Short-term Hydro Scheduling using Integer Programming: Management and Modelling Issues Andrew L. Kerr and E. Grant Read Department of Management University of Canterbury New Zealand
[email protected],
[email protected]
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.
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. Manager
OR/MS Expert
Software/Modelling Expert
Figure 1: Approaches to ‘the problem’ 1
Consider Figure 1, which depicts three perspectives of, or approaches to, a given problem. The manager is the 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 multiindustry and multi-discipline experience/knowledge; the motto for this individual could be “understand the problem and fit or devise a technique for it”. 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 in each case, even using their 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. and the manager has increased. 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 the 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, non-linearities in head effects, reserve provision, and uncertainty), and because a de-centralised decision making environment has not provided incentives to encourage optimisation at the organisational level [5]. With the advent of deregulated energy markets, decentralised management have 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 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 2
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 Integer Programming, which assume no prior information about the managerial decision problem. In this paper we present an overview of the problem (Section 2) and then some ideas for modifying standard Integer Programming to accommodate managerial needs (Section 3). Conclusions are presented in Section 4.
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 on 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: I
(1)USIP
Maximise
∑
ψ
i= 1
T i
∑ ∑ (α T
s iT −
t= 1 i= 1
Subject to (2) xit = xit − 1 + ui(+ )t − ui(− )t (3)
s = s t i
t− 1 i
+ n − q − w + t i
t i
I
t i
(+ ) i
(5)
u
t − ehi h
qit = xit − 1q$it +
∑ ( qcc
( + )t 0 ≤ qccim ≤Φ ( − )t 0 ≤ qccim ≤ Φ
(8)
git = xit − 1g$it +
( + )t im
( − )t − qccim )
∀ i, t
∀ i, t
× xit − 1 × xit − 1
∑ (∆( m= 1
∀ i, t ∀ i, t
M
m= 1 (+ ) im (− ) im M
)
h∈ B i
sitMIN ≤ sit ≤ sitMAX
(6) (7)
1
(− ) ( − )t i i
∑ (q ) + ∑ (w ) t − d hi h
h ∈ Ai
(4)
ui( + )t + α
+ ) im
( + )t ( − )t qccim + ∆ (im− ) qccim
∀ i, t, m ∀ i, t , m
)
∀ i, t
See [2] for the corresponding non-identical units formulation.
3
(9)
∑
git = Dt
∀t
i t i
(10) x integer The objective (1) maximises the value (ψ
T i
∀ i, t ) of storage ( s ) at the end of the T i
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 maximum of 2M segments. The piecewise segments allow ( + )t ( − )t positive qccim and negative qccim deviations from peak efficiency ( q$it ), where the level of peak efficiency for a given unit commitment is simply a multiple of peak efficiency for a single unit at the station (because the units are assumed to be identical). The bounds on the piecewise segments are scaled by xit − 1 in (6) and (7), where Φ (im+ )t and Φ (im− )t are the upper bounds on the segments for positive and negative deviations from peak efficiency (for a single unit). Generation is defined as a function of throughput in (8), similar to + ) − ) (5), with multipliers (slopes) ∆ (im and ∆ (im on the piecewise segment variables which reflect the inefficiency resulting from movement away from the efficient generation level ( g$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). Problem USIP can be implemented using standard Integer Programming 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. 2
Nodes can be defined as stations, reservoirs, head ponds, or dummy nodes, for example. Both [2] and [5] look at variations to this constraint, such as changing = to ≥ , and eliminating the constraint all together. Bothh consider generation as a variable and use generation revenue in the objective function. 3
4
Issues Management Integer Programming Unit aspects necessary accurate Feasible decisions necessary can provide Optimal decisions desirable can provide Implementability straightforward yes (relatively) Solution times fast please can be a problem Uncertainty desirable not explicitly considered Sensitivity information yes not easy 4 Complete integerisation wasted effort expensive 4 Complete optimality not necessary yes-with foresight 4 Complete feasibility not necessary yes-with foresight Unit significance first few units equal for all units Peaks and troughs important considered equally First few periods important considered equally Discrete time unnatural necessary Table 1: Management and Modelling Issues
Match? J J J J L L L Ka Kb Kc Kd Ke Kf Kg
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 which consider the complexities of the problem in an optimisation framework, and find ‘good’ solutions in a reasonable time frame.
3 Management/Integer Programming Compatibility Given that explicit modelling of unit commitment is necessary, to some level, for the short-term hydro scheduling problem, IP is an obvious and natural technique for this problem. Table 1 presents issues relating to the ability of standard Integer Programming 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 IP may provide an adequate solution technique, but for which standard IP assumptions may not provide a particularly good match to reality. Thus these areas may provide a productive focus when devising heuristics using the IP framework. There are three main areas where Integer Programming does not appear to satisfy managers’ needs: time discretisation, completeness, and focus, and performance in these areas could possibly be improved by modifying the form of USIP’s integer conditions. •
4
Completeness (a,b,c). The usual approach to USIP 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 model those periods accurately. Thus, the need to solve a problem to complete4 optimality and to use a solution that is completely4 feasible
Where ‘complete’ = for all periods and units.
5
is likely to be unnecessary. Thus, for example, we could relax USIP to allow xit ~ ~ to be integer ∀ i, ∀ t ≤ T and continuous ∀ i, ∀ t > T . •
Focus (d,e,f). 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, ∀ 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. Thus we can define xit to be integer ∀ i, ∀ t ∈ PT and continuous ∀ i, ∀ t ∉ PT . 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. So, xit can be defined as integer for the first few periods of the scheduling horizon and continuous for the remaining periods to reflect this.
•
Time discretisation (g). 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 [3]. Preliminary experiments using GAMS/CPLEX and the IP model described in [2] for the Waitaki Hydro System in New Zealand have investigated some of the ideas presented above perform on 24 period deterministic unit scheduling problems. They show
6
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.
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 EMRGWP-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, May-June 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, p Christchurch, New Zealand, 1996.
[10] D. C. Whybark, The Evolving Role of OR, Key Note Address, 32nd ORSNZ Annual Conference, Christchurch, New Zealand, 1996.
7
