Improving Risk Management RIESGOS-CM

Análisis, Gestión y Aplicaciones

P2009/ESP-1685

Technical Report 2011.7

Multiobjective Decision Support for the Kwanza River Management

Kiombo J. Marie, Javier Cano, David Ríos Insua

http://www.analisisderiesgos.org

Multiobjective Decision Support for the Kwanza

River Management

Kiombo J. Marie1, Javier Cano2, David Rıos Insua3

1Department of Economic Sciences, Agostinho Neto University, Angola2Department of Statistics and Operations Research, Rey Juan Carlos University,

Spain3Royal Academy of Sciences, Spain

Abstract

Water management has become a challenging problem worldwide,especially in developing countries, due to several reasons, including thegrowing scarcity of natural water resources; a demographic explosionin many urban settlements, followed by a rapid urbanization and in-dustrialization of neighboring areas; and the inherent increase in thedemand of energy and natural fresh water for human consumptionand agricultural developments. The Kwanza river, the longest andmost plentiful in Angola, is a paradigmatic case, as it is one of themain sources of wealth in that country. Many people base, direct orindirectly, their living on its water, and many economical activities,including energy production, depend also on its stream. We providea model for the multiobjective management of the Kwanza river. Theproblem is complicated by the need to take into account uncertainty invarious involved processes, to plan over a long period of time, and thepresence of several conflicting interests. We describe also a decisionsupport system implementing our model.

Keywords: Multipurpose river management; Expected utility; Fore-casting; Multiobjective decision analysis; Decision support system.

1 Introduction

With an area of 1,246,700 km2, Angola is a country in southwest Africa.The United Nations Program for Development (UNDP), through its HumanDevelopment Report (2009), estimated on 18,498,000 the total populationon Angola, a fourth of whom live in its capital, Luanda (slightly over onehalf if we consider its metropolitan area). The Kwanza is the main riverin Angola, with a total length of 960 km, flowing into the Atlantic Oceanjust south of Luanda. Its vast water resources could allow the development

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of water supply and hydroelectric projects for Angola, in general, and forLuanda, in particular.

However, only a minority of people living in Luanda have access to pipednatural fresh water. The rest of the population depends on trucks carryingwater from Kikuxi and Kifangondo plants, on the Kwanza and Bengo rivers,respectively. The capacity of water treatment of the Kikuxi plant, the mostimportant one, is quite limited, allowing, at best, for a daily per capita waterconsumption in Luanda of 40 liters, far below the 100 liters recommendedby the World Health Organization (WHO). The government of Angola aimsat increasing in the forthcoming years the proportion of population withaccess to drinking water from 57% to 100% in urban areas, and from 26%to 80% in rural areas.

Moreover, many UNDP Development Indicators show energy deficits inAngola. These emphasize the poor quality and reliability of the power sup-ply, restricted to urban centers, the limited financial and human resourcesavailable to cope with the process of rehabilitation, expansion and modern-ization of the electric infrastructure, and an inadequate pricing policy tocover production costs. The Kwanza river currently includes two main hy-dropower schemes, Capanda and Cambambe, but plans have been proposedfor seven additional schemes.

It is this situation of lack of water and power in Angola and, especially,in Luanda which motivated us to develop a decision analysis model to dealwith the multiobjective management of the Kwanza river. The analysis ofthis problem is further complicated by the presence of various sources ofuncertainty, mainly: the availability of water at Capanda and Cambambe,due to the random nature of inflows; the amount of water evaporated atboth reservoirs, which is nonnegligible; and the amount of water demandedat Luanda for urban and agricultural consumption. Besides, we need totake into account dynamical aspects, mainly those referring to constraintsinterrelating reservoir states over consecutive time periods, and the need tooperate over time. The good performance of the proposed model led us todevelop a decision support system for the multiobjective management of theKwanza river.

The structure of the paper is as follows. We first describe the environ-ment of the Kwanza river, highlighting its characteristics and uses. Then,in Section 3, we describe the structure of the multipurpose managementproblem of the Kwanza river. We next deal with the belief and preferencemodelling of our problem. In Section 6, we formulate our global decisionmaking problem, and present a Decision Support System in Section 7. Weend up with some discussion.

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2 The Kwanza River

The Kwanza river has its source in central Angola, southeast of Chitemboin the plateau of the Bie region. Its basin is divided into three sub-basins:

• The Lower Kwanza extends from the estuary, south of Luanda, to thecity of Dondo, in the province of Kwanza North, where the Cambambeplant is located, see Figure 1. This section is very flat, as the riverlevel changes only 50 m in 200 km, facilitating the navigation of smalland medium-size boats.

