4000 6000

0

0.2

0.4

0.6

0.8

1

week15

Storage Hm30 2000 4000 6000

0

0.2

0.4

0.6

0.8

1

week16

Storage Hm3

SDDP(1) SDDP(p) SDDPX(p,SWE) SDDPX(p,Fs)

Figure 3. Statistical distribution of the accumulated storages for the largest reservoirs for weeks 14, 15, and 16. Firstconfiguration.

Water Resources Research 10.1002/2017WR021701

PINA ET AL. SDDPX EXOGENOUS VARIABLES 8

80

capacity. During dry years, however, the differences between the statistical distributions become marginalbecause the SDDP formulations with and without exogenous variables generate similar policies (from adecision-making point-of-view, the exogenous variables become informationless). For the most upstreamreservoir (RPD), the large storage to inflow ratio implies that this reservoir does not spill 3 years out of 4.When spills do occur, SDDPX release decisions significantly reduce the discharges through the spillway,therefore having a positive repercussion on downstream infrastructure: the smaller reservoir (RCD) and therun-of-river power plant (CCS).

The exogenous hydrologic state variables also have an impact on the marginal water values, which corre-spond to the Lagrange multipliers associated with the mass balance equation (4). Figure 4 (right plots) com-pares the statistical distributions of the marginal water values in the system with the SDDP(1) andSDDPX(p; SWE; Pw ) formulations only (for clarity, the two intermediate formulations are not shown). As wecan see, SDDPX-derived policies yield larger water values across a wide range of hydrologic conditions. At

35 40 45 50 550

0.5

1

Marginal water values $/1000 m3

18 20 22 24 260

0.5

1

Marginal water values $/1000 m3

14 16 18 200

0.5

1

Marginal water values $/1000 m3

10 11 12 13 140

0.5

1

Marginal water values $/1000 m3

7.5 8 8.5 9 9.50

0.5

1

Marginal water values $/1000 m3

0 10 20 30 400

0.5

1

Spillage losses m3/s

Reservoir Passes-Dangereuses

0 20 40 600

0.5

1

Spillage losses m3/s

Reservoir Chute du Diable

0 20 40 60 800

0.5

1

Spillage losses m3/sNon

-Exc

eede

nce

prob

abili

ty [-

]

Chute-a-la-Savane

0 200 400 6000

0.5

1

Spillage losses m3/s

Reservoir Lac-St-Jean

0 100 200 3000

0.5

1

Spillage losses m3/s

Shipshaw

SDDP(1) SDDPX(p,SWE,Pw)

Figure 4. Statistical distributions of (left) spillage losses and (right) marginal water values. First configuration.

Water Resources Research 10.1002/2017WR021701

PINA ET AL. SDDPX EXOGENOUS VARIABLES 9

81

any given power plant, reservoir operators would therefore be willing to pay more for the same unit ofwater under SDDPX release policies because, as explained above, unproductive spills are reduced.

3.2. Joint Optimization of Physical and Financial AssetsLet us now move to the second configuration. Incorporating the secondary operating objectives (recreationand flood control in Lac-Saint-Jean Reservoir) and the load commitment reveals one main change withrespect to the first configuration: the gains in energy generation that we can expect with the SDDP(p) andSDDPX formulations are reduced (Table 3). With the most complex SDDPX formulation, the average annualgain is about 105 GWh in the first configuration. In the second configuration, the annual gain is reduced to51 GWh. It turns out that by constraining the system and forcing the operators to buy increasingly moreexpensive energy to meet the load, the resulting SDDPX decisions tend to hedge more against the hydro-logical risk: the operating policies tend to store more water, accepting immediate, shorter, energy deficitsto reduce the probability of greater, increasingly more costly, energy shortages in the future. Figure 5 showsthe average weekly storages associated to the most complex SDDPX formulation for both configurations.As discussed earlier, the higher storage levels in the second, more restrictive, configuration tend to increasethe unproductive spills throughout the system, therefore affecting the total energy output.

Compared to SDDP(1), the reduction in power output is however less important with SDDPX-derived poli-cies; the net energy purchases (i.e., differences between purchases and sales) are reduced by 6.6% and 6.7%when the exogenous variables are included (Table 3).

The results can also be analyzed in terms of the annual power efficiency of the system, which is definedhere as the ratio between the power produced by the system and the total outflow. Generally speaking, theaverage efficiency is improved when more hydrologic variables are incorporated in the state space vector(see Table 4). We can see that the power efficiency of the system actually increases when the storage levels

in the head reservoir are lowered. This apparently counter-intuitiveresult is due to the characteristics of the RT cascade where a signifi-cant portion of the energy is generated by downstream run-of-riverpower plants that are prone to large spills especially if the head reser-voir is also spilling. In a cascade with storage power plants, the conclu-sions would probably be different: power efficiency would increasewith the storage levels.

Figure 6 shows, for both configurations, the statistical distributions ofthe annual differences in power efficiency between SDDPX(p; SWE; Pw)and SDDP(1) formulations. As we can see, the system’s efficiency is

4 8 12 16 20 24 28 32 36 40 44 48 522000

3000

4000

5000

6000

7000

8000

9000

10000

11000

Weeks

Sto

rage

[Hm

3 ]

SDDPX(p,SWE,Pw) 1st Config

SDDPX(p,SWE,Pw) 2nd Config

Figure 5. Average weekly storages at reservoirs Passes-Dangereuses (RPD) and Lac-St-Jean (RLSJ)-FormulationSDDPX(p,SWE,Pw) for both configurations.

