STOCHASTIC OPTIMIZATION, INCLUDING CONDITIONAL
VALUE AT RISK CONSTRAINTS, OF LONG TERM ELECTRICITY TRADING
JESÚS M. VELÁSQUEZ BERMÚDEZ
DECISIONWARE LTDA.
jvelasquez@decisionware-ltd.com
presented in
SIMMAC XIII
INTERNATIONAL SYMPOSIUM ON MATHEMATICAL METHODS APPLIED TO THE SCIENCES
SUBMITTED TO THE SPECIAL VOLUME ON "OR MODELS FOR ENERGY POLICY, PLANNING
AND MANAGEMENT" OF ANNALS OF OPERATIONS RESEARCH
ABSTRACT
The necessary modeling to optimize electricity trading through standardized financial instruments and long
term bilateral contracts is analyzed based on the existing practical and theoretical developments in the area
of investment portfolios optimization. As a result, a linear stochastic optimization model is formulated.
Experimental results for a "simple" electricity market, integrated by a spot market and a non standardized
long term contracts market, are presented. The experiments compare different utility functions, including
Conditional Value-at-Risk constraints.
Keywords: Electricity Trading, Stochastic Optimization, Conditional Value-at-Risk Constraints
INTRODUCTION
The decisions associated with marketing electricity in the long run, through bilateral contracts or through
standardized financial instruments, as options and forwards, are equivalent to those which are taken when the
distribution portfolio is optimized. In the case of the electricity market exists, at least, an option of purchase,
or of sale, by each hour of the planning horizon.
The decision maker, buyer or seller, must face at least two risks: the price risk and the volumetric risk. The
combination of these two risks causes that the traditional tools oriented to the solution of simple cases will not
be effective upon facing this type of problems; more yet if we have in the mind that the relationship of the
previous risks with the hydro-meteorological processes, in the supply and in the demand side, establishes
strong and complex correlation structures. The decision process is complicated upon considering that the
negotiation modalities imply permanent commitments of purchase/sale electricity by long periods, what
increases the magnitude of the extreme risks.
The use of stochastic optimization models to support financial decisions is wide. Rachev and Tokat (2001)
present a summary of the different operations research methodologies that have been used to solve real
problems. Other paper that includes an overview is presented by Mulvey et al. (1997).
In electricity trading, the decisions are prices and quantities. The stochastic optimization models combined
with the appropriate measurement of the risks, as Value at Risk -VaR-, are an appropriate way for this, since
the primal variables are associated with the quantities, and the dual variables with the prices.
1.
MATHEMATICAL MODELING
In the following, it is considered a decision process in which the vector X represents the long term market
decisions, the vector Y the parameters associated with random scenario, the function f(X|Y) the income, or
costs, associated with a decision X since occurs the random condition Y, and p(Y) the probability distribution
function of Y.
1.1.
REFERENCE FRAMEWORK
A model to optimize the decisions of electricity trading is described. The modeling process must consider that
the regional regulatory entities have different conceptions about what "must be" a competitive electricity
market. Therefore, aspects of the models that depend on each specific market exist. In this document it is
assumed a "simple" market in which the agents can accomplish long term transactions under different
modalities and short term transactions in the spot market in which the agents should sell the surpluses, or buy
the shortages, of electricity that result as consequence of the transactions consolidation in the long term
market consider their supply and their demand.
In a long term commodity market the main factor of the decisions is the expectations about the future price of
the commodity in the spot market. If the probabilistic characteristics of the stochastic process that determines
the future spot price are known, it is possible to generate synthetic series of spot prices to use them in a
stochastic optimization model and thus to represent the random environment of the decision process.
There are many alternatives to generate spot prices in an electricity market. One of them it is to consider that
in a hydrothermal system the spot price depends on the hydro-climatic conditions, the existing industrial
infrastructure, the demand level and the "market regulation" and to obtain synthetic series of spot prices as
functions of the marginal costs of the demand equation in a minimum cost hydrothermal dispatch model
(Velásquez and Nieto 1999, Pereira and Campodómico 1995). Another possibility is to use equilibrium
models more oriented to competitive markets. Also it is possible to generate synthetic series using statistic
models and/or artificial intelligence methodologies taking as reference the historical series. This document
does not has as objective to analyze the mathematical models that can be used to determine long run
projections of the spot price. Obviously, the generation of this type of series makes part of a deep formal
study since that the "optimal" decisions depend on the knowledge of the random environment; if this
knowledge is weak, we can not wait "good" decisions.
