An Exact Multi-Plant Hydro Power Production
Function for Mid/Long Term Hydrothermal
Coordination
André Luiz Diniz
Electric Energy Research Center
Rio de Janeiro, Brazil
[email protected]
Ana Lucia Saboia
Electric Energy Research Center
Rio de Janeiro, Brazil,
[email protected]
Abstract – The generation of a hydroelectric power
plant is a nonlinear function of its storage and turbined
outflow. Nonlinear, mixed-integer or linear programming
models have been proposed to approximate this function in
the context of hydrothermal coordination, usually with one
function for each plant. This paper proposes a new exact
“Multi-plant hydro production function (MHPF)”, which
depends on the storage and discharge of a given reservoir
power plant and comprises not only its generation but also
the power output to all subsequent run-of-the-river plants
until the next reservoir. Based on this function we build a
piecewise linear model that can be embedded in stochastic
linear programs for optimal hydrothermal coordination.
We illustrate the proposed MHPF approach for real sets of
cascaded hydro plants of the Brazilian System.
Keywords: Hydroelectric power generation, linear programming, power generation planning.
I.
INTRODUCTION
The generation of a hydroelectric power plant is a
nonlinear function of its state (reservoir storage) and its
operation (turbined outflow and spillage) and is called
in the sequel as “individual Hydro Production Function”
(IHPF). Approximation models for this function for the
hydrothermal coordination problem have been applied
under different formulations, such as nonlinear [1],
mixed-integer [2] or linear programming [3]. In the
latter case, this function is usually approximated by
individual piecewise linear functions, one for each
plant, and embedded into the linear programs as a set of
linear inequalities for each hydro plant and time step.
The model above allows to accurately consider the
dependency of hydro plants generation on the water
head [3]. However, for long term hydrothermal planning
with a very long time horizon and a huge scenario tree,
the individual representation of each hydro plant - and
as consequence, of its production function - may lead to
impractical CPU times to solve the problem. For this
reason, equivalent energy reservoirs [4], [5] have been
considered so far to model the hydro cascaded in such
models [6]-[10].
Nevertheless, we note that in a hydro cascade the
system operation is defined only by the operation of
hydro plants with storage regulation capability (which
are simply called in the sequel as “reservoirs”), since the
run-of-the-river hydro plants will tend to release (pref-
R. M. Andrade
COPPE / UFRJ
Rio de Janeiro, Brazil
[email protected]
erably by turbining rather than spilling) all incoming
water, either from natural inflows or coming from the
operation of upstream plants.
In this sense, this paper proposes a new model for the
hydro production function, where we only build functions for reservoirs. This so called “Multi-plant hydro
production function (MHPF)” comprises not only each
reservoir but also all cascaded downstream run-of-the
river plants until (but not including) the next reservoir.
Such function gives the "exact"1 generation for this set
of plants, without approximations or pre-established
operation assumptions, based on the release of upstream
reservoir plant and the vector of natural inflows to runof-the-river plants. We also present how to extend the
piecewise linear model proposed in [3] for individual
hydro plants to the MHPF proposed in this paper. Such
MHPF model is useful to reduce the dimension of a
deterministic or a (more usual) stochastic mid-long term
hydrothermal coordination problem, since only hydro
plants with reservoirs need to be represented.
We illustrate the proposed approach for real sets of
cascaded hydro plants of the Brazilian System and perform an analysis of the accuracy and number of constraints obtained with such function. The results show
that we are able to produce as accurate values of hydro
generation as if an individual hydro production function
had been used, with a smaller number of constraints.
II.
INDIVIDUAL HYDRO PRODUCTION
FUNCTION (IHPF)
The generation
of a given hydro plant can be
expressed through expression (1) of its exact hydro
production function
:
=
= ( , , )=
ℎ ( )−ℎ ( + )−ℎ
(1)
where = 9.81 × 10 is a constant related to gravity and unit conversion, is the overall hydro plant efficiency2, , and are the storage volume, turbined and
spilled outflows, and the term in brackets consists in the
net water head ℎ, which is the difference between up1
Throughout this paper, we mean by "exact" generation the
values obtained by the analytical expressions (1)-(3).
