Energy Modelling Research Group
Department of Management
University of Canterbury
Short-term Hydro Scheduling
using Integer Programming:
Management and Modelling
Issues
Andrew L. Kerr, E. Grant Read
Department of Management
University of Canterbury
EMRG-WP-97-02
email:
[email protected]
[email protected]
No liability is accepted for errors of fact or opinion in this report whether or not due to
negligence on the part of the University, its contributors, employees or students.
Abstract
The scheduling of hydro stations has stochastic, integer, non-linear, and continuous
time aspects, with all the approaches described to date making some simplifying
assumption about one or other of these aspects. The (integer) unit commitment
decision has received relatively little attention, partly because the resulting problem
was deemed intractable given the potential gains in efficiency. However, with the
advent of deregulated energy markets, the implications of ignoring these integer
effects may be costly, and so they must be considered in some way. We discuss
some of the managerial and modelling issues relating to this problem and some
ideas for heuristics that incorporate management priorities into an Integer
Programming framework.
Short-term Hydro Scheduling using Integer Programming: Management and Modelling Issues
Contents
1.
Introduction .................................................................................................................1
2.
Unit Commitment/Dispatch Scheduling .................................................................4
3.
Management/MIP Compatibility..............................................................................7
4.
Conclusion .................................................................................................................10
References..........................................................................................................................11
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1. Introduction
Managers (decision makers) are often faced with what seem to be relatively
straightforward problems which ‘explode’when classical modelling approaches are
applied to them, and so standard approaches become less desirable, and possibly
intractable, for the purposes required. In situations such as this, solution methods
that produce satisfactory solutions in a reasonable time frame ie, heuristics, can
become an attractive alternative.
Figure 1: Approaches to ‘the problem’
Manager
OR/MS Analyst
Software Analyst
Consider Figure 1, which depicts three individuals involved in ‘solving’ a given
problem. The manager is the individual most involved with the problem and has
preferences about what the form of the solution should be, and insights into what
needs to be focussed on to achieve a satisfactory solution, but has less knowledge
about the formulation and solution algorithm techniques which could be applied;
the motto for the manager could be: “do the same as last time because we are too
busy to devise a different way to do it”. The traditional role of the ‘OR person’has
been split into two. The Software/Modelling Expert might have experience with
standard formulations, creating/modifying general purpose solution algorithms,
modelling methodology, commercial software, and IT issues, but have less
knowledge about this particular problem; the motto for this individual could be
“buy a good package and apply it”, or, as Whybark[10] would have it: “the model is
the answer”. The OR/MS analyst has some knowledge of problem and its context as
well as the techniques that can help to solve it, and might have multi-industry and
multi-discipline experience/knowledge; the motto for this individual could be
“understand the problem and fit or devise a technique for it”.
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Before the increase in availability and accessibility of computer technology, the main
source of problem solving assistance for the manager outside of the organisation was
the OR/MS analyst working in an academic or consulting environment. A new and
distinctive approach was often constructed by the analyst in each case, even using
his or her own code. As more generic software packages have become available, and
the computer technology required to support them has become more accessible, the
Software/Modelling expert has had an increasing role to play, with many problems
now being addressed using standard software using little or no ‘expert’contribution.
Our concern, here, is to consider the role which the OR analyst can play in
addressing a problem for which standard software has not yet proved particularly
tractable or particularly well adapted to the problems actually faced by decision
makers. We focus on the way in which analytical insights may be used to adapt
standard approaches so as to make them simultaneously more tractable and more
suitable to the decision making environment.
From a managerial perspective, the optimisation of short-term (24 hour) unit
commitment schedules has received relatively little attention for reasons of
complexity and solution time, because the potential gains from optimisation have
seemed small in relation to the complexities associated with it (eg. time
discretisation, unit ramping, rate of change, unit cusp curve discontinuities, nonlinearities in head effects, reserve provision, and uncertainty), and because a decentralised decision making environment has not provided incentives to encourage
optimisation at the organisational level [5]. With the advent of deregulated energy
markets, decentralised management has incentives to concentrate on such details as
integer effects in their own plant.
From the software/modelling perspective, solution techniques proposed to date
have tended to simplify the integer aspects, among others, and so there is scope for
developing techniques which can accommodate the problem's complexity while not
compromising the quality of the solutions, or having excessive solution times.