In the following sections is assumed that exist synthetic series of spot prices that represent the random
environment of the decision making process. Each synthetic series is associated to a possible future scenario
in a stochastic optimization model called as OPTMER (in Spanish "OPTmización del MERcadeo").
OPTMER is a stochastic linear optimization model that supports the decisions of an electricity generator, of a
distributor, of a trader and/or of a vertical integrated company. This document only consider the math formulation
for a trader that accomplishes purchase/sale transactions with multiple agents or clients. In the spot market the
trader purchase/sells the deficits/surpluses not covered in the long term market.
The agent attends a demand that it can be composed by regulated clients of mandatory attention and/or by
bilateral contracts subscribed with unregulated clients or with other agents. For simplicity, the math
formulation does not consider commercial differentiation factors between the clients, the contracts and/or the
agents.
According to the typical characteristics of the prices structure in the electricity markets the modeling is based
on the following concepts:
 Planning Periods: one, or several months
 Day Types: ordinary, Saturday and holidays.
 Load Blocks: a hour or a group of hours
 Contracts: Take or Pay and Options
 Negotiation Modalities:
 free blocks: irregular electricity sales for any load block
 monthly blocks: electricity blocks for all hours of a month, or a period.
 annual blocks: electricity blocks for all hours of the year, or the planning horizon.
1
 monthly blocks modulated by a load curve: electricity blocks for all hours of a month as a percentage
of the buyer load curve
 annual blocks modulated by a load curve: electricity blocks for all the hours of a year as a percentage
of the buyer load curve
 monthly options: options of electricity blocks for all hours of a month
 annual options: options of electricity blocks for all hours of a year
The previous negotiation modalities are some examples that can exist in the long term market. Each modality
implies a set of variables and restrictions especially oriented to describe the associated financial flows. To
include new modalities implies to include new equations and variables.
OPTMER can be used from two points of view:
 Physical: to determine the quantities to buy or to sell of a set of offers
 Economic: to determine prices and quantities to include in an offer that an agent will present to another agent.
In the first case the decisions are the quantities to contract in each hour under each negotiation modality, related
with the primal variables. In the second, the decisions are limits for the prices, called equilibrium prices, for those
which are convenient to accomplish the transactions, related with the dual variables. Based on a sensibility
analysis is possible to build agent's supply-demand curves for the long term market.
1.2.
MATH FORMULATION
To be short, the detailed math formulation is limited to present the parameters, the equations and the variables
related to purchase offers and ignores the possibility of sale offers, those which can be included using a
similar procedure to the described in the present document.
1.2.1.
DEFINITIONS
The index used in the model are t for period (month), d for type of day, b for load block, g for agent that sells
electricity and h for random scenario.