Table 4Average Power Efficiency of the System

Average efficiency (MW/m3 s21)

First configuration Second configuration

SDDP(1) 1.3976 1.3997SDDP(p) 1.4019 1.4014SDDPX(p, SWE) 1.4056 1.4033SDDPX(p; SWE; Pw ) 1.4060 1.4034

Water Resources Research 10.1002/2017WR021701

PINA ET AL. SDDPX EXOGENOUS VARIABLES 10

82

enhanced more than 65% of the time (the nonexceedance probability > 0.35) when the exogenous hydro-logic variables are taken into account. During the rest of the time, failed inflow forecasts lead to ill-informeddecisions which affect the overall efficiency of the system in the following months. Of interest is the factthat the variability of the efficiency gains/losses is less pronounced for the second configuration. In thatcase, both the upside and the downside of the hydrological risk associated with imperfect forecasts arepartly suppressed. As energy shortages are increasingly costly, i.e., compensated by purchases throughincreasingly more expensive contracts, the SDDPX release policies acts as a hedging mechanism therebyyielding more conservative release decisions over the entire spectrum of the hydrologic uncertainty.

4. Discussion and Conclusions

As the availability of various hydroclimatic information keeps increasing due to advances in environmentalmonitoring systems, there is a need for decision-making processes and tools that can process this information.SDDP has for many years been the most advantageous model for optimizing in a stochastic framework theoperating of large multiple-reservoir systems. We present a variant of the traditional SDDP algorithm in whichdifferent exogenous hydrologic variables such as snow water equivalent and/or sea surface temperature canbe included in the state space vector together with the past inflows. The incorporation of these exogenous var-iables relies on a built-in MPARX model to generate the inflows during both phases of the algorithm (backwardoptimization and forward simulation). The results are consistent with previous studies and show that goodinformation about future flows can result in more efficient hydropower system operations. This paper demon-strates that the SDDP model can show operators how to achieve such gains in large hydropower systems.

Appendix A: SDDPX Parameter Estimation

The appendix shows how to analytically derive the parameters of the hyperplanes used to approximate thefuture benefit function in SDDPX. As indicated in section 2.2, the main modification lies in the calculation ofthe hyperplanes’ parameters ut11; vt11; bt11, and Ct11 (see equation (5)). In particular, Ct11 which can bewritten as:

Ct11ht115ct11;1qt1ct11;2qt211 . . . 1ct11;pqðt2pÞ11

1ct11;p1jXðt2jÞ111 . . . 1ct11;p1bXðt2bÞ11

(A1)

According to the Karush-Kuhn-Tucker conditions for optimality, the derivative of the objective function withrespect to the state variables Si is given by:

-4 -2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

% Difference in Power Efficiency

Non

-Exc

eeda

nce

prob

abili

ty [-

]SDDP(1)

SDDPX(p,SWE,Pw) 1st config

SDDPX(p,SWE,Pw) 2nd config

Figure 6. % of difference in the power efficiency with respect to SDDP(1) formulation for both configurations.

Water Resources Research 10.1002/2017WR021701

PINA ET AL. SDDPX EXOGENOUS VARIABLES 11

83

@F@Si

52X

ki@gi

@Si(A2)

where ki is the dual information (Lagrange multiplier) of the convex optimization problem and gi is the con-straints (Kuhn & Tucker, 1951). Then, for the specific case at stage t and using constraints (4), (5), (8), and(10), the change of the one-stage objective function Ft respect to the state variables st; wt; ht , can be deter-mined by:

@Ft

@st5kw;t1

XH

h51

khhp;tw

ht11=2 (A3)

@Ft

@wt5ke;t (A4)

@Ft

@ht5kw;t

@qt

@ht1XL

l51

klc;t

@ Clt11ht11

� �@ht

(A5)

where kw;t; ke;t; klc;t , and kh

hp;t are, respectively, the vectors with the dual information associated to the massbalance (4), energy balance (10), the L cuts of the benefit-to-go function (5), and the H linear segments ofthe power functions (8).

Using (A1), we can calculate the derivatives in the above equation (A5) for each hydrologic variable½qt21; qt22,. . .,qt2p; Xt21; Xt2j,. . ., Xt2b�:

@ Ct11ht11ð Þqt21

5clt11;1

@qt

@qt211cl

t11;2

@ Ct11ht11ð Þqt22

5clt11;1

@qt

@qt221cl

t11;3

�

@ Ct11ht11ð Þqt2p

5clt11;1

@qt

@qt2p

@ Ct11ht11ð ÞXt2j

5clt11;1

@qt

@Xt2j1cl

t11;p1j11

�

@ Ct11ht11ð ÞXt2b

5clt11;1

@qt

@Xt2b

(A6)

Now, let us say that at stage t, s�

t ; w�

t , and h�

t 5 ½q�t21; q�

t22,. . ., q�

t2p; X�

t2j,. . ., X�

t2b� are sampled and, in orderto include the stochasticity of the problem, K vectors of inflows qK

t are generated using the MPARX(p,b)(equation (12)). Since Fk

t , which will be added to the expected-benefit-to-go function at stage ðt21Þ, can beapproximated by:

