Power Generation Planning using Scenario Aggregation

Transcription

1 81 32nd ORSNZ Conference Proceedings Power Generation Planning using Scenario Aggregation Kevin Broad Department of Engineering Science University of Auckland New Zealand bro v 1. auckland. ac. nz Abstract Electricity demand in New Zealand is predom inantly m et by hydro generation. The remaining demand is accounted for by therm al stations which generate electricity using costly fossil fuels. The hydro reservoirs are small on an international scale and can be emptied in a m atter of months. Furtherm ore, there is a large variance in weekly reservoir inflows. This makes it difficult to create efficient electricity generation schedules as it is unclear how much water can be used w ithout incurring future shortages or wastage. To produce a three m onth schedule th a t minimises the cost of therm al electricity generation, it is necessary to capture the uncertainty in the future reservoir inflows. This paper investigates the application of the progressive hedging algorithm to New Zealand s medium term hydro-therm al scheduling problem. Introd u ction Electricity generation management of a hydro-therm al scheme involves minim ising the use of therm al generation and hence the high therm al costs, whilst ensuring th at there is sufficient water storage available to enable the electricity dem and to be m et during periods of low inflow. The New Zealand system is particularly difficult to manage due to the high variability of reservoir inflows, relatively small reservoir capacities (equivalent to 12% of annual demand in 1992), the lim ited capacity of the inter-island direct current link and the country s isolated location preventing the im portation and exportation of electricity. The New Zealand power crisis of 1992 was the result of the lowest South Island reservoir inflows in sixty years occurring over a seven month period. This led to a shortage in the water available for electricity generation by some of the largest capacity hydro-electric power utilities in the country. The crisis led to the need for public electricity savings and costly generation initiatives by ECNZ, and highlighted the need for a better understanding of the New Zealand power system as well as improved models for generation planning.

2 32nd ORSNZ Conference Proceedings 82 It is im portant th at models for this problem capture the uncertainty in the hydro reservoir inflows. The approach presented in this paper uses a scenario aggregation technique and follows on from the work done by Dye [2], 1 The M odel T his section explores the structure of the medium term hydro-therm al scheduling problem. This is the problem of determining weekly storage targets for each of the main hydro reservoirs given determ inistic weekly electricity demands and stochastic reservoir inflows, whilst minimising therm al generation costs. 1.1 T he N ational Grid For the sake of experim entation, a simplified electricity network of fourteen transmission lines and buses has been implemented. A network model with gains is used which incorporates parallel arcs with increasing linear losses. These parallel arcs approxim ate the quadratic losses which occur in power transmission. Side constraints control the flow around cycles in the network, approxim ating the way power distributes itself in a power system. 1.2 R iver Chains River chains consist of an upstream reservoir, where water is stored, and a sequence of downstream hydro stations which can be used for power generation. W hen water is released from the reservoir, it flows down the river and through the turbines of the hydro stations, thus producing electricity. Alternatively, some or all of the water can be spilled around the hydro stations. The limited operational storage at each station and the traversal times for water flows are not significant in medium term planning. River chains can be modelled as linear networks, with side constraints which make the total power output from the hydro stations meet demand. 1.3 Therm al Stations As well as generating electricity using hydro generation, fossil fuels such as gas, oil and coal can be used. The model includes the six main therm al stations in New Zealand. These stations have varying operating costs depending on the station fuel type and efficiency. 1.4 E lectricity D em and Piecewise linear demand curves can provide a reasonable approxim ation to the load duration curves for each week of the model. This allows the variation of the electricity dem and within a week to be represented by a monotonically decreasing function as opposed to using a load curve representation which indicates the actual tim ing of dem and peaks during the week. W hile we are not concerned with the tim ing of decisions during the week, we still wish to model transm ission line limits and so it is im portant to have this intra-week representation of the load variation.

3 83 32nd ORSNZ Conference Proceedings 1.5 M od elling Stoch astic Inflows The uncertainty in the model arises from the inflows into the reservoirs and trib utary inflows along the river chains. Certain geographical com binations of these inflows are highly correlated and so a principal component analysis has been conducted to lower the number of random variables and to obtain uncorrelated ou t comes. Just two components are required to account for 71% of the variance. The third component accounted for less variance than a single inflow variable. An autoregressive model fitted to these two components, using historical data, provides forecasts for each component. Several outcomes with associated probabilities are determ ined so th at they give the same variance as the original forecast while being approxim ately normally distributed. Additional forecasts are made from each of these outcomes to produce a set of outcomes for the following week. This procedure continues for the duration of the model. 2 M odel D ecom position A determ inistic equivalent can be formulated which effectively involves replicating the model for each possible scenario (or inflow sequence) and adding constraints to ensure th at the model does not anticipate which scenario will occur. Models form ulated in this way tend to be inhibitively large, as is the case here, and require decomposition techniques to be used. 2.1 Possible M ethods The two main methods of decomposing stochastic programs are stage-wise decomposition and scenario decomposition, otherwise called scenario aggregation. Exam ples of stage-wise decomposition include stochastic dynamic program m ing (SDP) and stochastic Benders decomposition. These m ethods allow many scenarios to be considered, but restrict the model form. Restrictions include the discretization and aggregation of the state variables, as in the case of SDP. Also, separability of costs across stages is a lim itation which prevents the use of utility functions to model risk aversion, although clever techniques have been presented th a t can model risk aversion in SDP [1], Scenario aggregation is a technique which decomposes the determ inistic equivalent into scenario subproblems. The progressive hedging algorithm (PHA) can be used to find an optim al solution to the deterministic equivalent by repeatedly solving these subproblems using objective function updates to guide the solution process towards an optim ally hedged solution. The advantage of this approach is th at each scenario consists of the full determ inistic model over the entire tim e horizon. This allows the model structure to be exploited. Furtherm ore, a utility function can be used to evaluate tem poral returns or costs in a non-separable cost function over the stages of the model. 2.2 T he P rogressive H edging A lgorithm PHA iteratively solves scenario subproblems of the form: L (x s, w, x) = f ( x s) + wsx s + ^ { x s - x )2