The parameters used in the model are (in cursive letter):
 Deterministic parameters
Electricity prices ($/MWh)
PCBAg
Agent g price for annual blocks
PCBMt,g
Agent g price for monthly blocks in month t
PCLIt,d,b,g Agent g price for free blocks in month t type day d block b
PCMAg
Agent g price for modulate annual blocks
PCMMt,g
Agent g price for modulate monthly blocks in month t
PCOAg
Agent g price for annual options
PCSAg
Agent g strike price for annual options
PCOMt,g
Agent g price for monthly options in month t
PCSMt,g
Agent g strike price for monthly options in month t
Electricity quantities (MWh)
DVBAg
Agent g availability for annual blocks
DVMMt,g Agent g availability for modulate monthly blocks in month t
DVBMt,g
Agent g availability for monthly blocks in month t
DVMAg
Agent g availability for modulate annual blocks
DVLIt,d,b,g Agent g availability for free blocks in month t type day d block b
DVOAg
Agent g availability for annual options
DVOMt,g
Agent g availability for monthly options

Random Parameters (components of the vector Yh)
DEMt,d,b,h Demand (regulated plus contracts) in the month t type day d block b under random
condition h
PSPt,d,b,h
Spot price in month t type day d block b under random condition h
2
t,h
t,h
Coefficient associated with the payment capacity of the spot market in month t random
condition h
Coefficient associated with payment time of the spot market in month t random condition h
The variables used in the model are (in normal letter) :
 Long term market decisions (deterministic variables, components of X)
CLPt,d,b,g Total electricity bought to the agent g in month t type day d block b
CBAg
Electricity bought in annual blocks to the agent g
CBMt,g
Electricity bought in monthly blocks to the agent g in month t
CLIt,d,b,g Electricity bought in free blocks to the agent g in month t type day d block b
CMAg
Fraction of electricity demand bought in annual modulate blocks to the agent g
CMMt,g Fraction of electricity demand bought in monthly modulate blocks to the agent g in month t
COAg
Annual options of electricity blocks bought to the agent g
COMt,g Monthly options of electricity blocks bought to the agent g in month t

Simulated variables (random variables)
VMSt,d,b,h Sales in the spot market in month t type day d block b under random condition h
CMSt,d,b,h Purchases in the spot market in month t type day d block b under random condition h
1.2.2.
INCOME FUNCTION
The income are split into deterministic and stochastic. The deterministic part, d(X), does not dependent of the
random conditions and corresponds to commitments derived from the decisions in the long term market: the
costs of blocks of electricity and the costs of the options. d(X) can be expressed as
d(X) = g[ CBAg PCBA + COAg PCOAg + t CBMt,c PCBMt + COMt,g PCOMt,g +
db CLIt,d,b,g PCLIt,d,b,g + CMMt,g PCMMt,g DEMt,d,b + CMAg PCMAg DEMt,d,b ]
(1)
where X represents the decision vector related with long term market transactions. The stochastic income,
r(X|Yh), correspond to the purchases or sales in the spot market and to the exercise, or not, of the options.
They depend on the random condition h, and can be expressed as

COP(X|Yh): expenditures by exercises the options
COP(X|Yh)g t db Minimum (PCSAg ,PSPt,d,b,h) COAg + Minimum (PCSMt,g ,PSPt,d,b,h) COMt,g (2)


IMS(X|Yh): incomes by sales in the spot market
ISM(X|Yh) = t db Maximum(0, g CLPt,d,b,g - DEMt,d,b) PSPt,d,b,h t,h t,h

(3)
EMS(X|Yh): expenditures by purchases in the spot market 
ESM(X|Yh) = t db Maximum(0, DEMt,d,b - g CLPt,d,b,g) PSPt,d,b,h
(4)

where Yh represents the random parameters vector associated with the condition h. The income and the
expenditures in the spot market are considered independently due to the fact that would exist asymmetry in the
spot market payment conditions (t,h and t,h). The previous financial movements are caused in the future and
represent the risk of the decision. Their net value is r(X|Yh)
r(X|Yh) = ISM(X|Yh) - COP(X|Yh) -ESM(X|Yh)
(5)
The total income, deterministic plus stochastic, is
f(X|Yh) = d(X) + r(X|Yh)
3
(6)
If a stochastic linear programming model is formulated, the variables contained within the Maximum function
must be represented by a set of linear equations using the process that is described below. The following
expression is considered
Maximum [ 0 , z ] P
(7)
where the variable z is not restricted and it is represented as the difference of two positive variables
z = z + - z-
(8)
based on the previous change of variables we have
Maximum [ 0 , z ] P = z+ P
(9)
For a correct representation, it should be to guarantee that one of the two z-variables will be equal to zero,
what is procured in linear programming due to the colineality between z+ and z-. Based on the foregoing, the
following definitions can be considered
zt,d,b,h = VMSt,d,b,h - CMSt,d,b,h = g CLPt,d,b,g - DEMt,d,b
(10)
where CLPt,d,b,g represents the total purchases of electricity to the agent g in month t type day d block b under
random condition h and it is defined by the sum of negotiation modalities
CLPt,d,b,g = CBAg + CBMt,g + COAg + COMt,g + CLIt,d,b,g + CMMt,g DEMt,d,b + CMAg DEMt,d,b
Then
ISM(X|Yh)t db VMSt,d,b,h PSPt,d,b,h t,h t,h
(11)
(12)
ESM(X|Yh) = t db CMSt,d,b,h PSPt,d,b,h
(13)

The equations set {1, 2, 5, 6, 10, 11, 12, 13} constitute a linear system that describes the income/expenditures
that will have the agent and should make part of the optimization model; it will be called "the marketing
process constraints".