Standard Mixed-Integer Programming (MIP) ie, branch-and-bound, provides a
useful framework for accurately modelling the integer aspects of the unit
commitment problem, but has some incompatibilities with managerial needs in
other areas as discussed in Section 3.
From an OR/MS perspective, there is a need for integration of the managerial and
computational perspectives. We suggest that modifications based on analytical
insights can be made to a standard MIP approach so as to focus on the key
managerial aspects of the problem. Our goal is to produce solutions with reasonable
computational effort without sacrificing, and hopefully enhancing, managerial
acceptability. We are interested in evaluating this hypothesis by comparing the
performance of insight-influenced heuristics with techniques, such as standard MIP,
which assume no prior information about the managerial decision problem. In this
paper we present an overview of the problem (Section 2), some ideas for modifying
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Short-term Hydro Scheduling using Integer Programming: Management and Modelling Issues
standard MIP to accommodate manager needs and analytical insights (Section 3).
Conclusions are presented in Section 4.
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Short-term Hydro Scheduling using Integer Programming: Management and Modelling Issues
2. Unit Commitment/Dispatch Scheduling
Hydro systems usually consist of several hydro stations (each with several
generating units) connected by rivers and canals and supplied with water by storage
reservoirs, head ponds, and tributaries. Water stored in storage reservoirs has a
water value associated with it that represents the value of saving that water to
reduce shortage or thermal generation at some later stage. Head ponds are smaller
than storage reservoirs and store enough water to give some short-term flexibility to
power stations. The level of these ponds may affect the productivity of release via
‘head effects’, but this complication is ignored here. Water released from one station
will arrive at the next head pond in some later period, or periods.
For a given unit commitment, determining the optimal release/dispatch is relatively
straight forward using Linear Programming. Determining optimal unit commitment
schedules does not follow immediately from this type of analysis, because unit and
system restrictions mean that the potential profits/losses available if the unit were
committed must be traded off against the costs and restrictions that may result from
turning units off and on to meet the schedule. This process is complicated by the
fact that release decisions made at one station impact on the stations upstream and
downstream from it, an effect which may be further complicated by a variety of
other constraints, both physical and legal.
A (deterministic) mathematical formulation of the unit-scheduling problem for
identical units1 with a group generation target to be met is as follows:
(1) USIP Maximise
I
∑
ψ
T
i
i= 1
s iT −
∑ ∑ (α
T
I
t= 1 i= 1
(+ )
i
ui( + )t + α
(− ) ( − )t
i
i
u
)
Subject to
(2)
xit = xit − 1 + ui(+ )t − ui(− )t
(3)
s it = s ti − 1 + n ti − q it − w ti +
∀ i, t
∑ (q ) + ∑ (w )
t − d hi
h
h ∈ Ai
(4)
sitMIN ≤ sit ≤ sitMAX
(5)
qit = xit − 1q$it +
M
∑ ( qcc
( + )t
0 ≤ qccim
≤Φ
(+ )
im
∀ i, t
h∈ Bi
∀ i, t
( + )t
im
m= 1
(6)
t − ehi
h
( − )t
− qccim
)