4 32nd ORSNZ Conference Proceedings 84 where: x s is the decision variable for scenario s, where s S. ws is a dual multiplier for scenario s. /( ) is a convex cost function. The constant term, x, is defined by: where: x = Y. P*x s ses ps x*s is the probability of scenario s occurring, is the optim al solution from the previous iteration. These subproblem s have constraints defining the determ inistic model specific to each scenario s outcomes. A lg o rith m 1 Progressive hedging algorithm. k <- 0 < 0, for all s G S x < where is the scenario s component of the solution to the relaxed determ inistic equivalent. R e p e a t k 4 A: + 1 Solve L (x s, x k~l), subject to A x s b, for all s G S xk Z,esPsXk, <r- Wg-1 + r(x ks x k), for all s S Perform an optional penalty update, rk < /3rk~l + c, 0 > 0 U n til convergence Note th at Y^sesPsWs 0 at every iteration by virtue of the dual update. 3 M od ellin g Risk A version One advantage of PHA, over stage-wise decomposition m ethods, is th at all stages of a scenario are solved at the same time and can therefore be coupled by a nonseparable cost function. This is an im portant requirement for modelling risk aversion. The approach taken in this paper is th at of minimizing the expectation of a quadratic utility function, min Y,P s { f i x s) + 3 { f{ x s))2), ( 1) where (3 > 0. It is shown by Sarin and Weber [3] th at the above objective function actually

5 85 32nd ORSNZ Conference Proceedings minimises E [f{ x s)} + 0var[/(,)] + f3 E [f(xs)}2 (2) and by increasing the value of f3 we can increase our aversity to a large variance in objective costs and hence we minimise the overall risk. 4 Bounds on the D eterm inistic Equivalent One area th at is not explored in the PHA literature is th at of finding bounds for the objective function of the deterministic equivalent. A simple analysis shows th at by m aintaining the condition Y.sesPsWs = 0, the above approach is equivalent to using the augmented Lagrangian m ethod on the determ inistic equivalent. By removing the quadratic term and re-solving with the current duals, we are effectively solving the normal Lagrangian and hence forming a valid lower bound by using the standard result where L (\,x * ) < L(A,x*) < L(A*,x), (3) f(x* ) = L(X *,xm). (4 ) An upper bound is found by evaluating the expected value of any im plem entable (non-anticipative) solution achieved so far. C om putational experience indicates th at PHA has slow convergence, especially near optim ality. By monitoring the gap between these bounds we can choose to term inate the algorithm early with some indication of how close to optim ality our current implementable solution is. 5 P H A w ith Scenario Sam pling W hen the number of scenarios is large the PHA must solve many subproblems. To avoid this we may solve over a subset of the scenario space, term ed the active set, to provide an implementable solution for this subset. The rem aining scenario subproblems are then solved with the subproblem decision variables restricted to this implementable solution. Any infeasible scenario subproblems are added to the active set and the process is repeated until no infeasible subproblem s exist. Provided all subproblems are feasible, the expected value of all the subproblem objective values gives an upper bound. The expected value of the objective values for the active set problems and for the remaining scenario subproblems solved with no restriction on their decision variables provides a lower bound. The extreme scenarios (ones with very high or very low inflow sequences) are those which will have the biggest impact on the optim al solution. Therefore, they are the ones which should be included in the initial active set. To reduce the gap between the bounds, scenarios are sequentially added to the active set, while any scenario subproblems not contributing strongly to the im plementable solution are removed. The level of contribution of scenario subproblems can be identified by the m agnitude of their respective dual multipliers. A high absolute value indicates a

6 32nd ORSNZ Conference Proceedings 86 strong contribution to the implementable solution, while a low value indicates th at the subproblem is having very little impact. Therefore, we wish to add subproblems, not in the active set, which we anticipate will have large dual multipliers. One technique to achieve this is to estim ate dual multiplier values by interpolating the m ultipliers of sim ilar scenario subproblems in the active set. 6 C onclusions The advantages of the PHA are the ability to exploit the structure of the individual scenario subproblem s and the ease with which risk aversion can be included in the optim isation. Drawbacks include slow convergence and limited stochastic representation. This paper goes some way towards m itigating these disadvantages in the hope of m aking the model useful to decision makers. Future work will determ ine the effectiveness of the outlined techniques. A cknow ledgem ents I would like to thank ECNZ for their generous support and for supplying the necessary d a ta and information th at made this research possible. Also, thanks must go to my supervisor, Dr Andy Philpott, for his invaluable input into my work. R eferences [1] L. Blakeslee, T.A. Lone, Modelling Optimal Grain-marketing Decisions when Prices are Generated Autoregressively, European Review of Agricultural Economics, 22 (1995), pp [2] S. Dye, On a Flexible Model for New Zealand's Hydro-Thermal Electricity Generation System, PhD thesis, Massey University (1994). [3] R.K. Sarin, M. Weber, Risk-Value Models, European Journal of O perational Research 70 (1993) pp