1.2.3.
OTHER CONSTRAINTS
1.2.3.1. LOAD MODULATION
The variables related to blocks modulated by a load curve are demand fractions and they should be ranged
between 0 and 1
CMMt,g 
(14a)
CMAg
(14b)
1.2.3.2. AVAILABILITY TO SALE
Normally, the agents receive electricity offers in those which a price is associated to a quantity that the seller
is prepared to committing. This implies bounds for the quantities to buy
CBAg  DVBAg
CBMt,g  DVBMt,g
CMAg  DVMAg
CMMt,g  DVMMt,g
COAg  DVOAg
COMt,g  DVOMt,g
CLIt,d,b,g  DVLIt,d,b,g
4
(15a)
(15b)
(15c)
(15d)
(15e)
(15f)
(15g)
1.2.3.3. FINANCIAL CONSTRAINTS
Having in mind that the decisions of long term electricity marketing have as purpose the financial risks
hedging, it can be necessary, and/or convenient, to include in the model constraints associated with these
flows. This topic is not studied in this document and the reader is referred to other specialized articles, for
example Cariño and Ziemba (1998).
1.3.
UTILITY FUNCTIONS
In a stochastic optimization model exists multiple possibilities to determine the decisions utility function. At
least two functions can be considered to measure the kindness of the decision (yield measure): the revenue
expected value and the regret due to the decision.
1.3.1. MAXIMIZE THE EXPECTED INCOME
In this case the objective of the optimization will be maximize the expected income. This is
Maximize d(X) + h r(X|Yh) /NE
(16)
where NE represents the number of random conditions. This approach does not imply the rationalization of
the of the risk management, and in many cases is equivalent to a deterministic model based on the expected
value of the random parameters (Yh).
1.3.2. MAXIMIZE THE MINIMAL INCOME
To maximize the minimal income can be expressed as
Maximize { d(X) + Minimumh [ r(X|Yh) ] }
(17)
The previous objective function can be represented by the following formulation
{ Maximize d(X) + Rmin | Rmin r(X|Yh) ; h=1,NE }
(18)
where Rmin is a not restricted variable that it should be substitute by the difference of two positive variables
{ Maximize d(X) + Rmin+ - Rmin- | Rmin+ - Rmin-  r(X|Yh) ;
h=1,NE }
(19)
The previous problem join with the marketing process constraints is a linear programming problem.
1.3.3. MAXIMUM REGRET
The regret is the difference between the revenue of the decision X since occurred the random condition Yh
with the revenue associated with the optimum decision, X*(Yh), that must be taken if we a priori known that
Yh was going to occur. It can be expressed as

(X/Yh) = d(X*(Yh)) + r(X*(Yh)|Yh) - d(X) - r(X|Yh)
(20)
where d(X*(Yh)) + r(X*(Yh)|Yh) represents the maximum revenue under random condition h. X*(Yh) is the
solution to the deterministic problem of maximizing the revenue given the random condition h.
This case considers the minimization of the maximum regret, also known as Savage criterion (Raiffa 1968),
Minimize Maximumh [ (X/Yh) ]
The previous objective function must be represented based on a set of constraints
5
(21)
{ Minimize Rmax | Rmax  d(X*(Yh)) + r(X*(Yh)|Yh) - d(X) - r(X|Yh) ; h=1,NE }
(22)
where Rmax is a positive scalar. The previous problem join with the marketing process constraints is a linear
programming problem.
1.3.4. EXPECTED INCOME AND CVaR INCOME CONTRAINTS
The introduction of a risk constraint gives rationality to the expected value utility function. The risk measure
more known is the Value at Risk -VaR- that corresponds to the superior limit of an interval for the losses
associated with a portfolio at a given probability level.