• The Middle Kwanza, located between the municipality of Dondo andthe “Salto de Cavalo” area, is the “golden section” of the river, interms of hydropower generation, due to its steep slope. It has themain part of the river flow, a good climate, and a wide variety of soils.The Capanda plant is located close to the main city of Malanje, seeFigure 1.

• Finally, the Upper Kwanza spreads over almost 69% of the wholeKwanza river basin surface, and it is characterized by poor soils unattrac-tive for agriculture.

CAPANDA

CAMBAMBE

Figure 1: Lower section of the Kwanza river.

The climate in the Kwanza basin is essentially equatorial, with averageannual temperatures of 22◦C. The total annual precipitations in Angolafluctuate between 840 mm ·m−2 in the driest years and 1358 mm ·m−2 inthe most rainy ones. The average annual rainfall in the Kwanza basin isaround 1250 mm · m−2, and the average floods in the last hundred yearshave oscillated between 7900 and 9700 m3 · s−1, see [1].

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There are two power plants currently functioning along the Kwanza river,Capanda and Cambambe, located in the regions of Malanje and KwanzaNorth, respectively, see Figure 1. The slope between Capanda and Cam-bambe is about 800 meters, allowing for a potential capacity to producehydroelectrical energy of, approximately, 6700 MW, see [2].

The Cambambe hydroelectric plant was built at the end of the colonialtimes, in the 70’s. Since its inauguration, it has supplied electricity tothe north of Angola and the city of Luanda. After the independence in1975, with the abandonment of land by many settlers, and the onset ofarmed conflicts, the production of electricity began to suffer. In addition,lower levels in the river flow and scarcer rainfalls have also contributed toreduce the energy production at Cambambe. Its inflows oscillate between122 m3 · s−1 and 3740 m3 · s−1, being the average value around 750 m3 · s−1.It has four turbines, and its nominal installed capacity is 260 MW.

Regarding the plant at Capanda, its construction began in November1987 and was intended to be accomplished in two phases: (1) the installa-tion of two turbines of 130 MW each; (2) the installation of two additionalturbines, increasing the total power generation capacity up to 520 MW.However, Capanda was attacked and occupied during the war by UNITA’stroops, from 1992 to 1994. This fact delayed the conclusion of the works,which were resumed in 2000, after the civil war ended. The constructionof the plant finally concluded in 2004, and Capanda began to supply powerto Luanda, although with many difficulties, due to a defective transmissionnetwork.

3 Problem Structuring

We describe now the structure of the Kwanza river management problem.Following the standard decision analysis practice, see [3], we shall specifythe decision variables, the constraints, the sources of uncertainty and themanagement objectives, which stem from the multipurpose nature of thesystem. The subscript t of all variables refers to the t-th month, as we con-sider monthly planning within a multi-month management plan. SubscriptsP and B refer to Capanda and Cambambe, respectively. Unless otherwisestated, we express all magnitudes in International System units.

3.1 Decision variables

In our problem, the decision variables refer to water releases at month t forvarious purposes, u1Bt

, u2Bt, u1Pt

, u2Ptand uurbt , with the following meaning:

• u1Ptand u1Bt

represent the volumes of water released through turbinesin Capanda and Cambambe, respectively;

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• u2Ptand u2Bt

represent the volumes of water released through spillgatesin Capanda and Cambambe, respectively;

• uurbt refers to the volume of water, coming through the Kikuxi plant,for human consumption in Luanda.

3.2 Sources of uncertainty

There are several sources of uncertainty in our problem. These include: theinflows to Capanda and Cambambe, iPt

, iBt; the volumes of water evap-

orated in Capanda, and Cambambe, evPt, evBt

; and the volume of waterdemanded for human consumption in Luanda, durbt

. We shall develop fore-casting models for such quantities in Section 4.

3.3 Constraints

In our problem, we have to take into account several restrictions on thevalues of the decision variables or on related functions of them. Specifically,we can classify such constraints into four different types:

• Production constraints. They refer to the energy produced at Capandaand Cambambe plants, ePt

and eBt, respectively. Regarding Capanda,

because of the role of its regulating lagoon, the installed turbines mustoperate with a variable jump, which requires the level of the lagoonto vary between 890 and 947.3 m, in order to ensure a regulated flowof around 400 m3 · s−1, see [1]. The level of replacement will dependon the discharged flow through turbines, u1Pt

