1998 Long-Term Hydro Scheduling Based on Stochastic Models

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EPSOM’98, Zurich, September 23-25, 1998

Long-term Hydro Scheduling based on Stochastic Models

Mario Pereira Nora Campodónico Rafael Kelman

Power Systems Research Inc., PSRI Rio de Janeiro, [email protected], [email protected], [email protected]

Abstract: This paper describes some methodologies and tools being developed to address the

new challenges - and opportunities - posed by power sector restructuring in hydrothermalsystems: (a) optimal stochastic dispatch of multiple reservoir systems; (b) joint representation

of equipment outage and inflow uncertainty; (c) distortion of short-run marginal costs signals

when applied to cascaded plants with different owners; (d) economic efficiency and market

power issues in bid-based hydrothermal dispatch. The issues are illustrated with case studies

taken from the Colombian system.

Keywords: Hydrothermal Scheduling, Stochastic Optimization, Probabilistic Production

Costing, Market Power, Decentralized Dispatch.

1 Introduction

Electric utilities all over the world have been undergoing radical changes in their market and

regulatory structure. A basic trend in this restructuring process has been the replacement of

traditional expansion planning and operation procedures, based on centralized optimization, by

market-oriented approaches:

• Generators bid prices for their energy production (typically on an hourly basis for the next

day) in a Wholesale Energy Market – WEM. Units are then loaded by increasing price until

demand is met. Dispatched generators are remunerated on the basis of the system spot

price, which corresponds to the offer of the most expensive loaded unit.

• Instead of following an expansion schedule produced by a central planning agency, private

agents are free to decide on the construction of generating units and to compete for energy

sales contracts with utilities and individual customers. One of the key components in the

private investment decision is the forecast of WEM spot revenues for each plant, which are

then compared with the plant construction cost.

According to its proponents, one of the conceptually attractive aspects of the spot pricing

scheme is that, under perfect competition, it provides efficient economic signals for system

expansion, i.e. if the system is optimally dimensioned, the spot-based remuneration will match

investment costs plus operating expenses [1]. For similar reasons, it has also been argued that

the bidding scheme induces an efficient use of system resources in system dispatch.

However, the theoretical and practical validation of the above claims was primarily based on

thermal systems, and cannot be simply extrapolated to hydrothermal systems. The objective of

this paper is to describe some methodologies and tools being developed to address the new

challenges - and opportunities - posed by power sector restructuring in hydrothermal systems.




The paper is organized as follows. In section 2 we present an overview of hydrothermal

scheduling concepts and discuss the computational difficulty of finding an optimal strategy for

a multi-reservoir system, the so-called “curse of dimensionality”. We then describe a class of

solution procedures - stochastic dual dynamic programming - which is able to alleviate these

computational problems. In section 3, we discuss the integration of probabilistic production

costing models traditionally used in thermal system analysis into a hydrothermal scheduling

framework. In section 4, we analyze the distortion of economic signals resulting from WEM

spot prices when there are reservoirs in cascade, and describe an extended spot market where both energy and water are traded. Finally, in section 5 we address economic efficiency and

market power issues in bid-based hydrothermal dispatch.

2 Overview of Hydrothermal Scheduling

2.1 Purely Thermal Systems Characteristics

In purely thermal systems, the operating cost of each plant depends basically on its fuel cost.

Therefore, the scheduling problem is to determine the plant combination that minimizes the

total fuel cost required to meet the system load. In its simplest version, the scheduling problem

is formulated as:

z t = Min ∑ j=1

J

c( j) g t( j)

subject to (2.1)

∑ j=1

J

g t( j) = d t (2.1a)

g t ≤ g _

(2.1b)

where z t, c, d t, g t and g _ represent respectively the system operating cost in stage t , unit

operation costs, system load, power production and generation capacities. In turn, constraints

(2.1a) and (2.1b) represent respectively load supply and limits on generation capacity.

The thermal generation dispatch problem (2.1) can be solved by inspection: load generators by

increasing operating cost until demand is met. Although the actual scheduling problem is more

complex due to factors such as losses, transmission limitations, start-up costs, ramping rates

etc., the purely thermal scheduling problem retains some basic characteristics:

• it is decoupled in time, that is, an operating decision in stage t (e.g. this week) does not

affect next week’s operating decisions;• generating units have a direct operating cost, i.e. unit cost c( j) does not depend on the

output of the other system plants; besides that, plant operation does not affect the

generation capacity or availability of other plants; this provides a natural coordination

mechanism for energy purchase and sale




2.2 Hydrothermal System Characteristics

2.2.1 Time Dependence

Hydro plants can use the “free” energy stored in their reservoirs to meet demand, thus avoiding

fuel expenses with thermal units. However, the availability of this hydro energy is limited by

reservoir storage capacities. This introduces a relationship between the operative decision in a

given stage and the future consequences of this decision. For example, if the storedhydroelectric energy is used today, and a drought occurs, it may be necessary to use expensive

thermal generation in the future, or even interrupt the energy supply. If, on the other hand,

reservoir levels are kept high through a more intensive use of thermal generation, and high

inflows occur in the future, reservoirs may spill, which is a waste of energy and, therefore,

results in increased operation costs. Figure 2.1 illustrates the decision tree.

wet

dry

OK

deficitdry

wet

future inflows

usereservoirs

decision

do not usereservoirs

OK

consequencesoperating

spillage

Figure 2.1 - Decision Process for Hydrothermal Systems

In contrast with thermal systems, whose operation is decoupled in time, hydro system

operation is coupled in time, that is, a decision today affects operating costs in the future.