× xit − 1
∀ i, t
∀ i, t, m
1 See [2] for the corresponding non-identical units formulation.
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Short-term Hydro Scheduling using Integer Programming: Management and Modelling Issues
(7)
( − )t
0 ≤ qccim
≤Φ
(8)
git = xit − 1g$it +
(− )
im
∑ (∆(
M
m= 1
(9)
∑
× xit − 1
git = Dt
+ )
im
( + )t
( − )t
qccim
+ ∆ (im− ) qccim
∀ i, t , m
)
∀ i, t
∀t
i
(10)
xit integer
∀ i, t
The objective (1) maximises the value (ψ iT ) of storage ( siT ) at the end of the
scheduling horizon, less any costs incurred from switching units on and off at each
node2 (I) for each time period (T), where α (i + ) and α (i − ) are the unit switching costs
and u i( + )t and ui( − )t are variables corresponding to the number of units switched on or
off since the previous period at each station. The unit commitment in each period is
defined in (2) where xit is an integer variable which describes the number of units
operating at node i at the end of period t. Constraint (3) is an equilibrium flow
constraint, with inflows into each node including natural inflows ( nit ), and releases
(with delay time dhi ) and spill (with delay time ehi ) from upstream nodes ( Ai for
release and Bi for spill), while outflows are release ( qit ) and spill ( wit ). Storage
bounds are set in (4). Throughput is defined in (5) using piecewise linear
approximations to the approximately quadratic unit cusp curves that have a
( + )t
maximum of 2M segments. The piecewise segments allow positive qccim
and
( − )t
t
negative qccim deviations from peak efficiency ( q$i ), where the level of peak
efficiency for a given unit commitment is simply a multiple of peak efficiency for a
single unit at the station (because the units are assumed to be identical). The bounds
on the piecewise segments are scaled by xit − 1 in (6) and (7), where Φ (im+ )t and Φ (im− )t are
the upper bounds on the segments for positive and negative deviations from peak
efficiency (for a single unit). Generation is defined as a function of throughput in (8),
+ )
− )
similar to (5), with multipliers (slopes) ∆ (im
and ∆ (im
on the piecewise segment
variables which reflect the inefficiency resulting from movement away from the
efficient generation level ( g$it ). System generation targets ( Dt ) must be met exactly3
in each period(9). When applying USIP to ‘realistic’ system representations,
additional constraints might include release rate of change requirements, generation
ramping restrictions, and inter-station and intra-station unit coupling (see [2] for
examples of these).
2 Nodes can be defined as stations, reservoirs, head ponds, or dummy nodes, for example.
3 Both [2] and [5] look at variations to this constraint, such as changing = to ≥ , and eliminating the constraint all
together. Both consider generation as a variable and use generation revenue in the objective function.
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Problem USIP can be implemented using standard MIP software [2]. While
relatively quick to implement, flexible, and able to accommodate considerable detail
about the system in question, this approach can have unacceptable solution times
because of the number of variables required to model all the possible unit
commitments in realistically sized problems [6]. For this reason, specialised solution
approaches based on Lagrangian Relaxation have been implemented which
iteratively determine efficient schedules for specified prices, typically using DP to
dispatch each unit, and then adjust prices to guide the solution to a feasible unit
commitment schedule [4].
Alternatively, techniques such as Dynamic Programming and Stochastic Dynamic
Programming [8,9] have been applied to the real-time or continuous-time [1]
representations of the decision space. While these techniques can be implemented
and solved quickly for simple problems, they suffer computationally as the state
space of the problem increases with the number of units considered. In addition,
slight modifications to the formulation can be time consuming, and sometimes
impossible, to implement.
There does not appear to be a standard optimisation method which is flexible
enough to handle a formulation of a reasonable representation of a realistically sized
system while also being able to find an ‘optimal’ integer solution in a reasonable
time frame for real-time operations. Therefore, there is scope for developing
approaches that consider the complexities of the problem in an optimisation
framework, and find ‘good’solutions in a reasonable time frame.
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Short-term Hydro Scheduling using Integer Programming: Management and Modelling Issues
3. Management/MIP Compatibility
Given that explicit modelling of unit commitment is necessary, to some level, for the
short-term hydro scheduling problem, MIP is an obvious and natural technique for
this problem. Table 1 presents issues relating to the ability of standard MIP to meet
managerial requirements and rates them in terms of being satisfactory (J),
unsatisfactory (L), and mismatched (K), where mismatched areas are those in which
standard MIP may provide an adequate solution technique, but for which the
required assumptions may not provide a particularly good match to reality. Thus
these areas may provide a productive focus when devising heuristics using the MIP
framework.
Table 1: Management and Modelling Issues
Issues
Unit aspects
Feasible decisions
Optimal decisions
Implementability
Solution times
Uncertainty
Sensitivity information
Additional complexity
Management
Necessary
Necessary
Desirable
Straightforward
Fast please
Desirable
Yes
Desirable
Complete4 integerisation
Complete4 optimality
Complete4 feasibility
Unit significance
Peaks and troughs
First few periods
Discrete time
Wasted effort
Not necessary
Not necessary
First few units
Important
Important
Unnatural
MIP
Accurate
Can provide
Can provide
Yes (relatively)
Can be a problem
Not explicitly considered
Not easy
More difficult but may
decrease solution time
Expensive
Yes-with foresight
Yes-with foresight
Equal for all units
Considered equally
Considered equally
Necessary
Match?