, and can be determinedfrom the flow curve:

u1Pt= 1.2246 · (h′P − 844.5)1.252, (1)

where h′P represents the height of the Capanda dam. On the otherhand, the energy production function in Capanda can be expressed as

ePt= hP ·K · η · u1Pt

, (2)

where hP = h′P−844.5, η is the performance of each turbine, estimatedto be 0.92 for Capanda and Cambambe, and K = ρ · g, being ρ thewater density, equal to 1000 kg/m3, and g the gravity acceleration,equal to 9.81 m · s−2

Regarding Cambambe, its energy production function has the sameform than (2),

eBt= hB ·K · η · u1Bt

, (3)

where hB represents the water height of the Cambambe dam, estimatedas 86 m. Substituting the values of η and K in (2) and (3), and

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combining (1) and (2), we obtain the following expressions for theenergy production functions in Capanda and Cambambe, respectively:

ePt= 9025.2 · u1Pt

·

(u1Pt

1.2246

) 1

1.252

, (4a)

eBt= 776167.2 · u1Bt

. (4b)

• Constraints over water volumes released for various purposes. Thedesign of Capanda and Cambambe plants restricts the maximum andminimum volumes of water that can be released through turbines andspillgates. Regarding the turbines, although it would be desirable torelease as much water as possible through them, for energy productionreasons, the plants will have a maximum production capacity whichcannot be exceeded. The restrictions on the volume of water releasedthrough spillgates have to do more with situations in which the damlevel is too high, and water has to be released for security reasons,in order to avoid floods. According to the technical specifications ofCapanda we have that:

0 ≤ u1Pt≤ 400 m3 · s−1, (5a)

0 ≤ u2Pt≤ 8200 m3 · s−1. (5b)

In a similar way, for Cambambe, we have that:

0 ≤ u1Bt≤ 280 m3 · s−1, (6a)

0 ≤ u2Bt≤ 10540 m3 · s−1. (6b)

Besides, a minimum water flow allowing for river navigation needsto be guaranteed. To ensure this, the difference between the waterreleased in Cambambe, u1Bt

+ u2Bt, and the water used for human con-

sumption, uurbt , cannot be smaller than a certain value, specifically:

u1Bt+ u2Bt

− uurbt ≥ 110 m3 · s−1. (7)

Finally, the volume of water coming through the Kikuxi plant cannotexceed its water treatment capacity

0 ≤ uurbt ≤ 3.2 m3 · s−1. (8)

• Constraints on storage at reservoirs. These constraints reflect themaximum storage capacity of plants. If storage at Capanda is sPt

,and that at Cambambe is sBt

, then, the following conditions musthold:

0 ≤ sPt≤ 4.795 · 109 m3, (9a)

0 ≤ sBt≤ 1.02 · 108 m3. (9b)

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• Storage continuity conditions. These relationships express that thestorage at the next period is equal to the previous period storage, plusthe inflows, minus the releases (through turbines and spillgates), minusthe evaporation. In the case of Capanda, since it is a non regulatedupstream, the constraint is simply stated as

sPt+1= sPt

+ iPt−

(u1Pt

+ u2Pt

)− evPt

. (10)

On the other hand, the inflow to Cambambe, iBt, depend on the vol-

ume of water released upstream in Capanda, plus some incrementalflows. Note, however, that the flow from Capanda will take sometime τ to reach Cambambe. To estimate this traveling time, we haveused Manning’s equation for the average flow velocity [4] obtaining avalue v = 0.054 m · s−1. As the distance between both reservoirs, d,is approximately 140 km, we have obtained a value τ = 706.29 hours,approximately one month.

Therefore, the inflows at Cambambe at month t are the sum of thewater released through turbines and spillgates in Capanda at montht− 1, plus the incremental flow inct, i.e.

sBt+1= sBt

+(u1Pt−1

+ u2Pt−1

)+ inct −

(u1Bt

+ u2Bt

)− evBt

. (11)

To estimate the value of inct, let us assume, for a moment, the nonex-istence of Capanda. Then, the inflows at Cambambe would be the sumof inflows entering in Capanda in month t−1 plus the incremental flowinct

iBt= iPt−1

+ inct. (12)

The following equation describes a dynamic regression between theinflows at Capanda and Cambambe, reflecting a physical relationshipbetween inflows and basin sizes

iBt= αt + βt · iPt−1

. (13)

From (12) and (13), we get that

inct = αt + (βt − 1) · iPt−1. (14)

Substituting (14) in (11), we obtain, finally, the continuity constraintfor Cambambe:

sBt+1= sBt

+(u1Pt−1

+u2Pt−1

)+[αt+(βt−1)·iPt−1

]−(u1Bt

+u2Bt

)−evBt

.

(15)Coefficients αt and βt will be estimated using a dynamic linear model,as we shall explain in Section 4.2. Regarding the evaporations atCapanda and Cambambe, they will be estimated using Visentini’smethod [5] as we shall see in Section 4.4.

