7/27/2019 1998 Long-Term Hydro Scheduling Based on Stochastic Models http://slidepdf.com/reader/full/1998-long-term-hydro-scheduling-based-on-stochastic-models 1/22 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, Brazil [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 hydrothermal systems: (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.
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7/27/2019 1998 Long-Term Hydro Scheduling Based on Stochastic Models
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.
7/27/2019 1998 Long-Term Hydro Scheduling Based on Stochastic Models
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
7/27/2019 1998 Long-Term Hydro Scheduling Based on Stochastic Models
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.
7/27/2019 1998 Long-Term Hydro Scheduling Based on Stochastic Models
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.
7/27/2019 1998 Long-Term Hydro Scheduling Based on Stochastic Models
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]
7/27/2019 1998 Long-Term Hydro Scheduling Based on Stochastic Models
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.
7/27/2019 1998 Long-Term Hydro Scheduling Based on Stochastic Models
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.
7/27/2019 1998 Long-Term Hydro Scheduling Based on Stochastic Models
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
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.
7/27/2019 1998 Long-Term Hydro Scheduling Based on Stochastic Models
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.
7/27/2019 1998 Long-Term Hydro Scheduling Based on Stochastic Models
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.
7/27/2019 1998 Long-Term Hydro Scheduling Based on Stochastic Models
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].
7/27/2019 1998 Long-Term Hydro Scheduling Based on Stochastic Models
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