2
In unit commitment problems, it is important to use a
more accurate representation of each turbine and generation
efficiency factors.
stream (ℎ ) and downstream levels (ℎ ) after subtracting head losses ℎ . Such levels may be represented as polynomial functions3 of and the total release
= + [3]:
ℎ ( )=
+
+
+
+
(2a)
ℎ ( )=
+
+
+
+
(2b)
As explained in [3], for most of the plants the HPF
turns out to be a composition of a nearly concave function on variables and , and a convex function on ,
as shown in Fig. 1. For modeling purposes in linear
programming models, the first behavior is desirable
while the second one forces us to apply a linear approximation to ensure convexity.
plants, which are represented as triangles and circles,
respectively, in Fig. 2. The operation of all hydro plants
in each of these sets can be determined based solely on
the operation of the upstream reservoirs, since system
optimization will favor the run-of-the-river plants to
turbine all incoming water up to its maximum turbined
outflow. As a result, we can compute a unique "multiplant hydro production function (MHPF)” for each set,
as explained next. We note that such aggregated model
is more suitable for mid-long term planning, where time
discretization is longer (e.g., weekly/monthly) and some
aspects of hydro generation such as forbidden operating
zones and hill curves for turbine efficiency factors do
not need to be considered in detail.
MULTI HYDRO PLANT
run-of-thereservoir river plants
Fig. 2. Aggregation of hydro plants in a cascade, where a multiplant hydro production function can be defined.
(%)
(
/
(
/
Fig. 1. Illustrative example of the hydro production function and
its dependence on and (left) and (right).
Several models have been proposed to model the
HPF of a hydro plant, for a broader review we refer to
[3]. We are particularly interested in convex formulations for this function that are suitable to be used for
mid-long term power generation planning [6]-[11]. For
this reason, the benchmark for this work is the piecewise linear formulation proposed in [3]. We note that
due to the large size of such problems, multistage stochastic linear formulations are preferred and decomposition approaches such as nested Benders decomposition
(NBD) [12] or Stochastic dual dynamic programming
(SDDP) [13] are employed to solve the problem.
Even though the model in [3] provides a high accuracy to represent the hydro plants generation function, it
requires an individual representation of all plants in a
river basin. However, most of the recent applications of
NBD or SDDP to long term hydrothermal planning in
real large scale systems with an accurate representation
of uncertainties in the water inflows still make use of
modeling of equivalent reservoirs [4], in order to reduce
the problem dimensionality and allow the problem resolution in a reasonable CPU time [6]- [10]. In this sense,
the main motivation of this work is to propose a so
called "multi-plant hydro production function", which
allows a more concise representation of the river basins,
while still representing the individual characteristics of
the hydro plants and avoiding their aggregation in
equivalent reservoirs.
III.
MULTI-PLANT HYDRO PRODUCTION
FUNCTION (MHPF)
Based on the hydro basins topology, it is possible to
identify several sets of cascading plants composed of a
reservoir followed by one or more run-of-the river
3
Such functions are obtained based on topographical studies and should be periodically updated.
However, "V-shaped" configurations are very common in large river-basins, where a given plant has more
than one upstream reservoir. It is also possible to define
multi-plants in such case, which will contain more than
one upstream reservoir. Fig. 3 illustrates a reduction of a
small cascade with 13 hydro plants into a cascade with
only 5 multi-plants.
Fig. 3. Aggregation of a 13-hydro plant cascade into a set of 5multi-plant cascade.
We can compute the exact hydro production of a
multi-plant as a function of the discharge and storage of
the upstream plant, by summing up the generation of all
plants in the set, with the assumption that each plant
will only spill when the maximum turbined outflow has
been reached.
A. MHPF for a single upstream reservoir
We first consider a configuration such as "C" in Fig.