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3.4 Objectives

In our problem, we shall include various objective functions stemming fromthe various purposes of the Kwanza river system:

• The first objective refers to the maximization of the (combined) energyproduction in Capanda and Cambambe, mainly used in Luanda;

• The second and third objectives have to do with the minimizationof water releases through spillgates in Capanda and Cambambe, forreasons of equity, efficiency and safety;

• Finally, the fourth objective refers to minimizing water deficit for ur-ban consumption in Luanda.

We will build utility functions for these objectives in Section 5.

4 Forecasting Models

As we have mentioned, there are several sources of uncertainty in the Kwanzariver management problem. In this section, we describe the different fore-casting models used to deal with them. They have been framed within thedynamic linear modeling paradigm, see e.g. [6, 7, 8]. Models based on thisapproach allow us to decompose a time series into components that are eas-ily interpretable, such as trend, seasonality and regression. We have usedthe dlm R package [9], and adopted the notation therein.

To wit, let us denote by θt the state of the corresponding system atmonth t, and Yt be the observed variable, which we aim at forecasting. Fort ≥ 1, the system evolves through a set of stochastic linear equations asfollows

• The observation equation is expressed as

Yt = Ftθt + vt, vt ∼ N (0, Vt),

where Ft is a known matrix, and vt is a sequence of independent multi-variate normal random vectors, with zero mean and covariance matrix,Vt.

• The state equation of the system is

θt = Gtθt−1 + wt, wt ∼ N (0,Wt),

where Gt represents the state evolution matrix at time t, and wt isanother sequence of normal vectors, with zero mean and covariancematrices Wt.

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For t = 0, in the simplest case we assume a multivariate normal prior dis-tribution

θ0 ∼ N (m0,Σ0),

beingm0 the vector of means, and Σ0 the covariance matrix. The values of Vtand Wt can be specified using subjective or objective criteria. We have usedthe first approach, testing different possible values of Vt andWt with histor-ical data, and choosing the appropriate model using model selection criteriabased on the Median Absolute Deviation (MAD), or the Mean Absolute Per-centage Error (MAPE), see [8]. For each adjusted model, we have checkedwhether the assumptions of normality, independence and homoscedasticityhold, analyzing the residuals. All our models assume constant Ft, Gt, as wespecify below. Forecasting with this type of models is described in [6, 7, 8].

4.1 Forecasting model for Capanda inflows

We present now the forecasting model for the Capanda monthly inflows.Figure 2a describes the records of such inflows between January 2003 andDecember 2008.

Water inflow to Capanda

Time (months)

Flo

w (

m3 )

2003 2004 2005 2006 2007 2008 2009

1e+

093e

+09

(a) Original

Log−series of water inflow to Capanda

Time (months)

Flo

w

2003 2004 2005 2006 2007 2008 2009

20.0

21.0

22.0

(b) Log-transformed

Figure 2: Capanda monthly inflows between January 2003 and December2008.

As we can see, there is no trend in the data, and there is also a lack ofstationarity in variance. This leads us to apply logarithms to minimize thisphenomenon. We shall denote the transformed series by i

logPt

. Figure 2bplots the log-transformed series, which has less pronounced peaks.

As we can observe, the log-transformed series lacks a trend, and showsevidence of a seasonal pattern, as usual in monthly series of natural phenom-ena. In consequence, a model for local and seasonal level seems appropriateto represent the dynamic features of our data. We denote this model byM1.In this model, the state vector θt has dimension 12. The first term refers to

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the inflow level, whereas the other 11 terms refer to the seasonal part. (Theseasonal pattern is annual, with 12 months, but we constrain the seasonalterms to add up zero, so that there are 11 seasonal terms). We shall assumethat matrices Vt and Wt for model M1 are constant, taking the followingvalues:

ilogPt

=(1 1 0 · · · 0

)· θt + vt, vt ∼ N (0, 0.001) ,

θt =

1 0 0 0 · · · 00 −1 −1 −1 · · · −10 1 0 0 · · · 00 0 1 0 · · · 0...

......

.... . .

...0 0 0 · · · 1 0

· θt−1 +wt,

wt ∼ N

0,

109975.95 0 0 · · · 00 45.725 0 · · · 00 0 0.16 · · · 0...

......

. . ....

0 0 0 · · · 0.23

.

For simplicity, we have not shown the remaining diagonal elements of Wt,as they all have values close to zero. We use a noninformative prior forθ0. The predictive performance of the model is satisfactory, with low MADand MAPE. Besides, the analysis of residuals supports the hypothesis ofhomoscedasticity, independence and normality of the error terms.