Fkt � ul;k

t s�t 1vl;k

t w�t 1Cl;k

t h�t 1bl;k

t (A7)

the slopes ul;kt ; v

l;kt ;C

l;kt 5½cl;k

t;1; . . . ; cl;kt;p; c

l;kt;p1j; . . . ; cl;k

t;p1b� are determined for each hydrologic scenario k usingequations (A3), (A4), and (A6):

@Fkt

@stðjÞ5ul;k

t ðjÞ5kkw;tðjÞ1

XH

h51

kh;khp;tðjÞw

ht11ðjÞ=2 (A8)

@Fkt

@wtðcÞ5vl;k

t ðcÞ5kke;tðcÞ (A9)

Water Resources Research 10.1002/2017WR021701

PINA ET AL. SDDPX EXOGENOUS VARIABLES 12

84

@FKt

@qt21ðjÞ5cl;k

t;1ðjÞ5 kl;kw;tðjÞ1

XL

l51

kl;kc;tðjÞcl

t11;1ðjÞ !

@qt

@qt211XL

l51

kl;kc;tðjÞcl

t11;2ðjÞ

@FKt

@qt22ðjÞ5cl;k

t;2ðjÞ5 kl;kw;tðjÞ1

XL

l51

kl;kc;tðjÞcl

t11;1ðjÞ !

@qt

@qt221XL

l51

kl;kc;tðjÞcl

t11;3ðjÞ

�

@FKt

@qt2pðjÞ5cl;k

t;pðjÞ5 kl;kw;tðjÞ1

XL

l51

kl;kc;tðjÞcl

t11;1ðjÞ !

@qt

@qt2p

@FKt

@Xt2jðjÞ5cl;k

t;p1jðjÞ5 kl;kw;tðjÞ1

XL

l51

kl;kc;tðjÞcl

t11;1ðjÞ !

@qt

@Xt2j1XL

l51

kl;kc;tðjÞcl

t11;p1j11ðjÞ

�

@FKt

@Xt2b5cl;k

t;p1bðjÞ5 kl;kw;tðjÞ1

XL

l51

kl;kc;tðjÞcl

t11;1ðjÞ !

@qt

@Xt2b

(A10)

Defining gðjÞ as:

gtðjÞ5kl;kw;tðjÞ1

XL

l51

kl;kc;tðjÞcl

t11;1ðjÞ (A11)

and by using (12) to find the derivatives of qt respect to the hydrologic variables, the set of equation (A10)can be rewritten as:

cl;kt;1ðjÞ5gtðjÞ

rqt ðjÞrqt21ðjÞ

/t;1ðjÞ1XL

l51

kl;kc;tðjÞcl

t11;2ðjÞ

cl;kt;2ðjÞ5gtðjÞ

rqt ðjÞrqt22ðjÞ

/t;2ðjÞ1XL

l51

kl;kc;t11ðjÞcl

t11;3ðjÞ

cl;kt;pðjÞ5gtðjÞ

rqt ðjÞrqt2pðjÞ

/t;pðjÞ

�

cl;kt;p1jðjÞ5gtðjÞ

rXt ðjÞrXt2jðjÞ

#t;jðjÞ1XL

l51

kl;kc;t11ðjÞcl

t11;p1j11ðjÞ

cl;kt;p1bðjÞ5gtðjÞ

rXt ðjÞrXt2pðjÞ

#t;bðjÞ

(A12)

Taking the expectation over the K artificially generated flows, the vector of slopes ult; vl

t;1; clt;1;

clt;2,. . .,cl

t;p; clt;p1j,. . .,cl

t;p1b can be determined:

ultðjÞ5

1K

XK

k51

ul;kt ðjÞ (A13)

vltðcÞ5

1K

XK

k51

vl;kt ðcÞ (A14)

clt;arxðjÞ5

1K

XK

k51

cl;kt;arxðjÞ; (A15)

8arx51; 2; . . . ; p; p1j; . . . ; p1b

Finally, the constant term is given by:

Water Resources Research 10.1002/2017WR021701

PINA ET AL. SDDPX EXOGENOUS VARIABLES 13

85

blt5

1K

XK

k51

Fkt 2X

J

ultðjÞs

�

t ðjÞ2X

C

vltðcÞw

�

t ðcÞ . . .

2X

J

clt;1ðjÞq

�t21ðjÞ2

XJ

clt;2ðjÞq

�t22ðjÞ2

XJ

clt;pðjÞq

�t2pðjÞ . . .

2X

J

clt;p1jðjÞX

�

t2jðjÞ2 . . . 2X

J

clt;p1bðjÞX

�

t2bðjÞ

(A16)

ReferencesAhmad, A., El-Shafie, A., Razali, S. F. M., & Mohamad, Z. S. (2014). Reservoir optimization in water resources: A review. Water Resource Man-

agement, 28(11), 3391–3405. https://doi.org/10.1007/s11269-014-0700-5Archibald, T. W., Buchanan, C. S., McKinnon, K. I. M., & Thomas, L. C. (1999). Nested Benders decomposition and dynamic programming for

reservoir optimisation. Journal of the Operational Research Society, 50(5), 468–479. https://doi.org/10.1057/palgrave.jors.2600727Bellman, R. (1957). Dynamic programming (1 ed.). Princeton, NJ: Princeton University Press.Bras, R., Buchanan, R., & Curry, K. (1983). Real time adaptive closed loop control of reservoirs with the High Aswan Dam as a case study.