3, which is composed of 1 upstream reservoir and run
of the river plants. We use index 1 for the reservoir and
indices {2, … , + 1} for the run-of-the-river plants. Let
and
denote the volume and release of the reser( = 2,…, + 1) be the natural inflow
voir, and ,
and water intakes due to irrigation, evaporation4, etc, in
each run of the river plant , as in Fig. 4(a).
The turbined and spilled outflows of the reservoir can
be determined as follows:
4
Since the reservoir surface is constant for run-of-the-river
plants, their evaporation can be computed a priori.
{ , }
=
(3)
=
− .
By applying the usual assumption that run-of-the river plants spill only when the maximum turbined outflow
has been reached, their generation output are given by:
=
,
,
,
(4)
with:
(
)=
,
+∑
−∑
(
)=
+∑
−∑
−
(a)
(
),
(b)
Fig. 4. Variables that impact the generation of the multi-plant, with
only one (a) or several (b) upstream reservoirs.
where is a constant value for such plants and the term
under brackets accounts for the incoming water to each
plant, taking into account the release of upstream plants,
natural inflows and water intakes. The total generation
of the multi hydro plant is the sum of the outputs
of all the plants, which can be expressed as a function of
only storage and release
of the upstream reservoir,
as follows:
=
,
(
( , ) ==
), ( ) +
+ ∑
(
(5)
),
(
) B. MHPF for several upstream reservoirs
A similar procedure can be applied to obtain the production of the multi-plant in the more general case
where it is composed of upstream reservoirs followed
by run-of-the-river plants, as shown in Fig. 4(b):
=
=∑
+∑
(( ,
), ( ,
(
)+
( ,…,
,
), … , (
),
,
(
)) =
,…,
(6)
) ,
where the control variables for each plant are:
(
,…,
) = min
( ,…,
+∑
,
+∑
(
−
)=∑
+
(
−
) −
,…,
∑
)
).
(7a)
(7b)
Again, the generation of the multi-plant is a function
of only variables related to the upstream reservoirs.
B. Remarks
We make the following important remarks regarding
the proposed multi-plant hydro production function:
• the MHPF gives the exact generation (without approximations) for the set of plants, based on the storage
and the release of the upstream reservoirs. We emphasize that the only (quite reasonable) assumption is that
spillage occurs after maximum discharge is reached;
• The vector of natural inflows to run-of-the-river
plants is supposed to be known, whether in a deterministic approach or as a possible inflow scenario in a stochastic setting. Therefore, if a stochastic problem is
considered, the MHPF will be different for each scenario. In this sense, 'warm start' procedures to solve the
stochastic programming subproblems should take such
differences into account;
• The use of a MHPF in an optimal hydrothermal dispatch problem does not exclude the possibility of imposing individual operation constraints for the reservoir
or run-of-the-river plants, such as minimum/maximum
outflow and power generation. They can be modeled by
proper conditions when building the MHPF;
• The MHPF approach is also interesting for long
term time horizons, as a competitive and more accurate
model as compared to the usual approach of equivalent
reservoirs. In such context, it is necessary to build a
different model of all multi-plants in all inflows scenarios of each time step. However, this imposes only an
increase in memory, since in terms of CPU time these
functions (and corresponding piecewise linear models)
are computed very fast and only once, before starting a
NBD or SDDP solving procedure.
The drawback of the proposed MHPF is that it cannot
be applied when some aspects of hydrothermal scheduling are considered, as for example:
• in network constrained problems [14], where each
hydro plant injects power in a different bus of the electrical network. In such case, line flow limits or additional security constraints may constrain the maximum
generation of one or more plants in the set, and spillage
may occur before a plant reaches its maximum outflow,
thus violating the basic assumption of the model;
• in hydro unit commitment problems [2], [15], where
the generation of each unit of a plant is optimized. Actually, in that case even the individual hydro production
function (1) would not be appropriate, since it neither
takes into account the units status (on/off) nor allow a
proper representation of ramping constraints.
• when water delay times are considered [16], because
the release of the upstream plant reach the run-of-theriver plants in further time steps.
However, all these aspects are typical only for shortterm or real-time hydrothermal scheduling problems,
while the focus of this paper is the mid/log term hydrothermal planning, where the consideration of such constraints are of less interest.