4.2 Forecasting model for Cambambe inflows

We present now the prediction model for Cambambe inflows, based on thehistorical data available between 1944 and 1981, before the existence ofCapanda, which, we recall, is located upstream.

Figure 3a shows the absence of trend, although there is a pronouncedseasonality. Besides, it is clear that this time series is non stationary. Thiscan be mitigated by applying logarithms. The new series, denoted by ilogBt

,still shows fluctuations over time, and the seasonality previously observed,as shown in Figure 3b.

As the log-transformed data still present variability, and there is absenceof any trend, a local and seasonal model seems suitable for analyzing theCambambe inflows. We denote this model by M2.

ilogBt

=(1 1 0 · · · 0

)· θt + vt, vt ∼ N (0, 2) ,

10

Water inflow to Cambambe

Time (months)

Flo

w (

m3 )

1950 1960 1970 1980

0e+

004e

+09

(a) Original

Log−series of water inflow to Cambambe

Time (months)

Flo

w

1950 1960 1970 1980

1920

2122

(b) Log-transformed

Figure 3: Cambambe monthly inflows of January 1944 to December 1981.

θt =

1 0 0 0 · · · 00 −1 −1 −1 · · · −10 1 0 0 · · · 00 0 1 0 · · · 0...

......

.... . .

...0 0 0 · · · 1 0

· θt−1 + wt,

wt ∼ N

0,

1118.75 0 0 · · · 00 95.5 0 0 · · · 00 0 86.64 0 · · · 00 0 0 0.21 · · · 0...

......

.... . .

...0 0 0 0 · · · 0.17

.

Again, we do not shown all the diagonal elements in Wt, and we use anoninformative prior for θ0. The predictive performance of the model issatisfactory, with low MAD and MAPE. Besides, the analysis of the residualssupports the hypothesis of homoscedasticity, independence and normality ofthe error terms.

From model M2, we make predictions for the period 2003-2008, which,together with the actual data of inflows to Capanda, will be used as inputdata into our dynamic regression model (13). Figure 4a shows the behaviorof inflows to Cambambe (solid line) and Capanda (dotted line) between2003 and 2008. Both series present a similar qualitative behavior, makingreasonable the postulated dynamic regression model.

As stated, the analysis of the inflows between Capanda and Cambambe,using a dynamic linear model, requires the determination of αt and βt in (13).The maximum likelihood estimators of the variances of the observations, theintercept and the slope are 0.99, 0.99 and 0.071, respectively. The smoothed

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Time (months)

Flo

w (

m3)

2003 2004 2005 2006 2007 2008 2009

1e+

09

3e+

09

5e+

09

Cambambe Capanda

(a) Comparative inflows

5.4

57895

5.4

57905

Inte

rcept

0.5

1.0

1.5

2003 2004 2005 2006 2007 2008 2009

Slo

pe

Time (months)

(b) DLM coefficients

Figure 4: Relation between Capanda and Cambambe inflows.

coefficients of the model estimates in (13) are presented in Figure 4b.Specifically, the equations for the dynamic linear regression model are

iBt=

(1 iPt−1

)(αt

βt

)+ vt, vt ∼ N (0, 0.99) ,

with(αt

βt

)=

(1 10 1

)(αt−1

βt−1

)+ wt, wt ∼ N

(0,

(0.99 00 0.071

)).

We use noninformative priors for α0 and β0. The analysis of Figure 4bsuggests that the intercept αt and the slope βt change over time. Further-more, the slope estimates of the regression model indicate inflows elasticityto Cambambe, i.e., it reflects the impact on Cambambe inflows given a vari-ation in the inflows to Capanda. However, we should remark that, althoughthe upper part of Figure 4b suggests that the intercept varies with time,this variation is actually small.

4.3 Forecasting water demand in Luanda

We present now the analysis of water demand in Luanda. Figure 5a showsthat this time series is not stationary, showing an increasing trend overtime. To reduce variability, we apply logarithms to the observed demanddata, denoting the new series by d

logurbt

, and representing it in Figure 5b.The analysis of Figure 5b shows that the transformed data series of waterdemand is not stationary, and that there is still a trend. It also showsthe existence of a possible seasonality in the data under study. Therefore, alocal growth (linear trend) and seasonal model seems suitable to characterizewater demand in Luanda. We denote this model byM3. Its state vector hasdimension 13, being its first two elements related to the intercept and slope of

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Water demand in Luanda

Time (months)

Wat

er d

eman

d (m

3 )

2004 2005 2006 2007 2008 2009

2500

000

4000

000

(a) Original

Log−series of water demand in Luanda

Time (months)

Flo

w

2004 2005 2006 2007 2008 2009

14.8

15.0

15.2

15.4

(b) Log-transformed

Figure 5: Monthly water demand in Luanda.

the linear trend, and the remaining 11 to seasonal factors, respectively. Thevariance matrices of the system, Vt, and the observations, Wt, are assumedto be known, with structure as follows

dlogurbt

=(1 0 1 0 · · · 0

)· θt + vt, vt ∼ N (0, 2) ,

θt =

1 0 0 0 · · · 00 1 0 0 · · · 00 0 −1 −1 · · · −10 0 1 0 · · · 0...