Water Resources Research, 19(1), 33–52. https://doi.org/10.1029/WR019i001p00033Cerisola, S., Latorre, J. M., & Ramos, A. (2012). Stochastic dual dynamic programming applied to nonconvex hydrothermal models. European

Journal of Operational Research, 218(3), 687–697. https://doi.org/10.1016/j.ejor.2011.11.040Cot�e, P., Haguma, D., Leconte, R., & Krau, S. (2011). Stochastic optimisation of Hydro Quebec hydropower installations: A statistical compari-

son between SDP and SSDP methods. Canadian Journal of Civil Engineering, 1434(April), 1427–1434. https://doi.org/10.1139/l11-101Desreumaux, Q., Cot�e, P., & Leconte, R. (2014). Role of hydrologic information in stochastic dynamic programming: A case study of the Kemano

hydropower system in British Columbia. Canadian Journal of Civil Engineering, 41(9), 839–844. https://doi.org/10.1139/cjce-2013-0370Faber, B., & Stedinger, J. (2001). Reservoir optimization using sampling SDP with ensemble streamflow prediction (ESP) forecasts. Journal of

Hydrology, 249, 1113–1133. https://doi.org/10.1016/S0022-1694(01)00419-XGelati, E., Madsen, H., & Rosbjerg, D. (2013). Reservoir operation using El Ni~no forecasts case study of Daule Peripa and Baba, Ecuador.

Hydrological Sciences Journal, 59(8), 1559–1581. https://doi.org/10.1080/02626667.2013.831978Gjelsvik, A., Mo, B., & Haugstad, A. (2010). Long and medium term operations planning and stochastic modelling in hydro-dominated

power systems based on stochastic dual dynamic programming. In S. Rebennack et al. (Eds.), Handbook of power systems I, Energy sys-tems (pp. 33–55). Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-642-02493-1

Goor, Q., Kelman, R., & Tilmant, A. (2011). Optimal multipurpose multireservoir operation model with variable productivity of hydropowerplants. Journal of Water Resources Planning and Management, 137(3), 258–267. https://doi.org/10.1061/(ASCE)WR.1943-5452

Hamlet, A. F., & Lettenmaier, D. P. (1999). Columbia River streamflow forecasting based on ENSO and PDO climate signals. Journal of WaterResources Planning and Management, 125(6), 333–341.

Homen-de Mello, T., de Matos, V. L., & Finardi, E. C. (2011). Sampling strategies and stopping criteria for stochastic dual dynamic program-ming: A case study in long term hydrothermal scheduling. Energy Systems, 2(1), 1–31. https://doi.org/10.1007/s12667-011-0024-y

Karamouz, M., & Vasiliadis, H. V. (1992). Bayesian stochastic optimization of reservoir operation using uncertain forecasts. Water ResourcesResearch, 28(5), 1221–1232. https://doi.org/10.1029/92WR00103

Kelman, J., Stedinger, J., Cooper, L., Hsu, E., & Yuan, S. (1990). Sampling stochastic dynamic programming applied to reservoir operation.Water Resources Research, 26(3), 447–454. https://doi.org/10.1029/WR026i003p00447

Kim, Y., & Palmer, R. (1997). Value of seasonal flow forecasts in Bayesian stochastic programming. Journal of Water Resources Planning andManagement, 123(6), 327–335. https://doi.org/10.1061/(ASCE)0733-9496(1997)123:6(327)

Kim, Y-O., Eum, H., Lee, E., & Ko, I. (2007). Optimizing operational policies of a Korean multireservoir system using sampling stochasticdynamic programming with ensemble streamflow prediction. Journal of Water Resources Planning and Management, 133(1), 4–14.

Kristiansen, T. (2004). Financial risk management in the electric power industry using stochastic optimization. Advanced Modelling and Opti-mization, 6(2), 17–24.

Kuhn, H. W., & Tucker, A. W. (1951). Nonlinear programming. In Proceedings of the second Berkeley symposium on mathematical statisticsand probability (pp. 481–492). Berkeley, CA: University of California Press.

Kwon, H.-H., Brown, C., Xu, K., & Lall, U. (2009). Seasonal and annual maximum streamflow forecasting using climate information: Applicationto the Three Gorges Dam in the Yangtze River basin, Hydrological Sciences Journal, 54(3), 582–595. https://doi.org/10.1623/hysj.54.3.582

Labadie, J. W. (2004). Optimal operation of multireservoir systems: State-of-the-art review. Journal of Water Resources Planning and Man-agement, 130(2), 93–111. https://doi.org/10.1061/(ASCE)0733-9496(2004)130:2(93)

Ljung, L. (1999). System identification: Theory for the user. Englewood Cliffs, NJ: Prentice-Hall.Lohmann, T., Hering, A. S., & Rebennack, S. (2015). Spatio-temporal hydro forecasting of multireservoir inflows for hydro-thermal schedul-

ing. European Journal of Operational Research, 255(1), 243–258. https://doi.org/10.1016/j.ejor.2016.05.011Macian-Sorribes, H., Tilmant, A., & Pulido-Velazquez, M. (2016). Improving operating policies of large-scale surface-groundwater systems

through stochastic programming. Water Resources Research, 53, 1407–1423. https://doi.org/10.1002/2016WR019573Matos, V. L. D., & Finardi, E. C. (2012). A computational study of a stochastic optimization model for long term hydrothermal scheduling.