IV.
PIECEWISE LINEAR APPROXIMATION
FOR THE MHPF
The MHPF described previously is nonlinear and
probably non-concave in part of the ( × ) domain.
Even though nonlinear programming approaches [1] can
be employed to solve this problem, a linear program-
ming formulation is suitable in view of the large developments achieved so far in the stochastic linear programming literature [17]. In this sense, we also propose
in this paper an extension of the piecewise linear model
(PWL) that had been proposed in [3] for the individual
hydro production function (1) to the MHPF formulation
(5) and (6), in order to provide a convex approximation
of the feasible region below such functions.
Due to space limits, we do not provide details of how
to build this convex approximation here but rather refer
to section IV of our previous work [3]. However, there
are some differences in the procedures to obtain a PWL
approximation for the multi-plant hydro production
function proposed in this paper, in comparison with the
procedures to approximate an individual function.
• the MHPF is a function of (besides storage) the total
release of the upstream reservoir, instead of individual
turbined outflow and spillage variables. As a consequence, the PWL model for the case of a single upstream reservoir (section III.A) has only three dimensions and no secant approximation is necessary to model
spillage;
• the MHPF is non-differentiable at those points
where the maximum discharge of each plant in the set is
reached (see Fig. 7 later). In this sense, besides the uniform grid that is usually employed to generate discretization points of the true HPF (section IV.B of [3]), it is
useful to include at most ( + ) additional points related to maximum discharge of those plants5.
• if there is more than one upstream plant (section
III.B), the PWL approximation of the MHPF lies in the
space. Own experience by the authors has shown
ℜ
that computation of a convex hull becomes impractical6
for
≥3. Fortunately, such cases are not so common
and appear only three times in the real Brazilian hydro
topology. In such case we may apply an individual
model for the HPF of each upstream reservoir and a
MHPF for a set including only the run-of-the-river
plants.
After applying all these procedures, we end up with a
set of linear inequalities (8) that approximate the
generation for each multi-plant, as a function of decision variables and for the upstream reservoirs:
≤
( )
+∑
( )
( )
+∑
,
(8)
= 1, … , ,
which should be used together with the box constraint:
∑
≤
≤∑
.
(9)
A. Recovering the generation of each plant
The aim of using a MHPF is to allow the reduction of
the dimensionality of a deterministic or stochastic hydrothermal coordination optimization problem, by representing only variables associated to reservoirs with
regulation capability. However, once the optimal power
5
If the sum of water inflows up to plant is already greater
than its maximum discharge, such point will not be necessary.
6
for example, if only 3 discretization points are used for
and variables, up to 2.1×1016 hyperplanes may be evaluated.
output
of a given multi-plant is obtained, it is
necessary to extract the individual generations
for
each plant that composes such function.
We note that
would be obtained by the approximated model (8), with the storage and release for
all upstream reservoirs. Based on the values of such
, =
variables, we can compute the exact generation
1, … ,
by expressions (6) and (7), as well the overall
generation
of the multi-plant. We propose the
following algorithm to split the deviation Δ =
−
as evenly as possible among the set of all
= + plants of the multi-function, respecting their
individual power generation limits:
=
(initial generation of each
Initialization:
plant);
= |Δ | (amount of deviation to be allocated). If Δ > 0 do algorithm 1, else do algorithm 2.
Algorithm 1. Update index set Ω = { −
> 0} of plants that can still increase its generation.
Update each generation by = min{
,
+
/|Ω|}. Update
=
−∑
. If
= 0,
stop, otherwise repeat Algorithm 1.
Algorithm 2. Update index set Ω = { −
> 0} of plants that can still decrease its generation.
Update each generation by = max{
,
−
/|Ω|}. Update
= ∑
−
. If
= 0,
stop, otherwise repeat Algorithm 2.
Constraints (9) ensure that both algorithms yield a
feasible point {
, = 1, … } with ∑
=
.
V.
NUMERICAL RESULTS
We present numerical experiments of application of
the proposed multi-plant function for real hydro plant
data of the Brazilian system.