2.2.2 Immediate and Future Operating Costs

The tradeoff between immediate and future operating costs is illustrated in Figure 2.2.

immediateoperatingcost

futureoperatingcost

final storage

Figure 2.2 - Immediate and Future Costs versus Final Storage

The immediate cost function - ICF - is related to thermal generation costs in stage t . As the final

storage increases, less water is available for energy production in the stage; as a consequence,

more thermal generation is needed, and the immediate cost increases.




In turn, the future cost function - FCF - is associated with the expected thermal generation

expenses from stage t +1 to the end of the planning period. We see that the FCF decreases with

final storage, as more water becomes available for future use.

The FCF is calculated by simulating system operation in the future for different starting values

of initial storage and calculating the operating costs. The simulation horizon depends on the

system storage capacity. If the capacity is relatively small, as in the Spanish or Norwegian

system, the impact of a decision is diluted in several months. If the capacity is substantial, as inthe Brazilian system, the simulation horizon may reach five years. This simulation is made more

complex by the variability of inflows to reservoirs, which fluctuate seasonally, regionally, and

from year to year. In addition, inflow forecasts are generally inaccurate, in particular when

inflow comes from rainfall, not snowmelt. As a consequence, FCF calculation has to be carried

out on a probabilistic basis, i.e. using a large number of hydrological scenarios (dry, medium

and wet years etc.), as illustrated in Figure 2.3.

1 2 3 4 time

spillage

rationing

replacesthermalgeneration

max. storage

Figure 2.3 - FCF Calculation

In contrast with thermal plants, which have direct operating costs, hydro plants have an indirect

opportunity cost, associated to savings in displaced thermal generation now or in the future.

2.2.3 Water Values

The optimal use of stored water corresponds to the point that minimizes the sum of immediate

and future costs. As shown in Figure 2.4, this is also where the derivatives of ICF and FCF with

respect to storage become equal. These derivatives are known as water values.

ICF

FCF

final storage

water value

ICF + FCF

optimaldecision

Figure 2.4 - Optimal Hydro Scheduling

The optimal hydro dispatch is at the point which equalizes immediate and future water values.




2.3 Formulation of One-Stage Hydrothermal Dispatch

2.3.1 Objective Function

As seen above, the objective is to minimize the sum of immediate and future operating costs:

z t = Min ∑ j=1

J

c( j) g t( j) + αt+1(vt+1) (2.2)

The immediate cost in (2.2) is given by the thermal operating costs in stage t , ∑c( j) g t( j). In

turn, the future cost is represented by the function αt+1(vt+1), where vt+1 is the vector of

reservoir storage levels at the end of stage t (start of stage t+1). The operating constraints in

the stage are discussed next.

2.3.2 Water balance - As illustrated in Figure 2.5, the water balance equation relates storage

and outflow: reservoir storage at the end of stage t (beginning of stage t +1) is equal to initial

storage minus outflow volumes (turbined and spilled) plus inflow volumes (lateral inflow plus

releases from upstream plants):

vt+1(i) = vt(i) - ut(i) - st(i) + at(i) + ∑m∈U(i)

[ut(m) + st(m)] for i = 1,..., I (2.3)

where:

i indexes hydro plants (I number of hydro plants)

vt+1(i) stored volume in plant i at the end of stage t (decision variable)

vt(i) stored volume in plant i at the beginning of stage t (known value)

at(i) lateral streamflow arriving at plant i in stage t (known value)

ut(i) turbined outflow during stage t (decision variable)

st(i) spilled outflow volume in plant i during stage t (decision variable)

m∈U(i) set of plants immediately upstream of plant i

upstream

plant outflow

lateral inflow

outflow

Figure 2.5 - Reservoir Water Balance

2.3.3 Limits on Storage and Outflow

vt(i) ≤ v _

(i) for i = 1, ..., I (2.4)

ut(i) ≤ u _

(i) for i = 1, ..., I (2.5)

where v _

(i) and u _

(i) are respectively the maximum storage and turbine capacities.




2.3.4 Limits on Thermal Generation - same as in thermal dispatch (2.1):

g t( j) ≤ g _

( j) for j = 1, ... , J (2.6)

2.3.5 Load Supply

∑i=1

I

ρ(i) ut(i) +∑ j=1

J

g t( j) = d t (2.7)

where ρ(i) is the production coefficient of plant i (MWh/hm3) (known value).

2.4 Problem Solution and Marginal Costs

Problem (2.2)-(2.7) is usually solved by a linear programming (LP) algorithm. In addition to the

optimal scheduling decision, the LP scheme produces a set of simplex multipliers associated to

the problem constraints. These multipliers provide the marginal cost information required in

competitive schemes. In particular, the WEM spot price is the multiplier associated to the load

supply equation (2.7), and the water value of each hydro plant is the multiplier associated its

water balance equation (2.3).