The introduction of VaR constraints has been studied widely (Anderson and Ursayev 1999, Uryasev 2000). It
has been demonstrated that the appropriate form to consider VaR constraints is using the Conditional Valueat-Risk (CVaR) risk measure. CVaR is the expected loss exceeding Value-at-Risk and it is also known as
Mean Excess, Mean Shortfall, or Tail VaR.
The basic problem can be formulated as
{ Maximize d(X) + h r(X|Yh) /NE | (X)   }
(23)
where (X) represents for expected revenue since it is less than  (X). (X) represents the income level that
it can be exceeded with a probability .  is the lower bound for (X), that can be written as
(X) = (X) - (1-)-1 h=1,NE Maximum[0, (X)-f(X|Yh)]/NE
(24)
(X) can be expressed by a set of linear inequations
(X) = (X) - (1-)-1NEh=1,NE h
h(X) - f(X|Yh) h
h0 h
(25a)
(25b)
(25c)
where h represents the income deficit with respect to (X) if is taken the decision X and occurs the random
condition Yh. The model can be written as
{ Maximize d(X) + h r(X|Yh) /NE |
(X) - (1-)-1NEh=1,NE h 
h(X) - d(X) - r(X|Yh) h
h0 h }
(26a)
(26b)
(26c)
(26d)
The previous problem join with the marketing process constraints is a linear programming problem.
1.4.
EQUILIBRIUM PRICES
The model generates information about the opportunity cost/benefit of the different negotiation modalities;
then it can be used to determine the equilibrium price of purchase/sale electricity for each modality. This price
corresponds to an indifferent price that maintains the utility level of the agent, and serves as reference to
establishes the convenience or not of a negotiation.
The prices, economic variables, are obtained from the dual variables of the model. It should be consider that
these prices are function of the quantity and that if we wished to have a demand curve it should be done a
sensibility analysis.
2.
EXPERIMENTAL RESULTS
6
Below it is presented a hypothetical case of a buyer agent that evaluates an electricity purchase competitive
process in which participate multiple seller agents. The planning horizon is one year and the numbers are
related with a "realistic" case in the Colombian Electricity Market. As additional condition, it is assumed that
the agent limits the total of annual purchases to his annual demand (which is assumed deterministic), that
implies
tdb CLPt,d,b tdb DEMt,d,b
2.1.
PARAMETERS
2.1.1.
ELECTRICITY DEMAND
(27)
In this case the demand is considered as a deterministic parameter. The buyer agent attends a demand that has
only one day type with the a typical load curve . Additionally it is known the aggregate monthly demand for
the planning horizon. The hourly demand is calculated combining the aggregate monthly demand with the
typical load curve. Alternatively, the hourly demand can be calculated taking into account the contracts that
attends the agent.
MW
270
1
260
0.5
250
0
240
1
7
13
19
01/01/00
05/01/00
09/01/00
Hour
Month
Figure 1. Normalized Load Curve
Figure 2. Aggregate Demand
2.1.2. OFFERS
The buyer agent has received quotes of eleven sellers in five modalities. The numerical details of the quotes
are omitted, but they are available in Velasquez (2001).
TABLE 1. OFFERS MODALITIES
2.1.3.
SELLER
Annual Blocks
Monthly Blocks
Annual Blocks
Modulated
Monthly Blocks
Modulated
CHBG
CHVG
CRLG
CTFG
EBSG
EEBG
EPMG
EPSG
GCLG
ISGG
TRMG
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Free Blocks
X
X
X
X
SPOT PRICE
The information about the future variability of the spot price is summarized in ten synthetic series. As
reference, the price average in the spot market is 53345 $/MWh (Colombian pesos by megawatt-hour),
superior to the price average in the long term offers, near to 40000 $/MWh.
7
$/MHh
175000
150000
125000
100000
75000
50000
25000
0
Figure 3. Spot Price Synthetic Series
2.2.
RESULTS
OPTMER is used to determine the optimum policy of agent purchases. The following utility functions were
analyzed: maximum cost, expected cost, maximum regret and expected cost with CVaR cost constraint. The
results are presented in two stages: initially some detailed results of two traditional utility functions, the
expected cost and the maximum cost, are compared; thereinafter, global results are presented for the others
utility functions.