A. Detailed Analysis of a specific MHPF
First we perform a detailed analysis of the MHPF
function proposed in this paper for a set of four hydro
plants (one reservoir and three run-of-the-river plants),
as shown in Fig. 5. We list in Tables I and II the main
hydro plant and reservoir data, respectively, to compute
the production function of this set of plants. Each line in
Table II shows one coefficient of the forebay or tailrace
level polynomial (depending on the column) of each
reservoir. Storage limits in the reservoir are = 1540
hm3 and = 5040 hm3.
M. Moraes
( = )
Estreito
( = )
Jaguara
( = )
Igarapava
( = )
Fig. 5. Set of cascaded plants in the illustrative example.
The graph of the exact hydro production function for
this set of plants as a function of storage and outflow of
the upstream reservoir is shown in Fig. 6. We also present in Fig. 7 a sensitivity analysis of the generation in
this set as the inflow in each run-of-the-river plant is
increased, for fixed pairs ( , ) and natural inflow
values for the other plants. In each case, the nondifferentiable points appear when the discharge of any
of the remaining downstream plants achieves the value
related to its maximum generation or turbined capacity.
Beyond such point, an increase in inflow will cause
spillage in that plant, thus sharply reducing the local
efficiency of the multi-plant.
TABLE I.
POWER GENERATION DATA - ILLUSTRATIVE EXAMPLE.
Turbine
Efficiency
(MW/(m3/s))
Plant
(% or
)
’
⁄ )
(
)
(
=1
=1
0.0083
0.80%
1328.0
478.0
0.0088
1.50%
2028.0
1104.0
=2
0.0089
1.34%
1076.0
424.0
=3
0.0090
0.30 m
1480.0
210.0
-
-
5912.0
2216.0
Total
TABLE II.
Coef
POLYNOMIAL DATA - ILLUSTRATIVE EXAMPLE
(2a)
=1
(2b)
=1
=2
=3
=4
0
6.42e-2
6.19e+2
5.57 e+2
5.10e+2
1
8.09e-03
1.73e-3
1.22e-3
1.77e-3
7.82e-4
2
-3.70e-07
-4.89e-8
-7.51e-8
-7.40e-8
-1.04e-7
3
-7.11e-11
-
2.16e-12
1.12e-11
-
4
9.12e-15
-
-
-4.80e-16
-
4.94e+2
not incur in a significant loss of accuracy as far as the
detailed operation of each plant is concerned.
TABLE III.
Turbined Outflow Q (% maximum)
V
25%
50%
0%
-0.30 0.07
25%
0.22
0.12
50%
0.64
0.05
75%
0.64
0.05
100% 0.64
0.05
min. absolute deviation
max. absolute deviation
Hydro generation (MW)
4
1
5
6
2
7
16
2
3
In Table III we show the deviations between the exact
function and the proposed piecewise linear model for
this MHPF function, along a discretization grid on the
storage × outflow domain for the upstream reservoir.
The results show that the proposed MHPF model yields
very accurate generation values, as if each individual
production function (IHPF) had been considered. These
results illustrate that the aggregate model indeed does
26
11
47
17
48
18
156
72
6
38
8
39
27
49
19
169
77
28
12
9
32
9
10
40
42
31
13
21
249
15
50
43
51
12
172
20
22
173
33
61
16
71
78
82
174
52
10
11
155
14
15
4
8
Fig. 7. Sensitivity analysis of the multi-plant output as the inflow
in each run-of-the-river plant increases.