International Journal of Electrical Power & Energy Systems, 43(1), 1443–1452. https://doi.org/10.1016/j.ijepes.2012.06.021Mo, B., Gjelsvik, A., & Grundt, A. (2001). Integrated risk management of hydro power scheduling and contract management. IEEE Transac-

tions on Power Systems, 16(2), 216–221. https://doi.org/10.1109/59.918289New York Independent System Operator (NYISO) (2017). Hourly custom report-Time-Weighted/Integrated Real-Time LBMP-Reference Bus.

Rensselaer, NY: New York Independent System Operator. Retrieved fromhttp://www.nyiso.comOliveira, R., & Loucks, D. P. (1997). Operating rules for multireservoir systems. Water Resources Research, 33(4), 839–852. https://doi.org/10.

1029/96WR03745Pereira, M. (1989). Optimal stochastic operations scheduling of large hydroelectric systems. International Journal of Electrical Power &

Energy Systems, 11(3), 161–169. https://doi.org/10.1016/0142-0615(89)90025-2Pereira, M. V. F., & Pinto, L. M. V. G. (1991). Multi-stage stochastic optimization applied to energy planning. Mathematical Programming,

52(1–3), 359–375. https://doi.org/10.1007/BF01582895

AcknowledgmentsAuthors acknowledge the CRDPJproject funded by NSERC and Rio TintoAlcan, and NSERC Strategic NetworkProject FloodNet. We also thank thereviewers for their constructivecomments and suggestions. We thankDr. Charles Roug�e and Dr. Sara S�eguinfor their valuable discussions.Supporting data are not publiclyavailable and belong to Rio Tinto, theprivate company operating the powersystem used to illustrate the proposedmethodology. These data can berequested by contacting the thirdauthor (pascal.cote@riotinto.com).

Water Resources Research 10.1002/2017WR021701

PINA ET AL. SDDPX EXOGENOUS VARIABLES 14

86

Pereira-Cardenal, S. J., Mo, B., Gjelsvik, A., Riegels, N. D., Arnbjerg-Nielsen, K., & Bauer-Gottwein, P. (2016). Joint optimization of regionalwater-power systems. Advances in Water Resources, 92, 200–207. https://doi.org/10.1016/j.advwatres.2016.04.004

Pina, J., Tilmant, A., & Anctil, F. (2016). Horizontal approach to assess the impact of climate change on water resources systems. Journal ofWater Resources Planning and Management, 143, 04016081-1. https://doi.org/10.1061/(ASCE)WR.1943-5452.0000737

Poorsepahy-Samian, H., Espanmanesh, V., & Zahraie, B. (2016). Improved inflow modeling in stochastic dual dynamic programming. Journalof Water Resources Planning and Management, 142, 04016065-1. https://doi.org/10.1061/(ASCE)WR.1943-5452.0000713

Pritchard, G. (2015). Stochastic inflow modeling for hydropower scheduling problems. European Journal of Operational Research, 246, 496–504. https://doi.org/10.1016/j.ejor.2015.05.022

Rani, D., & Moreira, M. M. (2009). Simulation optimization modeling: A survey and potential application in reservoir systems operation.Water Resource Management, 24(6), 1107–1138. https://doi.org/10.1007/s11269-009-9488-0

Raso, L., Malaterre, P. O., & Bader, J. C. (2017). Effective streamflow process modeling for optimal reservoir operation using stochastic dualdynamic programming. Journal of Water Resources Planning and Management, 143, 04017003-1. https://doi.org/10.1061/(ASCE)WR.1943-5452.0000746

Rotting, T., & Gjelsvik, A. (1992). Stochastic dual dynamic programming for seasonal scheduling in the Norwegian power system. IEEE Trans-actions on Power Systems, 7(1), 273–279. https://doi.org/10.1109/59.141714

Shapiro, A., Tekaya, W., da Costa, J. P., & Soares, M. P. (2013). Risk neutral and risk averse Stochastic Dual Dynamic Programming method.European Journal of Operational Research, 224(2), 375–391. https://doi.org/10.1016/j.ejor.2012.08.022

Stedinger, J. R., Sule, B. F., & Loucks, D. P. (1984). Stochastic dynamic programming models for reservoir operation optimization. WaterResources Research, 20(11), 1499–1505. https://doi.org/10.1029/WR020i011p01499

Tejada-Guibert, J. A., Johnson, S. A., & Stedinger, J. R. (1995). The value of hydrologic information in stochastic dynamic programming mod-els of a multireservoir system. Water Resources Research, 31(10), 2571–2579. https://doi.org/10.1029/95WR02172

Tilmant, A., & Kelman, R. (2007). A stochastic approach to analyze trade-offs and risks associated with large-scale water resources systems.Water Resources Research, 43, W06425. https://doi.org/10.1029/2006WR005094

Tilmant, A., & Kinzelbach, W. (2012). The cost of noncooperation in international river basins. Water Resources Research, 48, W01503.https://doi.org/10.1029/2011WR011034

Tilmant, A., Pinte, D., & Goor, Q. (2008). Assessing marginal water values in multipurpose multireservoir systems via stochastic program-ming. Water Resources Research, 44, W12431. https://doi.org/10.1029/2008WR007024

Yeh, W. W.-G. (1985). Reservoir management and operations models: A state-of-the-art review. Water Resources Research, 21(12),1797–1818. https://doi.org/10.1029/WR021i012p01797