2.2.1. MAXIMUM COST VERSUS EXPECTED COST
2.2.1.1. TOTAL COSTS
Below it is presented the comparative analysis of the results obtained for these cases. The monetary unit is
thousands of Colombian pesos ($**3)
TABLE 2. TOTAL COSTS ($**3)
LONG TERM
SPOT M ARKET
STATISTIC
TOTAL
PURCHASES
NET SALES
UTILITY FUNCTION: MAXIMUM COST
MEAN
77159.4
43632.3
33527.2
DEVIATION
0
5615.8
5615.8
MAXIMUM
77159.4
56835.2
37588.1
MINIMUM
77159.4
39571.3
20324.2
UTILITY FUNCTION: EXPECTED COST
MEAN
834960
1076067
-241107
DEVIATION
0
462877.7
462877.7
MAXIMUM
834960
1707938
455757
MINIMUM
834960
379203
-872978
Below the distributions of the costs are presented
%
100
MAXIMUM COST
75
50
25
0
-806541
EXPECTED COST
-540794
-275047
-9300
256447
$**3
Figure 4. Costs Distribution. Maximum Cost versus Expected Cost
8
An analysis of the previous graph concludes that the expected cost policy implies risk prone positions. In this
case, it can be verified that the results are equivalent to solve a deterministic optimization problem
using the expected spot price. Alternatively, minimizing the maximum cost the agent rationalizes his
purchases in the long term limiting them to the minimal possible risk, but it is a policy totally risk averse.
2.2.1.2. EQUILIBRIUM PRICES
The equilibrium prices, based on the dual variables, indicate balance prices, in the sense that produce an
utility equal to zero if negotiations are accomplished at that price in the indicated modality. The equilibrium
prices based on the expected cost permit to assume much more risk and therefore they are but large.
42000
MAXIMUM COST
$/MWh
$/MWh
80000
EXPECTED COST
EXPECTED COST
MAXIMUM COST
38000
50000
20000
34000
Ene-99
Mar-99
May-99
Jul-99
Sep-99
Nov-99
Ene-99
Figure 5. Equilibrium Prices - Monthly Blocks
Mar-99
May-99
Jul-99
Sep-99
Nov-99
Figure 6. Equilibrium Prices - Modulate Monthly Blocks
2.2.1.3. PURCHASES
Below the results related to the quantities to negotiate in some modalities are presented. The optimum
purchases for annual blocks are in Table 3.
AGENT
CHBG
EPMG
EPSG
ISGG
TABLE 3. OPTIMUM PURCHASES ANNUAL BLOCKS
OFFER
POWER PURCHASES
SHADOW PRICE
QUANTITY-PRICE
(MW)
($/MWh)
POWER
PRICE
MINI
EXPECTED
MINI
EXPECTED
(MW)
$/MWh
MAX
VALUE
MAX
VALUE
540
40960
0
540
950
41600
0
950
35211
55854
70
37800
0
70
100
39700
0
100
The equilibrium prices are established using the reduced costs, "simplex multipliers", associated with the
upper bound of the purchase variables. It is evident the greater arrangement to pay of the expected cost policy.
Below the quantities to negotiate for the modality annual modulate blocks are presented.
AGENT
CHBG
EBSG
EEBG
EPMG
GCLG
ISGG
TABLE 4. OPTIMUM PURCHASES MONTHLY BLOCKS
OFFER
PURCHASES
SHADOW PRICE
QUANTITY-PRICE
DEMAND PERCENTAGE (%)
($/MWh)
PERCENTAGE
PRICE
MINI
EXPECTED
MINI
EXPECTED
(%)
U$/MWh
MAX
VALUE
MAX
VALUE
50
44733
0
0
100
37154
0
0
60
41560
0
0
1324
1355
60
40332
0
0
100
37632
0
0
50
38612
0
0
9
The reason of the equilibrium prices so low is that this modality commits all annual demand of the buyer
agent, subtracting to him flexibility to capture earnings in other negotiation modalities in the long term market
and/or in the spot market. This is coherent with the principle that the portfolio optimality is the diversification
and not the concentration, unless the prices offered are totally outside of market.