J
J
J
J
L
L
L
J
A
L
K
B
K
C
K
D
K
E
K
F
K
G
K
H
There are three main areas where MIP does not appear to satisfy manager’s needs:
additional complexity, completeness, focus, and time discretisation. Performance in
these areas could possibly be improved by modifying the form of USIP’s integer
conditions.
• Additional complexity (A). Modelling aspects of the problem such as spinning
reserve provision and rough running ranges will provide a higher quality
solution. This will complicate the formulation but, by restricting the range of
4 Where ‘complete’= for all periods and units.
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feasible unit commitments, may actually decrease the computational effort
required to find optimal schedules.
• Completeness (B,C,D). The usual approach toUSIP is to integerise all integer aspects
as in (10). In the latter part of the scheduling horizon, though, system parameters such as
generation targets and inflows are only likely to be estimates, and there is no need to
4
model those periods accurately. Thus, the need to solve a problem to complete
4
optimality and to use a solution that is completely feasible is likely to be unnecessary.
• Focus (E,F,G). Standard IP considers all periods and unit commitments to be equally
important. Three areas that should receive more focus in the optimisation procedure are:
• Significant Units. The form of the unit efficiency curves is such that stations have
significantly more flexibility when several (>2) units are committed than when a few
units (<=2) are committed. Thus, for example, we could add constraints/variables to
USIP so that unit commitments are integer ∀ i ≤ 2, ∀ t and continuous ∀ i > 2, ∀ t .
There may be other reasons for scheduling some units before others such as unit
capacity, operating inflexibility, and reserve provision, although these are more easily
implemented using a non-identical units formulation, as in [2].
• Peak and trough periods. The peak and trough demand periods PT
( ) are important
because at these times the system will be at its extreme unit commitments for the day,
and so we know what the range of unit commitments is likely to be between these
periods. Note that this eliminates the need to model switch costs in non-peak/trough
periods. See [6] for a similar strategy applied to thermal unit scheduling.
• First few periods. The first few periods in the scheduling horizon are likely to have
considerably less uncertainty than later periods and uncertainty will increase as we
look further away.
• Time discretisation (H). In reality, rivers do not flow at the same rate for an hour and
then suddenly change to a new rate, nor do units get switched on or off on the hour.
There is a continuous transition between these system states, but system generation
targets are specified in half-hour blocks, which is the obvious form of time discretisation
for an IP approach. Thus, there is tension between the need to model both continuous and
discrete time aspects. One way of partially addressing these issues is to define periods to
be different lengths, depending on the importance of the periods in question. But, in the
peak/trough model discussed above, consideration could be given to having unit
performance for a given peak/trough constant, expressed as a continuous function of, say,
the length of time the unit is committed, rather than as the sum of performance in discrete
periods.
Some of these ideas can actually be implemented in an IP framework with relative
ease. In [3], heuristics based on B, C, F, and G were implemented using
GAMS/CPLEX and the MIP model described in [2] for daily deterministic unit
scheduling in a river chain. A number of ‘naïve’ heuristics were implemented to
provide an additional benchmark for comparison to the MIP solutions because the
latter approach would be expected to have large solution times due to the
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requirement to prove optimality. An example of a naive heuristic is to solve the LPrelaxation, round up all commitment variable values, fix, and re-solve the resulting
LP5. Heuristics based on B and C were also termed naive. Examples of these are:
accepting the first integer solution; or accepting the first solution with an objective
within $x of the maximum possible objective value. Heuristics based on F were
termed SAM heuristics, with the basic principle being to solve the MIP with xit
integer ∀ i, ∀ t ∈ PT and continuous ∀ i, ∀ t ∉ PT (where PT is the set of peak and
trough periods in the daily demand profile), fixing xit ∀ i, ∀ t ∈ PT to the solution
values, then re-solving for xit ∀ i, ∀ t ∉ PT and all other variables. Heuristics based
on G were described as partial integerisation (PI) heuristics and were implemented
~
~
by solving the MIP with xit integer ∀ i, ∀ t ≤ T and continuous ∀ i, ∀ t > T , fixing xit
~
~
∀ i, ∀ t ≤ T to the solution values, then re-solving to optimise xit ∀ i, ∀ t > T and
releases for ∀ i, ∀ t .