Zahraie, B., & Karamouz, M. (2004). Hydropower reservoirs operation: A time decomposition approach. Scientia Iranica, 11(1–2), 92–103.Zhao, T., Zhao, J., & Yang, D. (2014). Improved dynamic programming for hydropower reservoir operation. Journal of Water Resources Plan-

ning and Management, 140(March), 365–374. https://doi.org/10.1061/(ASCE)WR1943-5452.0000343

Water Resources Research 10.1002/2017WR021701

PINA ET AL. SDDPX EXOGENOUS VARIABLES 15

87

Appendix C

Autoregressive modeling

C.1 Multisite periodic autoregressive model MPAR

Autoregressive (AR) models have been extensively used in hydrology and water resources since

1960’s. Its popularity and attractiveness rely on the simplest formulation and its intuitive type of

time dependence, variables values at time t are dependent to the preceding values. First introduce by

Thomas and Fiering (1962) and later by Box and Jenkins (1970). AR models can be represented as

models with constant parameters, parameters varying with time and combination of both. The models

with constant parameters are often implemented for modeling time series on the annual basis. Models

with periodic parameters are usually used with time series of intervals that are fraction of the year

(e.g. seasons, months, weeks, etc.) Salas et al (1980). The latter models, are referred as periodic AR,

PAR(p) models, and the periodicity may be in the mean, variance and/or autoregressive parameters.

Finally, since analysis in water systems involves time series at various geographic locations mod-

els which evaluate the spatial correlation are required, then let us introduce the multivariate spacial

models Multisite PAR, MPAR.

The multi site ( j) periodic process can be modeled by an autoregressive model of order p represented

by:

(qt( j)−µqt ( j)

σqt ( j)

)=

p

∑i=1

φi,t( j)

(qt−i( j)−µqt−i( j)

σqt−i( j)

)+ εt( j) (C.1)

where qt is the time dependent variable for year v and time t, with t=1,2,...,52 weeks. µqt and σqt are

the periodic mean and standard deviation of qt , respectively, φi,t( j) are the autoregressive parameters

of the p order periodic model, and ξt is a time independent stochastic noise with mean zero and

periodic variance σ2ε,t( j).

88

C.1.1 Parameter estimation

Moments

The set of parameters to be determined in the model are: µt( j), σt( j), φi,t( j) and σ2ε,t( j). Given N

years of historical incremental reservoirs inflows data qv,t( j) available at site j, the parameters µqt ( j),

σqt ( j) can be estimated by the first and second moment as:

µqt ( j) = E[qt( j)

]≈ 1

N

N

∑v=1

qv,t( j) (C.2)

σ2qt( j) = E

[q2

t ( j)]≈ 1

N−1

N

∑v=1

(qv,t( j)−µt( j))2 (C.3)

defining the vector Zt( j) [N×1] of normalized incremental inflows as:

Zt( j) =

(qv,t( j)−µqt ( j)

σqt ( j)

), ∀v = 1 . . .N (C.4)

by using equation C.1 the normalized inflows can be expressed as:

Zt( j) = φ1,t( j)Zt−1( j)+ · · ·+φp,t( j)Zt−p( j)+ εt( j) (C.5)

The temporal Lag-k autocorrelation ρt(k) between qv,t and qv,t−k for each season t is obtained by

multiplying C.5 by Zt−k and taking the expectation term by term:

E[Z( j)Zt−k( j)] = φ1,t( j)E[Zt−1( j)Zt−k( j)]+ · · ·+φp,t( j)E[Zt−p( j)Zt−k( j)]+E[εt( j)Zt−k( j)] (C.6)

ρt (k) = φ1,tρt−1(|k−1|)+ · · ·+φp,tρt−p(|k−1|) (C.7)

Equation C.7 constitutes a set of p linear equations, for k=1,2,...,p

ρt (1) = φ1,tρt−1(0)+φ2,tρt−2(1)+ · · ·+φp,tρt−p(1−p)

ρt (2) = φ1,tρt−1(1)+φ2,tρt−2(0)+ · · ·+φp,tρt−p(2−p)

... (C.8)

ρt (p) = φt,1ρt−1(p−1)+φt,2ρt−2(p−2)+ · · ·+φt,pρt−p(0)

89

which may be written for each j as:

ρt (1)

ρt (2)...

ρt (p)

=

1 ρt−1(1) . . . ρt−1(p−1)

ρt−1(1) 1 . . . ρt−2(p−2)...

......

ρt−1(p−1) ρt−2(p−2) . . . 1

φ1,t

φ2,t...

φp,t

(C.9)

in a matrix form:

ρ t( j) =COV q,t( j) ·Φt( j) (C.10)

where COV q,t( j) is the periodic covariance and Φt( j) and ρ t( j) respectively, are the vectors of autore-

gressive coefficients of the MPAR(P) and autocorrelation coefficients. Then φi,t( j) can be calculated

as:

Φt( j) =COV q,t( j)−1 ·ρ t( j) (C.11)

According to Salas et al (1980) the residual variance for each t, σ2ε,t( j) can be estimated as a function

of the periodic autoregressive coefficients φi,t( j) and the periodic autocorrelation coefficients ρi,t( j).