2.5 Calculation of Future Cost Function - SDP Recursion

The future cost function in each stage is calculated through a stochastic dynamic programming

(SDP) recursion:

a) for each stage t (typically a week or month) define a set of system states, for example,

reservoir levels at 100%, 90%, etc. until 0%. Figure 2.6 illustrates the system state

definition for a single reservoir. Note that the initial state (i.e. storage levels at the

beginning of the first stage) is assumed to be known.

1 2 T-1 T

system states

(initial storage level)

for stage T

initial

state

Figure 2.6 - Definition of System States

b) start with the last stage, T, and solve the one-stage hydrothermal dispatch problem (2.2)-

(2.7) assuming that the initial reservoir storage corresponds to the first storage level

selected in step (a) - for example, 100%. Because we are at the last stage, assume that the

future cost function is zero. Also, because of inflow uncertainty, the hydro scheduling

problem is successively solved for N different inflow scenarios, i.e. different possible values

for inflows in that stage. The procedure is illustrated in Figure 2.7.




1 2 T-1 T

one-stage operationsubproblem - inflow scenario 1

one-stage operationsubproblem - inflow scenario 2

one-stage operationsubproblem - inflow scenario N

Figure 2.7 - Optimal Strategy Calculation - Last Stage

c) Calculate the expected operation cost associated to storage level 100% as the mean of the

N one-stage subproblem costs. This will be the first point of the expected future cost

function for stage T-1, i.e. αT(vT). Repeat the calculation of expected operation costs for

the remaining states in stage T. Interpolate the costs between calculated stages, and

produce the FCF αT(vT) for stage T-1, as illustrated in Figure 2.8.

1 2 T-1 Tcost

FCF for stage T-1

Figure 2.8 - Calculation of the FCF for Stage T-1

d) The process is then repeated for all selected states in stage T-1, T-2 etc. as illustrated in

Figure 2.9. Note that the objective is now to minimize the immediate operation cost in

stage T-1 plus the expected future cost, given by the previously calculated FCF.

1 2 T-1 future cost

minimize immediate cost in T-1

+ expected future cost

storage in T

Figure 2.9- Calculation of Operation Costs for Stage T-1 and FCF for stage T-2

The final result of the SDP scheme (a)-(d) is the set of future cost functions αt+1(vt+1) for each

stage t . Note that the calculation of this function requires the representation of joint system

operation, with full knowledge of the storage state and inflows of all hydro plants in the

system. In other words, the FCF is a non-separable function of hydro plant states.




2.6 SDP Scheme Limitations

The SDP scheme is straightforward to implement and has been used for several years in most

hydro-dominated countries (e.g. [2],[3]). However, due to the need to enumerate all the

combinations of initial storage values, computational effort increases exponentially with the

number of reservoirs, the well-known “curse of dimensionality” of dynamic programming. For

this reason, it has been necessary to resort to approximations such as the aggregation of system

reservoirs into one reservoir that represents the energy production capability of the cascade [3]and the use of partial dynamic programming schemes (typically, calculation of separate future

cost functions for each basin) [4]-[7].

When all plants belonged to state-owned utilities, those approximate schemes were felt to be

satisfactory, because plant revenues usually came from long-term contracts, and eventual

differences in individual plant generation with respect to an ideal dispatch would cancel out in

the long-run. However, the implementation of a competitive environment raised a series of

concerns:

• in contrast with thermal systems, where spot price calculation is straightforward and easy

to interpret, hydrothermal spot prices are difficult to explain and to audit (as shown above,they reflect the expected opportunity cost along several inflow scenarios and stages)

• because plant revenues depend both on spot prices and on individual generation, there is a

greater need for detailed system modeling, which prevents the use of aggregation schemes

For these reasons, there has been a renewed interest in the development of stochastic

optimization algorithms able to handle detailed hydrothermal system dispatch. We will describe

one approach, stochastic dual dynamic programming [8]-[10], which has been used in several

countries in South and Central America, plus USA, New Zealand, Spain and Norway1.. An

alternative approach, based on Lagrangian relaxation, is described in [12].

2.7 The Dual Dynamic Programming Scheme

The Dual DP scheme is based on the observation that the FCF can be represented as a piecewise

linear function, i.e. there is no need to create an interpolated table. Furthermore, it is shown

that the slope of the FCF around a given point corresponds to the expected water values which,

as seen in section 2.4, are given by the simplex multipliers associated to the water balance

equations. Figure 2.10 illustrates the Dual DP calculation of expected operation cost and FCF

slope for the last stage, initial state = 100% (step (c) of the traditional DP procedure)

1 2 T-1 Tcost

expected operation cost

slope = derivative of op. cost

with respect to storage

Figure 2.10 - Dual DP - Calculation of First FCF Segment

1 A related scheme, called constructive dynamic programming, has been applied to the Australian system [11]




Figure 2.11 illustrates the calculation of operation cost and FCF slopes for each state in stage T.