2.2.2. OTHER UTILITY FUNCTIONS
Below the comparative results for all the studied utility functions are presented. The analysis is concentrated
in the comparison of the distribution of the total costs. Detailed results are available in Velasquez 2001.
2.2.2.1. MAXIMUM REGRET
Below it is presented the cost distribution when is used as decision criterion the maximum regret
%
30
20
10
0
-611235
-243451
124333
$**3
Figure 7. Cost Distribution. Utility Function: Maximum Regret
Table 5 contains the comparative analysis of the results obtained in the three cases. The monetary unit is
millions of Colombian pesos ($**6)
TABLE 5. COMPARATIVE ANALYSIS OF UTILITY FUNCTIONS
($**6)
REGRET
TOTAL COST
RANDOM
SCENARIO
OPTIMUM
COST
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
-885
-21
-16
31
16
-200
-18
-894
-445
-745
-318
390
31
-894
926
MEAN
DEVIATION
MAXIMUM
MINIMUM
RANGE
MINI
MAX
MAX
REGRET
EXPECTED
VALUE
MINI
MAX
MAX
REGRET
EXPECTED
VALUE
921
41
54
6
20
236
49
929
483
772
351
390
929
6
923
274
43
75
277
40
58
24
277
137
210
142
108
277
24
253
22
68
100
424
66
11
41
21
10
2
77
126
424
2
422
36
20
38
38
36
36
31
35
38
28
34
6
38
20
17
-611
22
59
308
56
-141
7
-618
-308
-534
-176
324
308
-618
926
-864
47
84
456
82
-189
24
-873
-435
-743
-241
463
456
-873
1329
10
The results indicate that the policy of the maximum regret is an intermediate point between the expected cost
and the maximum cost. However, in this case, the maximum regret presents a high propensity to the risk. The
equilibrium prices are coherent with the previous affirmation. The next graph compares the equilibrium prices
for monthly blocks.
MAXIMUM COST
MAXIMUM REGRET
EXPECTED COST
$/MWh
80000
50000
20000
Ene-99
Mar-99
May-99
Jul-99
Sep-99
Nov-99
Figure 8. Equilibrium Prices - Monthly Blocks
2.2.2.2. EXPECTED COST INCLUDING CVAR CONSTRAINT
The decision making process based in to minimize the expected cost including CVaR constraint to a given
probability level was studied. Experiments with several probability levels and several limits for the CVaR are
presented. The results obtained in four cases in those which CVaR limit is fixed in 50 millions of Colombian
pesos and the probability level of not to exceed this limit is varied are presented in the next table.
RANDOM
SCENARIO
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
MEAN
DEVIATION
MAXIMUM
MINIMUM
RANGE
TABLE 6. COMPARATIVE ANALYSIS WITH FIXED CVAR LIMIT
TOTAL COST ($**6)
CVAR LIMIT = 50 ($**6)
OPTIMUM
PROBABILITY OF NOT TO EXCEED LIMIT - -
COST
0.0
0.60
0.75
0.95
-885
-864
-295
-130
-13
-21
47
-12
-6
8
-16
84
-2
-1
10
31
456
186
102
50
16
82
28
23
29
-200
-189
-74
-38
8
-18
24
-13
-8
6
-894
-873
-298
-132
-14
-445
-435
-130
-45
13
-745
-743
-257
-123
-21
-318
-241
-87
-36
8
390
463
158
75
21
31
456
186
102
50
-894
-873
-298
-132
-21
926
1329
484
234
71
The results are coherent with the theory: the increase in the probability level reduces the cost volatility. The
solution for  =0.0 corresponds to the case of minimizing the unrestricted expected cost. Below the
comparative analysis of six cases in those which is fixed the probability level and the CVaR limit is varied are
presented
11
TABLE 7. COMPARATIVE ANALYSIS WITH PROBABILITY LEVEL FIXED
TOTAL COST ($**6)
= 0.95
RANDOM
MINI
EXPECTED
CVAR LIMIT ($**6)
SCENARIO
MAX
VALUE
38
40
50
75
100
200
1988
36
34
22
-13
-85
-148
-372
-864
1989
20
14
14
8
5
8
7
47
1990
38
25
22
10
10
20
30
84
1991
38
38
40
50
75
100
200
456
1992
36
32
32
29
27
30
38
82
1993
36
30
26
8
-20
-39
-93
-189
1994
31
22
19
6
1
1
-3
24
1995
35
33
22
-14
-86
-149
-376
-873
1996
38
38
31
13
-24
-59
-179
-435
1997
28
26
15
-21
-87
-143
-332
-743
MEAN
34
29
24
8
-18
-38
-108
-241
DEVIATION
6
8
8
21
54
86
199
463
MAXIMUM
38
38
40
50
75
100
200
456
MINIMUM
20
14
14
-21
-87
-149
-376
-873
RANGE
17
24
26
71
162
249
576
1329
Again, the results are coherent with the theory, when the CVaR limit is reduced the solution tends to the
maximum cost solution, when the limit increases the solution tends to the expected cost solution.