σ2ε,t( j) = 1−

p

∑j=1

φ j,t( j)ρ j,t( j) (C.12)

Least square parameter estimation

The least square estimation method is based on finding the estimates φi,t( j) so that the sum of the

squared differences between the observed Zt( j) values and the expected values Zt( j)= φ1,t( j)Zt−1( j)+

· · ·+ φp,t( j)Zt−p( j), is minimized.

minφt

{∑(Zt( j)− Zt( j))2

}(C.13)

Let us first say that Zt can be expressed in the matrix form

Zt( j) = Xt( j)Φt( j) (C.14)

90

where Xt( j) contains [Zt−1( j), . . . ,Zt−p( j)] and Φt( j) = [φ1,t( j), . . . , φp,t( j)] then from equation C.13

the objective function can be written as:

Θ( j) = ∑(Zt( j)−Xt( j)Φt( j))2

= ∑εt( j)2 = εᵀt ( j)εt( j)

= [Zt( j)−Xt( j)Φt( j)]ᵀ[Zt( j)−Xt( j)Φt( j)] (C.15)

To find the minimum, all partial derivatives of the sum respect to the estimates must be equal to zero:

∂Θ

∂Φt=

∂

∂Φt[Zt −XtΦt ]

ᵀ[Zt −XtΦt ] =−Xtᵀ[Zt −XtΦt ] = 0 (C.16)

Φt( j) = [Xtᵀ( j)Xt( j)]−1Xt

ᵀ( j)Zt( j) (C.17)

which represents the same correlation cross-correlation structure identified in equation C.11.φ1,t

φ2,t...

φp,t

=

∑Zt−1

2∑Zt−1Zt−2 . . . ∑Zt−1Zt−p

∑Zt−1Zt−2 ∑Zt−22 . . .

......

...

∑Zt−1Zt−p . . . ∑Zt−p2

−1

∑Zt−1Zt

∑Zt−2Zt...

∑Zt−pZt

(C.18)

Moreover, the Hessian matrix ∂ 2Θ

∂Φ2 must be positive definite

∂ 2Θ

∂Φ2 = Xtᵀ( j)Xt( j) positive definite (C.19)

The unbiased estimate of the variance of the model can be constructed as:

σ2ε,t( j) =

1N− p

εᵀ( j)ε( j) (C.20)

C.1.2 Stochastic noise

Assuming that the noise εt( j) follows a 3-parameters (µv( j), σv( j) and κt( j)) log normal distribution:

fεt( j) =1

(εt( j)−κt( j))√

2πσv( j)e−0.5

(log(εt ( j)−κt ( j))−µv( j)

σv( j)

)2

(C.21)

91

with mean µεt and variance σ2ε,t

µεt ( j) = κt( j)+ e

(µv( j)+ σ2

v ( j)2

)

σ2ε,t( j) = e2(µv( j)+σ2

v ( j))+ e(2µv( j)+σ2v ( j)) (C.22)

to ensure non-negative inflows qt( j)> 0 from equation C.1 the lower bound κt( j) is defined as:

εt( j)>−µqt ( j)σqt ( j)

−p

∑i=1

φi,t( j)

(qt−i( j)−µqt−i( j)

σqt−i( j)

)= κt( j) (C.23)

and the parameters µv( j) and σv( j) are determined as:

µv( j) = 0.5Log σ2ε,t

Λ( j)(Λ( j)−1) (C.24)

σv( j) =√

log(Λ( j)) (C.25)

Λt( j) = 1+ σ2ε,t

κ2t ( j) (C.26)

Finally the standardized stochastic noise is estimated as:

Vt( j) =log(εt( j)−κt( j))−µv( j)

σv( j)(C.27)

C.1.3 Spatial cross correlation

The spatial statistical dependence of reservoir inflows is introduced by the lag0-covariances and cross

variances of the standardized stochastic noise. Defining the vector Vt as the collection of independent

standardized noises of each j node of the system Vt = [Vt(1), . . .Vt(J)], the spatial model can be written

as:

Vt = AtWt (C.28)

where Wt is a column vector of J independent elements consisting of white noises, normally distributed

with zero mean and variance equal to 1. The estimation of matrix A can be obtained from the Cholesky

factorization of the covariance matrix of standardized noise at each node j:

At Aᵀt =Cov(Vt) (C.29)

92

C.1.4 Example parameter estimation Gatineau River Basin

The set of parameters to be determined in the hydrologic model equation C.1 are: the mean µt( j),

standard deviationσt( j), autoregressive parameters φi,t( j) and the variance of the residuals σ2ε,t( j).

Given the N=28 years of weekly incremental inflows data qv,t( j) available at each site j (see: Table

3.1, Figure3.2). For each node j and each week the parameters are determined as follow.

Mean and standard deviation

Figure C.1 displays the parameters µqt ( j), σqt ( j) estimated by the first and second moment equation

C.2 and C.3. The nival regime can be described by a very high discharge during spring-summer after

the snow melt season (weeks 12-24). Great variability during spring and autumn and low flows during

winter.