The resulting piecewise cost surface is the FCF αT(vT) for stage T-1.

1 2 T-1 Tcost

piecewise future cost

surface for stage T-1

Figure 2.11 - Calculation of a Piecewise FCF for Stage T-1

In addition to the analytical representation of the FCF, the Dual DP scheme uses an iterative

simulation/optimization scheme to select only those states which are relevant for the scheduling

decisions. As a consequence, it becomes possible to solve the stochastic scheduling problem

for a large number of reservoirs with a reasonable computational effort.

2.8 Case Study

Figure 2.12 illustrates the hydro configuration for the Colombian system (80% hydro, 40 hydro

plants, with a total installed capacity ≈ 11 GW). The stochastic operation policy for 60 months

using a Dual DP scheme was calculated in approximately 4h (300 Mhz Pentium II processor).

V T

Troneras

TV

Guadalupe 3

Guadalupe 4

Miraflores

Tenche

V

V

T

Guatape

Jaguas

San Carlos

Playas

T V

Calderas

T

Insula

V T

VT

V T

Esmeralda

CampoalegreChinchina

Sanfrancisco

T

Estrella

V

T V

Prado

T

Prado 4

V

Riomayo

Betania

TV

VT

Porce 2

Niquia

T V

Latasajera

RioGrande 1

Quebradona

Laguaca

TV

Desafran

T V

Paraiso

Muna

V

Canoas

Salto

Colegio

T

Bomb-Mu

Laguneta

Chivor

Florida

Calima

Salvajina

Bajoanchicay

Altoanchicay

Urra 1

Guavio

Figure 2.12 - Colombian system configuration




3. Analytical Representation of Plant Outages

The hydrothermal scheduling scheme described in the previous section represents equipment

outages in a simplified way, usually as a derating of plant capacity. This simplified

representation is reasonable for hydro-dominated systems (where thermal plants are base-

loaded and hydro plants are responsible for peaking) but becomes less acceptable as thermal

participation increases, which is the current trend in most countries. Also, many countries use a

“capacity payment” as an incentive to the construction of peak generation reserve, which is based on the probabilistic evaluation of the plant’s contribution to supply reliability [13].

Therefore, it has become necessary to incorporate an analytical representation of forced

outages into the hydrothermal scheduling framework.

3.1 Probabilistic Hydrothermal Dispatch - Single Hydro Plant

We will initially analyze a system composed of J thermal plants and one hydro plant. The one-

stage dispatch (2.2)-(2.7) is rewritten as:

Min ct(ρut) + αt+1(vt+1)

subject to (3.1)

vt+1 = vt - ut - st + at (3.1a)

vt+1 ≤ v _

(3.1b)

ut ≤ u _

(3.1c)

where ct(ρut) represents the thermal operating cost as a function of the hydro generation

decision. This function is implicitly calculated as:

ct(ρut) = Min ∑ j=1

J

c( j) g t( j)

subject to (3.2)

∑ j=1

J

g t( j) = d t - ρut (3.2a)

g t ≤ g _

(3.2b)

Our objective is to transform ct(ρut) into a probabilistic production costing (PPC) model [14-

15] which calculates the expected thermal operation cost, taking into account equipment

outages and load fluctuations. The following scheme [16] is used to construct this extended

curve, based on the successive application of the convolution scheme proposed in [17]:

a) solve the PPC with the hydro plant represented as a dummy thermal plant at the last position in the loading order, that is: {T1, T2, ... , TJ, H}. Calculate the expected energy

generated by the hydro and thermal plants, and the corresponding system operation costs.

b) solve the PPC with the hydro plant at the first position in the loading order, that is: {H, T1,

T2, ... , TJ}. Calculate the expected energy generated by the hydro and thermal plants, and

the corresponding system operation costs. Figure 3.1 illustrates both calculations.




T1

T2

T3

T4

H

H

T1

T2

T3

T4

Figure 3.1 - Initial PPC Calculation

c) calculate the expected hydro generation and system operation cost associated to each

intermediate loading point, e.g. {T1, H, T2, ... , TJ}. Note that it is not necessary to carry out

additional PPC runs. The mean generation of T1 comes from the PPC run in (a), in which the

hydro was last in the loading order. The reason is that the expected generation of a given

plant does not depend on which plants come after in the loading order. In turn, the mean

generation of each of the remaining plants i.e. {T2, ... , TJ}. comes from the PPC run in (b).

The reason is that the expected generation of a given plant does not depend on the loading

order of the previous plants. Finally, the expected hydro generation is calculated as the

difference between the expected demand and the expected thermal generations. The

procedure is shown in Figure 3.2.

T1

T1T2

T2T3

T3T4

T4H

H

T1

T2

T3

T4

H

Figure 3.2 - Intermediate Points in the Cost Curve

d) Plot the expected hydro generation values obtained in (a) and (b) and the intermediate

values calculated in step (c) as the breakpoints of a piecewise linear cost × hydro energy

curve, illustrated in Figure 3.3.