......

.... . .

...0 0 0 · · · 1 0

· θt−1 + wt,

wt ∼ N

0,

328.75 0 0 0 0 0 · · · 00 15.5 0 0 0 0 · · · 00 0 10.25 0 0 0 · · · 00 0 0 30.38 0 0 · · · 00 0 0 0 45.64 0 · · · 00 0 0 0 0 0.17 · · · 0...

......

......

.... . .

...0 0 0 0 0 0 · · · 0.07

.

Again, the remaining diagonal elements in Wt are close to zero and are notshown. We use a noninformative prior for θ0. The predictive performance ofthe model is satisfactory, with low MAD and MAPE. Besides, the analysisof the residuals supports the hypothesis of homoscedasticity, independenceand normality of the error terms.

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4.4 Evaporation forecast

Regarding the volume of water evaporated in Capanda and Cambambe, wehave used Visentini’s approach [5] in which the evaporation depends onlyon the surface water temperature. Table 1 presents monthly data on theaverage evaporation heights and temperature in Capanda and Cambambe

Table 1: Average evaporation height (m) and temperatures (◦C) in Capandaand Cambambe

Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Capanda 0.154 0.154 0.165 0.145 0.154 0.149 0.149 0.151 0.141 0.147 0.142 0.139Cambambe 0.174 0.173 0.169 0.139 0.132 0.108 0.104 0.116 0.145 0.155 0.158 0.158Temperature 21.8 22.3 22.3 22.0 21.3 19.7 19.5 20.8 21.8 21.9 21.4 21.6

Based on the data in Table 1, we can obtain evaporation models forCapanda and Cambambe, applying a nonlinear regression model of the form

ev = a · T b, (16)

where a and b are parameters to be estimated. We have taken logarithmsin both sides of (16), and applied an ordinary least squares procedure toestimate a = 0.10, and b = 0.13 for Capanda, and a′ = 1.90 · 10−6, andb′ = 3.66 for Cambambe.

5 Preference Modeling

We formulate now the utility functions associated with the objectives de-scribed in Section 3.4. Let us designate by

• g1, the component utility function associated with energy productionin Capanda and Cambambe, et = eBt

+ ePt;

• g2 and g3, the component utility functions associated with water re-leases through spillgates in Capanda and Cambambe, respectively;

• g4, the component utility function associated with water deficit forhuman consumption in Luanda.

We aim at

max g1 = f1(et),

min g2 = f2(u2Bt),

min g3 = f3(u2Pt),

min g4 = f4(δurbt),

where δurbt = max(durbt − uurbt , 0) represents the water deficit in Luanda.

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We have used the probability equivalent method [10] to assess severalvalues for each component utility function and then fit an appropriate curvethrough least squares. As an example, Table 2 shows various levels of energyproduced and their assessed utilities.

Table 2: Values for the utility function on energy produced

Consequences (Gwh) 0 100 200 300 500 600 650 700 774

Utilities 0 0.512 0.64 0.80 0.96 0.992 0.9984 0.99968 1

Applying least-squares to the data in Table 2, we get the following expressionfor the utility function over energy production:

g1(et) = 1.004 − 0.991 · exp(−0.012 · et),

which is plotted in Figure 6a. Note that the utility function is increasingand suggests risk aversion.

0 200 400 600 800

0.0

0.2

0.4

0.6

0.8

1.0

Energy (GWh)

Util

ities

(a) g1(et)

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Spillgates releases in Cambambe (m3·s−1)

Util

ities

(b) g2(u2Bt)

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

Spillgates releases in Capanda (m3·s−1)

Util

ities

(c) g3(u2Pt)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

Water deficit in Luanda (m3·s−1)

Util

ities

(d) g4(δurbt)

Figure 6: Utility functions

We proceed in a similar way with the other three objectives. For g2,Figure 6b shows the utilities associated with spillgate releases in Capanda.