37
14
7
Natural inflows (m3/s)
75%
100%
150% 200%
0.07
0.07
0.07
0.07
0.11
0.11
0.18
0.16
-0.04
0.15
0.25
0.56
-0.04
0.19
0.27
0.76
-0.04
0.25
0.44
1.03
0.89 MW
5.73 MW (V = 0%, Q= 10%).
B. Overall analysis for several cascades
Now we perform a broader analysis taking into account 51 cascaded hydro plants, which compose part of
the current Brazilian system. We split “V-shaped” configurations into two or more parallel cascades in order
to avoid having more than one upstream reservoir for
the multi-plant. This was done because the extension of
our 3-dimensional convex hull algorithm of [3] to five
or more dimensions does not present satisfactory results
so far. As shown in Fig. 8, such plants are aggregated in
23 multi-plants, where the number of plants in each set
depends on the type of the hydro plants (reservoir or
run-of-the river plants). We note that whenever there are
consecutive reservoirs in a given cascade, we have a
multi-plant with only one reservoir, which is similar to
the individual HPF described in section II and presented
in [3], except for the model of turbined outflow and
spillage as a single release variable .
The plants ID are the official ones used in the centralized planning of the Brazilian system, which is performed by the Independent system operator with the
optimization models described in [6]. All the characteristic data for those plants can be accessed through the
ISO´s web page (http://www.ons.org.br).
1
Fig. 6. Hydro generation of the set (M. de Moraes, Estreito,
Jaguara and Igarapava) as a function of operation of M. Moraes.
DEVIATIONS BETWEEN THE MHPF AND THE PWL
APPROXIMATION OF THIS FUNCTION
111
23
112
62
113
63
114
Fig. 8. Overall hydro cascade for part of the Brazilian system.
In order to allow an assessment of our model for different system conditions, we considered twelve different
inflow data for the run-of-the river plants (which are
necessary to compute the MHPF) corresponding to their
monthly long-term average values extracted from the
historic record. Therefore we built 12 different MHPFs
as well as 12 piecewise linear models to approximate
them, for each multi-plant.
As mentioned previously, the MHPF described in
section III provides the exact generation of the set of
plants as a function of the release of the upstream reservoir. However, in order to allow taking into account this
function in a hydrothermal dispatch optimization problem, the piecewise linear approximation described in
section IV was computed for all multi-plants. For comparison purposes, we also computed the piecewise linear
approximation for each individual hydro production
function (IHPF) [3].
The number of discretization points for storage was 5
in both MHPF and IHPF models, with a window width
of 20% around the initial storage (see Table V in [3]).
For the turbined outflow we considered 5 discretization
points for the IHPF, which was shown in [3] to provide
a good trade-off between accuracy and CPU time to
solve an underlying hydrothermal scheduling problem.
As for the MHPF we used only 3 points in the grid for
two reasons: (i) additional discretization points will
automatically be included due to maximum discharge
limits, as explained in the second bullet of section IV;
(ii) since the MHPF tends to present more points in its
convex hull as compared to the IHFP (due to being a
composition of different individual functions), using the
same number of breakpoints in both models will lead to
a much higher number of constraints in the MHPF.
1) Assessment of the deviations between the piecewise linear model and the exact functions.
We then randomly selected a number of “operation
points” ( , ) for the upstream reservoir in each set
and computed the exact generation levels and the approximated generation yielded by the piecewise linear
functions, both for the aggregated and the individual
case. We show in Table IV the relative deviations of the
approximated model as compared to the values of the
exact function, for both the original IHPF and the proposed MHPF functions. The deviation of the IHPF was
computed based on the sum of the exact and approximated generations for all individual plants in the set.
TABLE IV. TABLE IV. MEAN RELATIVE DEVIATION OF THE MHPF
AND IHPF MODELS, AS COMPARED TO THEIR RESPECTIVE EXACT
FUNCTIONS.
Multi% dev.
% dev.
Multi% dev.
% dev.
plant ID
(MHPF)
(IHPF) plant ID (MHPF)
(IHPF)
1
14
0.35
0.40
1.27
0.62
2
15
0.77
0.73
0.14
0.06
3
16
0.29
0.54
0.07
0.00
4
17
0.73
0.62
0.24
0.31
5
18
5.54
4.67
8.93
10.25
6
19
0.13
0.33
3.33
3.20
7
20
0.31
0.07
0.36
0.20
8
21
0.00
0.00
1.37
0.49
9
22
1.21
0.99
0.37
0.42
10
23
0.23
0.12
0.21
0.82
11
MEAN
1.12%
1.22%
0.66
0.73
12
MAX
8.27%
10.15%
0.02
0.08
13
1.55
0.00
It can be seen that the proposed MHPF model provides similar results and in some cases even outperforms the individual model. In the overall average anal-
ysis taking into account all multi-plants, deviations of
the MHPF are have a quality only around than 10%
poorer than the IHPF model. It is important to note that
the purpose of the MHPF proposed in this paper is NOT
to obtain a more accurate model than [3], but rather to
obtain a level of accuracy as close as possible to it, but
modeling the cascaded hydro plants multi-plants, in
order to reduce the size of the problem.