HydroGeneration

C1

C2

C3

C4

C0

E(operation cost)

H0 H1 H2 H3 H4

Figure 3.3 - Cost ×× Hydro Generation Curve




The curve in Figure 3.3 corresponds to the desired probabilistic version of ct(ρut), and can be

used to solve the one-stage hydrothermal dispatch taking into account equipment outages and

load variation.

3.2 Multiple Hydro Limited Plants

A similar procedure could in principle be applied to construct a multi-dimensional cost × hydro

energy curve for a system with I hydro plants (all at the bottom, one at the top, the remainder at the bottom etc.). Note, however, that we would need to carry out PPC runs for all 2I

combinations of hydro plants at the top and bottom of the loading order, which becomescomputationally infeasible if the number of reservoirs is large (e.g. the Brazilian system hasmore than 60 plants).

This problem can be solved by generating only the part of the curve corresponding to theoptimal hydro generation targets [18, 19]. From LP theory, we know that c(ρut) is a piecewiselinear function of the I-dimensional turbined outflow vector ut,. Therefore, it can be representedas a convex combination of its breakpoints. The probabilistic scheduling problem (3.1) isrewritten as2:

Min ∑k =1

K

λk [ct(ρut)]k + αt+1(vt+1)

subject to (3.3)

vt+1 = vt - ∑k =1

K

λk [ut]k - st + at (3.3a)

vt+1 ≤ v _

(3.3b)

∑k =1

K

λk [ut]k ≤ u

_ (3.3c)

∑k =1

K

λk = 1 (3.3d)

1 ≥ λk ≥ 0 (3.3e)

where:

K number of breakpoints in the piecewise cost × hydro energy curve[ct(ρut)]

k expected thermal operating cost at the k -th breakpoint[ut]

k turbined outflow vector (k-th breakpoint)λk decision variable that represents the convex combination of breakpoints

Note that the decision variables in problem (3.3) are vt+1, st and the convex combination factors{λk }. The turbined outflows are obtained implicitly from the convex combination of breakpoints.

Problem (3.3) is solved by Dantzig-Wolfe decomposition [20] which iteratively generates the"relevant" columns for the LP problem, called Dantzig-Wolfe master problem. Figure 3.4illustrates the DW scheme [18].

2 For notational simplicity, the same symbols vt, ut, st etc. used in the one-reservoir example (3.1) now

represent I-dimensional vectors of storage, outflow, inflow etc. in problem (3.3). Also for simplicity, we did not

represent the water balance constraints for the more general case of reservoirs in cascade.




MASTER PROBLEM

Min E(Operating Cost) + Future cost

• water balance equations(relate turbining and storage)

• hydroelectric energy equations(relate turbining and hydroelectric production )

PROBABILISTIC DISPATCH

Hydrogeneration

implicit costs

(loading order)

Cost

Hydro generation

E(Operating Cost)function of the hydro

generation

cost

Storage

Future Cost as

a functionof the storage

Figure 3.4 - Integrated Hydro Scheduling - PPC Scheme

At the first iteration, a relaxed version of problem (3.3), with just one breakpoint (K =1) andone variable λk is solved. A shadow cost for energy production in each hydro plant is obtainedfrom the simplex multiplier associated to the water balance constraint (3.3a). This cost is thenused to determine the loading order of that plant in the PPC scheme. Next, the PPC problem issolved, and a new breakpoint is generated. This point is added to the master problem, and the process is restarted. This decomposition scheme allows efficient solution algorithms - PPC andstochastic DP - to be jointly used without substantial modifications in the original codes.

3.3 Case Study

The decomposition scheme was applied to a configuration of the Colombian generation system,

composed of 29 hydro and 50 thermal plants. Each hydro plant got a monthly energy target,

produced by the hydrothermal scheduling model. The load duration curve was represented by

six load levels, as shown in Figure 3.5 below.

0

1000

2000

3000

4000

5000

6000

7000

8000

50 100 150 200 250 300 350 400 450 500 550 600 650 700 hours

MW

Figure 3.5 Load Duration Curve - Colombian system




The multi-dimensional cost × hydro generation curve of a system composed of J thermal and I

hydro plants can have up to (J+2)I breakpoints. For the Colombian system, this corresponds to5229 ≈ 1050, which obviously prevents the use of the explicit enumeration scheme presented inSection 3.1. The decomposition procedure described in Section 3.2 obtained the optimalsolution in 114 iterations, that is, only 114 breakpoints were generated. Figure 3.6 shows theevolution of expected operation cost (plus penalties for hydro target violations) along theiterations.

0

100

200

300

400

500

1 11 21 31 41 51 61 71 81 91 101 111

iter

M$

c.pen.

e[c.oper]

Figure 3.6 - Expected operation cost ×× iteration

The total CPU time was 14.10 seconds (Pentium 166 MHz, 32 Mbytes). The mean solution

time of each master problem was 0.10s; each PPC subproblem solution took 0.02 s.