4 8 12 16 20 24 28 32 36 40 44 48 520

10

20

30

40

50

60Node 1: Cabonga

[Hm

3]

4 8 12 16 20 24 28 32 36 40 44 48 520

50

100

150

200Node 3: Paugan

[Hm

3]

weeks

4 8 12 16 20 24 28 32 36 40 44 48 520

100

200

300

400

500Node 2: Baskatong

[Hm

3]

4 8 12 16 20 24 28 32 36 40 44 48 520

5

10

15

20

25

30Node 4: Chelsea

[Hm

3]

weeks

µq

t

σq

t

µq

t

σq

t

µq

t

σq

t

µq

t

σq

t

Figure C.1: Mean and standard deviation

Autoregressive parameters

As an example, let us take week 15 of the inflow series at the Baskatong reservoir and let us explore

the correlation structure with 12 preceding weeks. The autoregressive parameters of model for all p

orders are determined. To identify the order of the autoregressive model, four strategies are evaluated:

i) partial autocorrelation function, ii) the evolution of the mean square error, iii) parsimony of the

models and iv) the stationarity conditions

93

The partial autocorrelation function is another way of representing the time dependence structure, and

it is useful for helping to identify the type of order of the model. In the set of equations C.9 or C.17,

the partial autocorrelation is given by the last coefficient φp,t , and is practical zero beyond a particular

lag, if lies inside the 1−α probability limits Box and Jenkins (1970).[−u1− α

2√N

;u1− α

2√N

](C.30)

where uα is the quantile of the standard normal distribution α the probability level (α = 5%), and N

is the sample size.

The principle of parsimony in model building can be considered into account by using the mathemat-

ical formulation Akaike Information Criterion (AIC) or Bayesian information criterion (BIC). Under

this criterion the model which gives the minimum AIC and BIC is the one selected.

AIC = N · ln(σ2ε,t( j))+2p+1 (C.31)

BIC = N · ln(σ2ε,t( j))+ p · log(N) (C.32)

Figure C.2 displays for each order p, (1) the partial correlogram φk and the 95% confidence interval,

(2) the mean square error MSE, (3) AIC and (4) BIC. From the collection of figures we can perceived

that models of order p =2, p =3 and p =6 are good candidates to be selected. The partial correlogram

reveals strong correlation between inflows of week 15 and week 12 (p =3) and week 9 (p =6). Like-

wise, except for the order p =4, the evolution of the MSE displays the important reduction in the error

function while increasing the order of the autoregressive model until p =6. Finally, the mathematical

information criterion AIC and BIC, displays local minimum values for model of order p =2 and p =6.

One further check on the parameters is to assure that σ2ε,t is always greater than zero. (Salas et al,

1980).

Stationary conditions

The stationary conditions are satisfied if the roots of the characteristic equation C.33 lie inside the unit

circle. That is, |ui|< 1, i= 1,...,p.

up−φ1,t( j)up−1−φ2,t( j)up−2−·· ·−φp,t( j) = 0 (C.33)

If we isolate the autoregressive parameters and we evaluate the stationary conditions by calculating

the roots of the polynomial characteristic equation C.33. From table C.1 one can see that the first root

of the model of order p =6 is outside the unite circle∣∣uy∣∣> 1 .

94

1 2 3 4 5 6 7 8 9 10 11 12-0.5

0

0.5

1

PACF

( k)

Order(p)

(1)

1 2 3 4 5 6 7 8 9 10 11 120.2

0.3

0.4

0.5

0.6

0.7

Order(p)

MSE

(2)

1 2 3 4 5 6 7 8 9 10 11 12-15

-10

-5

0

5

10

Order(p)

AIC

(3)

1 2 3 4 5 6 7 8 9 10 11 12-10-505

10152025

Order(p)BIC

(4)

Figure C.2: (1) Partial correlogram PACF (2) Mean square error MSE (3) AIC (4) BIC

Table C.1: Autoregressive parameters φp,t and polynomial characteristic roots ui

p φp,t(2) roots ui

p =1 0.5643 0.5643

p =3 0.5289 0.9293-0.1212 -0.2002 +0.6731i

0.4583 -0.2002 - 0.6731i

p =6 0.4585 1.0330-0.0423 0.5289+ 0.6850i

0.3524 0.5289- 0.6850i

-0.0595 -0.4000+0.8875i

-0.2044 -0.4000-0.8875i

0.6102 -0.8323

Hydrologic modeling

Once the model is parametrized, it can be used to generate different hydrologic scenarios. For example

in Figure C3 and C4 we can see the average and standard deviation of 25 synthetically generated

inflow series at each node of the system using the MPAR(p) model. Generally speaking, we can see

that mean and standard deviation are pretty much conserved.

95

4 8 12 16 20 24 28 32 36 40 44 48 520

100

200

300

400

500

Node 2: BaskatongAverage

[hm

3]

Historical

Simulated

4 8 12 16 20 24 28 32 36 40 44 48 520

50

100

150

200

250Standard deviation

[hm

3]

weeks

Historical

Simulated

4 8 12 16 20 24 28 32 36 40 44 48 5210

20

30

40

50

60

Node 1: CabongaAverage

[hm

3]

Historical

Simulated

4 8 12 16 20 24 28 32 36 40 44 48 520

5

10

15

20Standard deviation

[hm

3]

weeks

Historical

Simulated

Figure C.3: Cabonga and Baskatong

4 8 12 16 20 24 28 32 36 40 44 48 520

50

100

150

200

Node 3: PauganAverage

[hm

3]

Historical

Simulated

4 8 12 16 20 24 28 32 36 40 44 48 520

10

20

30

40

50

60

70Standard deviation

[hm

3]

weeks

Historical

Simulated

4 8 12 16 20 24 28 32 36 40 44 48 520

5

10

15

20

25

30

Node 4: ChelseaAverage

[hm

3]

Historical

Simulated

4 8 12 16 20 24 28 32 36 40 44 48 520

2

4

6

8

10Standard deviation

[hm

3]

weeks

Historical

Simulated

Figure C.4: Paugan and Chelsea

96