Experiments were performed using 3 water values, 3 switch costs, and 3 different
market structures (27 instances in all) and the heuristics were evaluated on the basis
of solution time and difference from the optimal objective values. Solution times for
the MIP ranged from 40 seconds to 40 minutes, while the average solution time was
around 7 minutes. The heuristics generally had less variable and faster solution
times. In terms of objective values, most of the heuristics found solutions with
objective values trivially close to the optimal objective values.
The SAM heuristic which integerised the peak and trough periods (SAM-PT)
performed the best out of all the heuristics in terms of objective value differences,
differing by an average of $3 out of about $150,000 and produced the optimal
objective value (after the second solve) for 24 of the 27 instances. SAM-PT was about
twice as fast as the MIP, with an average solution time of 221 seconds. The PI
heuristic objective values also differed by relatively small amounts from the optimal
objective values, with TD-6 (integerise the first 6 periods) differing by $18 and TD-12
by $32, and both had average solution times of around 200 seconds. As a basis for
comparison, the naïve ‘round up’heuristic had a solution time of about 20 seconds
and an average optimal objective difference of $4,200.
These preliminary results are certainly promising, and indicate that further
investigation of these heuristic ideas is worthwhile.
5 Rounding unit commitment variables values to the nearest integer value and re-solving produced infeasible
solutions for some target driven instances.
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4. Conclusion
Joint consideration of management and modelling issues may lead to better
approaches to problems such as short term hydro unit commitment which are too
complex to be solved when all aspects are considered.
By incorporating
management priorities into mathematical modelling techniques, it is hoped that
schedules can be produced which are more managerially acceptable and less
computationally intensive, while still being based on a realistic system
representation. Preliminary experiments using heuristics based on the ideas
presented in the previous section indicate that solution time can be reduced
markedly while not compromising solution quality.
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References
[1]
M. Craddock, A Continuous-time Model for Optimal Hydro-electric Scheduling, PhD
thesis, University of Auckland, 1996.
[2]
J. A. George, E. G. Read, R. E. Rosenthal, and A. L. Kerr,
Optimal Scheduling of Hydro
Stations: An Integer Programming Model, EMRG Working Paper EMRG-WP-95-07,
1995.
[3]
A. L. Kerr, Hydro Scheduling Heuristics: Implementations of SAM and PI Heuristics in
a Deterministic Integer Programming Framework, System/Data Descriptions, and
GAMS Code Listing, EMRG Working Paper EMRG-WP-97-01, Department of
Management, University of Canterbury, New Zealand, 1997.
[4]
J. A. Muckstadt and S. A. Koenig, An Application of Lagrangian Relaxation to
Scheduling in Power-Generation Systems, Operations Research, Vol. 25, No. 3, MayJune 1977.
[5]
E. G. Read, OR Modelling for a Deregulated Electricity Sector, International
Transactions in Operational Research, Vol. 3 No. 2, pp 129-137, 1996.
[6]
E. G. Read and A. L. Kerr, Scheduling of Thermal Stations: A Structured Analytical
Method, EMRG Contract Report EMRG-CR-94-04, Department of Management,
University of Canterbury, New Zealand, 1994.
[7]
E. G. Read and A. L. Kerr, The Waitaki Hydro Development: A Comparison of
Experimental Results from Integer Programming and Heuristic Approaches, EMRG
Contract Report EMRG-CR-95-02, Department of Management, University of
Canterbury, New Zealand, 1995.
[8]
S. Takriti, J. R. Birge, and E. Long, A stochastic model for the unit commitment
problem, IEEE Transactions on Power Systems, 1995.
[9]
H. Waterer, Hydro-electric Unit Commitment Subject to Uncertain Demand,
Proceedings of 32nd ORSNZ Annual Conference, pp 69-74, Christchurch, New
Zealand, 1996.
[10] D. C. Whybark, The Evolving Role of OR, Key Note Address, 32nd ORSNZ Annual
Conference, Christchurch, New Zealand, 1996.
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