2) Assessment of the number of constraints of the
PWL model
Probably the main goal of the MHPF is to provide a
more concise set of variables and constraints for the
hydrothermal coordination problem in comparison with
the individual model, which is widely used in short/mid
term planning but becomes prohibitive for long-term
planning with an extended time horizon and a detailed
modeling of uncertainties. Therefore we compare in
Table V the average number of constraints (linear inequalities) in both MHPF and IHPF models, for each of
the 23 multi-plants.
TABLE V.
Multiplant ID
1
2
3
4
5
6
7
8
9
10
11
12
NUMBER OF CONSTRAINTS OF THE MHPF AND IHPF
MODELS.
# constraints # constraints Multi- # constraints # constraints
plant ID
(MHPF)
(IHPF)
(IHPF)
(MHPF)
6.0
7.0
10.2
13.3
7.0
9.0
9.0
5.0
11.0
7.0
12.0
11.3
11.0
11.0
31.0
19.0
14.0
15.0
28.0
11.0
15.0
11.0
19.0
15.0
13
14
15
16
17
18
19
20
21
22
23
MEAN
7.0
10.5
9.3
11.1
9.0
7.0
7.0
8.7
8.8
11.1
8.4
8.94
14.0
18.0
27.0
19.0
29.0
11.0
11.0
11.0
19.0
19.0
26.0
17.57
We can see that the proposed MHPF reduces the
number of constraints to represent the hydro production
function by roughly 50%, in relation to the individual
model. Therefore one major advantage of this approach
is the drastic reduction in the size of the resulting linear
programming hydrothermal dispatch problem, with a
very small loss of accuracy, as seen in Table IV. We
note that additional reductions in the hydrothermal dispatch problem by using the multi-plant representation of
the hydro cascade are possible, since hydro balance
equations and some operation constraints can also be
represented by multi-plants.
Finally, we mention that if we had used 5 breakpoints
for turbined outflow in our MHPF model (plus the maximum discharge breakpoints), the average number of
constraints would have jumped to 21.51, which would
turn the aggregated less favorable. This is the reason
why a reduced number of breakpoints is recommended
to such aggregated model.
VI.
CONCLUSIONS
This paper presented a new multi-plant model for the
production function of hydro plants (MHPF) that comprises several cascaded reservoirs in a single function.
This approach is able to represent the individual charac-
teristics of the hydro plants while providing a reduction
in the number of linear constraints to approximate the
function as compared to the traditional approach of
having individual models for each plant (IHPF). Due to
the reduced number of constraints, the use of such multi-plant models tends to yield lower CPU times when
solving an underlying hydrothermal coordination problem. Numerical results with real data for a large number
of hydro plants of the Brazilian system also show that
our MHPF does not lead to an extra loss of accuracy in
representing the generation of the hydro plant as compared to the IHPF model.
In particular, the proposed approach is suitable for
long-term planning, as a more accurate approach to
represent hydro plants as compared to usual equivalent
energy reservoir model. It is important to note that environmental constraints have forced most of the new hydro plants to be run-of-the river plants, making the use
of multi-plant hydro production function models even
more suitable.
Future work by the authors will comprise the inclusion of the MHPF developed in this work in the stochastic long-term hydrothermal planning problem for large
systems, as well as adjustments in the modeling of several constraints of the problem to adapt them to this
multi-plant framework. In this sense we can verify the
practical advantages of this multi-plant model from the
computation point of view of solving large-scale stochastic hydrothermal planning problems.
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