4. Economic Signals for Hydro Plants in Cascade

4.1 Distortions in Spot Signals

As discussed in the Introduction, one of the attractive features of the spot pricing scheme is to

provide efficient economic signals. In particular, if the system is optimally dimensioned, thespot-based remuneration should match investment costs plus operating expenses. This pricing

efficiency is easily demonstrated for thermal systems and, by analogy, would also seem to apply

to hydro plants. However, as illustrated next, the situation becomes more complex when there

are hydro plants in cascade. Figure 4.1 shows a system composed of a “pure” reservoir, that is,

with no associated generation, upstream of a run-of-the-river plant.

downstream

regulation1

energy

sale

2

Figure 4.1 - Hydro Plants in Cascade

This reservoir brings an obvious benefit to the system, by regulating the inflow to the

downstream plant, and thus increasing its energy production capability. However, under the

spot pricing scheme, which remunerates only the energy generated, the upstream reservoir

would receive no compensation, and the downstream plant would retain all benefits. In other

words, there is a clear distortion in the allocation of economic benefits.




4.2 Water and Energy Markets

The reasons for this pricing distortion is that two commodities are being traded in a

hydrothermal system:

• water - commercialized by system reservoirs;

• electric energy - commercialized by thermal plants and turbine/generator sets.

In other words, the reservoir is an economic agent that “purchases” water in the wet periods -

when it is cheap - and stores it until the dry periods - when it has a high opportunity cost. In

turn, the turbine/generator set purchases this water from the reservoir and transforms it in

energy for sale to the WEM. Because the compensations associated to water transactions are

ignored, downstream plants capture the rent that should have been allocated to upstream

reservoirs 3. An extension of the spot market to take into account both aspects is presented

next.

4.3 Representation of Upstream Economic Agents

Let the hydrothermal dispatch for the two-hydro plant system of Figure 4.1 be represented

below:

Min ∑ j=1

J

c( j) g t( j) + αt+1(vt+1)

subject to (4.1)

vt+1(1) = vt(1) - st(1) + at(1) (4.1a)

ut(2) = at(2) + st(1) (4.1b)

∑

j=1

J

g t( j) + ρ ut(2) = d t (4.1c)

vt+1 ≤ v _

(4.1d)

where decision variables (generation, turbined volume, spillage etc.) are as defined previously.

Eqs. (4.1a) and (4.1b) represent the water balance for both the reservoir and run of the river

plant. For notational simplicity, we assume that the upstream reservoir has no turbining

capacity - i.e. it only spills - whereas the downstream plant has no capacity limit, i.e. it

generates as much as required. (these assumptions will be relaxed later). Rewriting (4.1a) in

terms of its outflow, we have:

3

This distortion is not relevant if all hydro plants in a cascade belong to the same agent, as the totalremuneration will be correct. However, there are many countries where this is not the case, such as Colombia,

Chile, Spain and Brazil. In the Brazilian system, for example, there are as many as six utilities sharing plants

along the same river. In both Chile and Colombia, which use a spot pricing scheme, utilities owning plants in

the same cascade are now in court, claiming recognition of upstream benefits. In Argentina, the issue was

sidestepped because the hydro plants were sold in auctions to private agents. As buyers took into account the

future plant revenues under the spot pricing scheme, upstream plants got price offers which were smaller than

their actual construction cost. In turn, the sale price of downstream plants exceeded their cost. The total revenue

from the sales was therefore correct, and the future revenue for the new owners became compatible with their

remuneration requirements. Of course, this “market solution” can only be applied to existing plants belonging

to a sole owner (the government, in this case). Also, the problem of signaling the construction of new hydro

plants still persists.




st(1) = at(1) + ∆vt (4.2)

where ∆vt = vt(1) - vt+1(1) represents the reservoir storage variation in stage t . Under a spot

pricing scheme, we know that hydro remuneration would be πd×ρut(2), where πdt is the spot

price for stage t (simplex multiplier associated to the load supply equation (4.1c)). Replacing

(4.2) into (4.1b), and multiplying both sides by πd×ρ, we obtain:

πd×ρ ut(2) = πd×ρ[at(2) + at(1)] + πd×ρ∆vt (4.3)

Equation (4.3) shows that the plant remuneration can be divided into a component that

corresponds to the total natural outflow arriving at the plant (i.e. the outflow that would have

arrived without upstream regulation) plus a term that represents the effect of upstream

regulation. This suggests that the second term should be credited to the upstream reservoir 4. In

other words, the reservoir can be seen as an economic agent that purchases water in wet

periods - when it is cheap - and stores it in order to sell it in dry periods - when it is expensive.

It is also intuitive that the clearing price for purchase and sale of water should be the water

value, i.e. the shadow price associated to the water balance constraints. In fact, the general

expression for hydro remuneration in each stage is [21]:

a) reservoirs collect from the system (or pay to the system) an amount πh×∆vt, where πh is

the water value at the reservoir site.

b) hydro plants pay to the system (or collect from the system) an amount ∆πh×(ut + st - qt)

where ∆πh is the difference between water values at the plant site and immediately

downstream, whereas qt represents the total natural inflow at the plant.

Expressions (a)-(b) apply in the general case, e.g. if turbines at their limits or reservoirs arespilling.

4.4 Case Study

The extended spot concept was used in Colombia to calculate the compensation thatdownstream plants in Figure 4.2 should pay to upstream reservoirs for their regulation [21].