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The analytic expression for the utility function is

g2(u2Pt) = −0.045 + 1.098 · exp(−0.475 · u2Pt

).

g3 refers to releases through spillgates in Cambambe, as shown in Figure 6c.Its analytical expression is:

g3(u2Bt) = −0.007 + 1.016 · exp(−0.455 · u2Bt

).

Finally, g4, plotted in Figure 6d, represents the utility over water deficit forurban consumption in Luanda, and has the expression

g4(δt) = −0.007 + 1.058 · exp(−1.232 · δt).

We assume that the utility function is weighted additive, see [11]. Therefore,the global utility function has the expression

g(dect, unct) = w1 · g1(et) + w2 · g2(u2Pt) + w3 · g3(u

2Bt) + w4 · g4(δt),

where dec represent the five decisions to be made (the water released throughturbines and spillgates in Capanda and Cambambe, u1Bt

, u2Bt, u1Pt

, u2Pt; and

the water delivered to Kikuxi plant, uurbt), and unc designates the sourceof uncertainty (inflows to reservoirs, iBt

, iPt; evaporations, evBt

, evPt; and

water demand for human consumption, durbt). The values wi, i = 1, 2, 3, 4,are the weights associated with each component utility function. To as-sess them, we have used the swing weights method [12] which provides(0.66, 0.04, 0.03, 0.27). Then, if we aim at planning monthly within a T -month horizon, the utility function may be described as

g(dec, unc) =

T∑

t=1

g(dect, unct).

6 Optimization Problem

We now combine the above information to formulate our problem as find-ing, for each month, the controls dec =

[(u1Bt

, . . . , u1Bt+T

),(u2Bt

, . . . , u2Bt+T

),(

u1Pt, . . . , u1Pt+T

),(u2Pt

, . . . , u2Pt+T

),(uurbt , . . . , uurbt+T

)]which maximize the

expected utility

ψ(dec) =

∫· · ·

∫g(dec, unc) dunc, (17)

subject to constraints (4), (5), (6), (7), (8), (9), (10) and (15).The objective function (17) cannot be expressed in explicit form. Thus,

we use an MC approximation for the objective and, if necessary, its gradient.Explicitly, the MC approximation of (17) is:

ψ(dec) =1

N·

N∑

k=1

g(dec, unc(k)),

16

where{unc(k)

}N

k=1is a sample from the forecasting models described in

Section 4, over the period {t, . . . , t + T}. We have used an interior-pointalgorithm as implemented in Matlab 7.7.0 on Windows for the optimizationproblem. There were two factors that limited computations: the numberT of months considered and the size N of the MC approximation. Weperformed a simple computational study to ascertain how big T and N

could we cope with, with results expressed in Table 3

Table 3: Computational time (min) for various T and N

N/T 2 3 4 5 6 7 8 9 10 11 12

1000 24 39 63 98 142 201 288 399 533 714 8992000 70 90 155 240 439 640 911 1217 1636 2110 · · ·3000 95 120 325 455 580 · · ·4000 120 170 480 620 · · ·5000 148 396 919 · · ·

As a consequence, we finally implemented the approach shown in Figure7 for T = 12 and N = 1000. We first run the forecasting model and, then,obtain the optimal controls that maximize the expected utility (17) for theforthcoming T months. We only implement the controls for the first period,dect. We collect new data, update the forecasting models, and repeat theprocess for the next period t+ 1.

Forecast

t, t + 1, . . . , t + T

Optimize

t, t+ 1, . . . , t + TImplement dect

Collect

new data

t ←− t + 1Update

forecast model

Figure 7: Basic scheme of the Kwanza management model.

After performing the above scheme for a period of 12 months, we plotin Figure 8 the maximum expected utility controls. As we can observe inFigures 8a and 8b, our strategy tends to keep the values of water releasedthrough spillgates reasonably low, whereas the volumes of water releasedthrough turbines are close to the maximum capacity of the plants, ensuring,in this way, a nearly optimal use of water resources. Regarding Figure 8c,we can see that the problem of water scarcity in Luanda is far from beingsolved, as only about 50% of the nominal capacity of the Kikuxi plant wouldbe actually used even with the optimal controls. However, this problem hasmore to do with deficiencies on the pipes and the dilapidated state of the

17

Kikuxi plant. Indeed, we are currently studying an optimal investment planto improve the performance of that plant.

2e+

084e

+08

6e+

088e

+08

1e+

09

Water releases in Capanda

Period

Flo

w (

m3 )

Jan09 Mar09 May09 Jul09 Sep09 Nov09

Through turbinesThrough spillgatesMaximum flow through turbines

(a) Capanda releases1e

+08

3e+

085e

+08

7e+

08

Water releases in Cambambe

PeriodF

low

(m

3 )

Jan09 Mar09 May09 Jul09 Sep09 Nov09

Through turbinesThrough spillgatesMaximum flow through turbines

(b) Cambambe releases

Figure 8: Optimal controls.