3.
CONCLUSIONS
It can be concluded that:
 The decisions based on the simple analysis of the expected cost put on serious danger the financial
stability of the agents. This fact includes the use of stochastic optimization models in the decision making
process.
 The policy of minimizing the maximum cost seems to extremely risk averse.
 The weakness of the policy of the maximum regret is that does not control the risk level, which it is a
result of the optimization process
 The incorporation of CVaR constraints in the policy of optimizing the expected costs is an interesting
alternative, since it permits to integrate the optimization of the expected revenue of the decisions with the
risk level that the decision maker wishes and/or can to assume. It can be asserted that assigning the
appropriate parameters this alternative represents anyone of the others utility functions, and therefore
covers all the cases, since it handles explicitly the risk levels and the expected utility that imply the
decisions.
 Finally, the results prove that the existing theory for the strategic managing of financial risks is valid and
coherent mathematically, and it can be extended to the electricity markets composed by a spot market and
a long term market.
REFERENCES
Andersson, F. and Uryasev, S. "Credit Risk Optimization with Conditional Value-at-Risk Criterion".
Research Report #99-9, Center for Applied Optimization, Dept. of Industrial and Systems Engineering,
University of Florida, 1999.
Cariño D. R. and Ziemba W. T. “Formulation of the Rusell-Yasuda Kasai Financial Planing Model”.
Operations Research Vol No. 4 (1998).
Edgeworth, F.Y., 1888, “The Mathematical Theory of Banking”, Journal of the Royal Statistical Society,
51/1, March, pp. 113-127
J.P.Morgan, 1996, “RiskMetrics™: Technical Document”, 4th ed., New York NY
Mulvey, J.M., Rosembaum, D.P. and Shetty, B. "Strategic Financial Risk Management and Operation
Research". European Journal of Operations Research 97 (1997) 1-16.
Pereira, M.V.F. y Campodómico N.V. “Modelo de Despacho HidroTérmico con Restricciones de
Transmisión. Technical Specifications. Agosto 1995.
12
Rachev, S. And Tokat, Y. "Asset and Liability Management: Recent Advances". Working Paper Institute of
Statistics and Mathematical Economics, School of Economics, University of Karlsruhe, Kollegium am
Schloss, Bau II, 20.12, R210, Post-fach 6980, D-76128, Karlsruhe, Germany. e-mail: rachev@lsoe.unikarls-ruhe.de. 2000.
Raiffa, H., (1968), Decision Analysis, Addison-Wesley, Reading, Mass.
Uryasev, S. "Conditional Value-at-Risk: Optimization Algorithms and Applications". Working Paper. Dept.
of Industrial and Systems Engineering, University of Florida, 2000.
Velásquez, J. y Nieto P., G. “Sistema SHTG/COLOMBIA. Despacho Integrado de Electricidad y Gas.
Versión Colombia”. Mundo Eléctrico Colombiano (Enero 1999), (http://www.decisionware-ltd.com).
Velásquez, J. "Optimización del Mercado de Energía a Largo Plazo". Documento de Trabajo DW-035-01.
2001 (http://www.decisionware-ltd.com).
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