RioGuatape

RioNegro

Rio SanCarlos

Guatape

~

Calderas

RioCalderas

Jaguas

~

Playas

~

~

S.Carlos

~

T,V

T

V

TV

V T

T,V

Figure 4.2 - Reservoir Compensation Example

4 Note that ∆vt can be either positive (depletion) or negative (fill-up). If it is depleting, this means that the

reservoir is selling its stored water to the system, and should thus be remunerated. If it is filling up, this means

it is purchasing water from the system, and should therefore pay for it.




5. Competitive Bidding in Hydrothermal Systems

As mentioned in the Introduction, many systems have implemented a decentralized dispatch

based on generator price bids. In this section, we initially formulate the bid-based dispatch for

thermal systems and then extend it to hydro systems.

5.1 Bidding Schemes - Thermal Systems

5.1.1 Bid-Based Dispatch

Every day, generators provide a set of hourly generation prices and available capacities. Based

on this data and on an hourly load forecast, the following economic dispatch is carried out:

z = Min ∑h=1

H

∑ j=1

J

λhj× g hj Multiplier

subject to (5.1)

∑

j=1

J

g hj = d h πdh (5.1a)

g hj ≤ g _

hj (5.1b)

for h = 1, ..., H; for j = 1, ..., J

where:

h indexes load blocks - typically hours (H is the number of blocks)

z total system operating cost

λhj operating price of generator j in load block h ($/MWh); note that the hourly price of

a generator may be different from its “true” unit operating cost , represented as c j.

g hj energy production of generator j during load block h (MWh) g _

hj maximum generation of j in load block h (MWh)

5.1.2 Net Spot Revenues

As discussed previously, each plant receives a gross revenue given by the product of system

spot price and its energy production. The net revenue of each plant, represented by R j,

corresponds to the difference between its spot revenue and its “true” operating cost c j:

R j = ∑h=1

H

(πdh – c j)× g hj for j = 1, ..., J (5.2)

The net revenue of a generation enterprise, which may be a utility or an independent power

producer (IPP), is given by the sum of revenues from plants under its control:

R k = ∑ j∈Ek

R j for k = 1, ..., K (5.3)

where:

k indexes the enterprises; K number of enterprises

R k net revenue of enterprise k

j∈Ek set of plants in enterprise k




5.1.3 Bidding Strategies

The objective of each enterprise k is to determine a set of hourly price vectors λhk = {λhj, j∈Ek }

that maximize its net revenue:

R k (λk ) = Max ∑h=1

H

∑ j∈Ek

[πdh(λhk ) - c j]× g hj(λhk ) (5.4)

where πdh(λhk ) and g hj(λhk ) represent the system spot price and plant generation resulting from

system dispatch (5.1) in hour h when the price vector is λhk .

5.1.4 Bidding under Uncertainty5

The bidding problem complexity is compounded by the fact that the calculation of πdh(λhk ) and

ghj(λhk ) in (5.4) depends on the knowledge of price vectors for all enterprises, as well as their

generation availability and system load values. However, this information is not available to any

single enterprise at the time of its bid. Therefore, the bidding strategy has to take into account

the uncertainty around these values. One approach to solve this problem is to define a set of

scenarios for the unknown values, and maximize the expected net revenue over all scenarios:

ER hk (λhk ) = Max ∑ s=1

S

psR s

hk (λhk ) (5.5)

where:

ER hk (λhk ) expected net revenue of enterprise k in hour h s indexes the scenarios (S number of scenarios) ps probability of scenario s

R s

hk (λhk ) revenue of enterprise k in scenario s, hour h

5.1.5 Solution Approaches

Problem (5.5) has to be solved as a global optimization scheme where, for each trial value of

{λk }, we calculate the expected net revenue over all scenarios, as illustrated in Figure 5.1.

price offer

system dispatchscenario # 1

system dispatchscenario # 2

system dispatchscenario # S

λk

+expected revenue

R 1

k (λk )

R S

k (λk )

R 2

k (λk )

Figure 5.1 - Bidding Strategy under Uncertainty

5 because each hour in (5.4) can be optimized separately, the discussions that follow refer to a given hour h.




As illustrated in Figure 5.2, the enterprise tries to balance the benefit of increasing revenues (by

increasing bid prices) and the risk of not being dispatched [23]. This problem is difficult to

solve analytically because it is non-convex and it is not possible to obtain gradient information.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

104 106 108 110 112 114 116 118 120

Energy price offer ($/MWh)

Probability

345

350

355

360

365

370

375

380

385

Proffit ($)

Dispatch probability Net revenue

Figure 5.2 - Tradeoff between price, revenue and dispatch frequency

5.1.6 Nash-Cournot Equilibrium

Generation availability and system load scenarios can be generated based on outage statistics

for each plant type and on load forecasting models. However, the creation of price offer

scenarios for the remaining enterprises has to take into account that those enterprises are active

agents, i.e they also wish to maximize their own net revenues and, therefore, are trying to

anticipate the other enterprises’ price strategies. This problem can be handled through an

iterative process, where each agent optimizes in turn its bidding strategy given the (hidden)

strategies of the remaining enterprises. The resulting set of prices corresponds to a Nash-

Cournot equilibrium [22].