4e+

065e

+06

6e+

067e

+06

8e+

06

Water released for human consumption

Period

Flo

w (

m3 )

Jan09 Mar09 May09 Jul09 Sep09 Nov09

Water releasedMaximum treatment capacity

(c) Human consumption

Figure 8: Optimal controls.

The outputs of the system, compare very favorably with those of the cur-rent management, as plotted in Figure 9. As we can observe in Figure 9a,the energy production levels with the optimal planning strategy improve theactual data registered for 2009. It is specially relevant the value of June, inwhich the energy production dropped on about 20% with respect to the pre-vious months due to an extraordinary dry rainy season. However, with ourmodel, this water shortage could have been predicted using the forecastingmodels, and a more adequate policy on the water releases would maintainthe energy production level. Similar results are observed on January andDecember, and with the optimal planning it would have been possible tomitigate such abrupt production falls. In general terms, we may say thatthe attained energy production is higher and smoother. Regarding Figure9b, we can see that the deficit on water for human consumption would have

18

been considerably reduced on average if we had implemented the optimalcontrols, when compared with the current management. This opens up theissue of fair distribution of water when this is a scarce resource, for whichseveral solutions have been proposed in the last years see e.g. [13, 14], andreferences therein.

7.0e

+12

8.0e

+12

9.0e

+12

Energy produced

Period

Ene

rgy

(J)

Jan09 Mar09 May09 Jul09 Sep09 Nov09

Optimal planningCurrent planning

(a) Energy produced

2600

000

2800

000

3000

000

Water deficit

Period

Flo

w (

m3 )

Jan09 Mar09 May09 Jul09 Sep09 Nov09

Optimal planningCurrent planning

(b) Water deficit

Figure 9: Outputs of the system.

7 Decision Support System for the Management

of the Kwanza River

Given the relevance of the proposed model, we have designed a simple Deci-sion Support System (DSS) to facilitate operational decisions at the Kwanzariver main installations (Capanda, Cambambe, Kikuxi). The system is de-veloped in Java, with calls to R programs that implement the forecastingmodels and Matlab programs that perform the optimization. The interfaceis also developed in Java. The basic loop performed by our DSS has beenplotted in Figure 7.

The architecture of the DSS is composed of three layers, as shown inFigure 10:

• The Front End layer, developed in Java, includes a Java GUI for R(JGR) interface;

• The Computing layer, which calls to R and Matlab programs; and

• The Data layer, exchanging plain and XML files.

At each time step, the system provides the optimal controls (water releasesthrough turbines and spillgates, and water released for human consumptionin Luanda), and outputs (water demanded and water deficit in Luanda, and

19

Figure 10: DSS architecture.

energy production in Capanda and Cambambe). The system also providestracking and monitoring of other variables of interest, such as reservoir levelsat Capanda and Cambambe, and navigation level at the lower Kwanza river.

As an additional feature, the system also assesses unforeseen events,such as unexpected high inflows to facilitate interventions in that kind ofsituations. Thus, our DSS is endowed with a management by exceptionprinciple, see [6].

8 Discussion

We have provided a model for the monthly operational management of theKwanza river. The main difficulties in the problem derive from the mul-tiobjective nature stemming from the multipurpose nature of the system(hydropower, flood protection, urban consumption); the many sources ofuncertainty present (inflows, evaporations, water demand); and the needto operate over time and respecting the constraints. There are also severalcomputational limitations which we have dealt with MC approximations andlimiting the planning horizons.

The results obtained compare favorably with respect to current manage-ment in terms of more energy produced, less water released through spill-

20

gates and less water deficit. As a consequence, we have developed a simpleDSS to support the Kwanza river operations. The DSS facilitates the re-quired data collection, forecasting, optimization and monitoring tasks.

The Kwanza river has an enormous potential in hydropower generationand we are currently studying what would be an optimal investment plan toexpand the capacity of its hydropower scheme and improve the transmissionlines. This is very important, given the growing electricity demand in a fastdeveloping country as Angola is.

Acknowledgements

Work supported by the Spanish Ministry of Science and Innovation pro-grams MTM2009-14087-C04-01 and RIESGOS, the Government of MadridRIESGOS-CM program S2009/ESP-1685, and the AECID URJC-U11N pro-ject on Decision Engineering.

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[11] R. T. Clemen, Making Hard Decisions: An Introduction to DecisionAnalysis, 2nd edn. (Duxbury Press, Belmont, 1996).

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