5.2 Bidding Schemes - Hydrothermal Systems [23]

5.2.1 Hourly Bids in a Hydrothermal System

The optimal bid for enterprise k in stage t is given by the solution of the following problem:

R tk = Max ∑h=1

H

[ ∑ j∈Ek

(πdh(λhk )-c j)× g hj(λhk )+ ∑i∈Ek

(πdh(λhk )× g hi(λhk ))] + R t+1,k (vt+1) (5.6)

where:R tk immediate net revenue of enterprise k in stage t

πdh system spot price in hour h (depends on price bid vector λhk )

g hj generation of thermal plant j in hour h (depends on price bid vector λhk )

g hi generation of hydro plant i in hour h (depends on price bid vector λhk )

R t+1,k future net revenue of enterprise k (depends on final storage vector vt+1)

Note that the calculation of hourly bids for day t is based on the tradeoff between immediate

revenues for hydro plants and their future revenues, given by function R t+1,k . This is similar to

the tradeoff between immediate and future costs in centralized dispatch.




5.2.2 Immediate Revenue Calculation

Given a set of hourly prices {λhk }, the immediate revenue calculation is identical to the purely

thermal case, where hourly spot prices {πdh} and plant generation { g h} are obtained from the

economic dispatch solution.

5.2.3 Future Revenue Calculation

Given the hourly generation of hydro plants { g hi}, the future revenue R t+1,k (vt+1) is evaluated

through the following procedure:

• initialize v0 = vt (reservoir storage vector at the beginning of stage t )

• repeat for each hour h = 1, ..., H

• repeat for each hydro plant i = 1, ..., I (from upstream to downstream)

• update storage level:

vh+1(i) = vh(i) - g hi/ρi + ah(i) + ∑m∈U(i)

(uh(m) + sh(m))

where: g hi/ρi turbined outflow volume of plant i in hour h

ah(i) lateral inflow volume to plant i in hour h

U(i) set of plants immediately upstream of plant i

• spilled outflow: sh(i) = Min{0, vh+1(i) - v _

(i)}

• storage limits: vh+1(i) = Min{v _

(i), vh+1(i)}

• set vt+1 = vH+1 and calculate future revenue FR t,k = R t+1,k (vt+1)

5.2.4 Calculation of Expected Future Revenue Function for each Stage

In the previous derivations, we assumed that the expected future revenue function for stage t ,R tk (vt), was known. This function is calculated through a stochastic dynamic programming

recursion, similar to the one used for the centralized hydrothermal dispatch.

• repeat for t = T, T-1, ..., 1

• repeat for each storage vector vt = v1

t, v2

t, ..., vM

t

• initialize future revenue function R tk (vt) ← ∝

• repeat for each trial bid vector λk = λ1

k , ..., λL

k

• calculate the expected total revenue ETR tk for initial storage vector vt and trial

bid vector λk using the procedure of sections 5.2.3 and 5.2.4

• update the optimal solution value R tk (vt) ← Max{R tk (vt), ETR tk }

5.2.5 Nash-Cournot Equilibrium

As in the purely thermal case, this is achieved by introducing an additional loop in the

stochastic DP recursion, where the agents iteratively adjust their price strategies. This

equilibrium calculation is carried out for each storage vector and for each stage. The result is a

set of future revenue functions {R tk (vt)} for k = 1, ..., K . A DP-based solution approach with

one hydro plant is described in [24]; a simplified solution scheme with multiple plants, but only

one stage, is described in [25].




5.3 Efficiency of Bidding Scheme

One important question is whether the hydrothermal bidding scheme described in the previous

sections is efficient, i.e. whether it is possible to achieve the same operating efficiency as an

ideal centralized dispatch. In contrast with thermal systems, where this efficiency can be

achieved if generators bid their true operating costs, a hydrothermal bidding system is

inherently inefficient, for the following reasons:

• the optimal hydro plant bid from the point of view of global system optimization is the

plant’s opportunity cost; however, the accurate calculation of this cost depends on the

knowledge of which thermal plants are available and of their operating costs; this

information is not available at bidding time (note that this information is available in the

case of a centralized dispatch)

• as discussed previously, the FCF calculation requires the knowledge of all reservoir states

and operating decisions; this information is not available in a bidding scheme

It is important to emphasize that these limitations do not imply that the bid-based system

dispatch will be necessarily inefficient, the point being made is that the results will be system-

dependent.

6. Conclusions

• the calculation of operating policy and spot prices in a hydrothermal system is a complex

computational procedure; a class of solution algorithms based on Dual dynamic

programming has been successfully applied to large-scale systems in several countries

• decomposition schemes can be used to integrate probabilistic production costing models

traditionally used in thermal system analysis into a hydrothermal scheduling framework

• the economic signals resulting from WEM spot prices are incorrect when applied to hydro

plants in cascade; it is necessary to extend the spot market concept to trade both energyand water

• a hydrothermal bidding system is inherently inefficient compared with a centralized dispatch

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Documents

1998 Long-Term Hydro Scheduling Based on Stochastic Models