Power Plant Valuation
An even more challenging operational constraint is a hard
bound limitation to the number of starts, or alternatively, that start costs increase with the number of starts made before. This can be used to avoid an otherwise quite destructive number of starts, e.g. every working day. As a consequence of such a hard bound constraint, the owner of a plant will try to minimize the number of startups. In any case, the plant operator needs to keep track of two quite different statistics: 1. How many hours is the plant already offline, or how many hours is it already producing? This is a statistic (‘state’ in the dynamic programming language) which looks at the recent history of actions.
2. How many times did the plant start already this year? This is a statistic of a different nature, looking back until the beginning of the year.
Whatever the model implementation is, an optimisation
tries to search for the optimal dispatch taking into account the current and future spark spread levels, the current ‘short-history’ state, and the current ‘long- history state’.
max {
3. As 2, but assuming no perfect foresight. This means the plant operator has to judge the likely level of spark spreads in the future and take decisions which are inevitably suboptimal in certain situations. We use Least-Squares Monte Carlo to establish this strategy.
The paper of Tseng and Barz (2002) is the primary academic
reference for a description of a dynamic programming approach to handle short-history constraints of up/down times. The picture below shows the main logic. The model distinguishes between different states (y-axis) that the plant can reach at different hours (x-axis). In the shown example with 2 hours minimum on-time and 2 hours minimum off-time, if the plant is up (‘on’) for 2 hours or more it can stay ‘on’ or go ‘off’; if it is off for 1 hour (idem if on for 1 hour), the operator has no choice other than to stay off for one more hour. At each node, the model compares the continuation values (F) for the alternatives it has and chooses the maximum. For example, when at the end of hour t the plant is already off for 2 hours or more, the choice is:
F on 1hr + Margin – Start Costs t+1 – Shadow Costs “Plant switched on in hour t+1” F off 2hr
t+1
Solution Approaches to Handle Start Constraints In this paragraph we will demonstrate three different solution approaches, each using backward valuation dynamic programming. We show the pros and cons of each, but before doing so, a fourth alternative is worth mentioning. Mixed- integer linear programming (MILP) is a common alternative in which the objective (profit maximization) and constraints are written down in a set of linear equations. While this approach is relatively easy to implement, it is often a factor 5-20 slower than a dynamic programming approach due to the large number (at least 8,760 per year) of decision variables. Even more, MILP ignores any price uncertainty, which is part of our third approach below. These approaches are: 1. A dynamic programming approach for the ‘short-history’ constraint, and a ‘shadow cost’ for the ‘long-history’ constraint to limit the number of starts. This approach assumes perfect foresight about future price levels.
2. A dynamic programming approach for both types of constraints, meaning we have an additional dimension to keep track of the number of previously made starts. This approach assumes perfect foresight about future price levels.
“Plant switched on in hour t+1”
Solution approach 1 means we run through the dynamic
program a couple of times. In round 1 we only include the real costs for starting up (Start Costs, e.g. fuel-related) and set the shadow costs for starts to zero (Shadow Costs = 0). If the number of starts is below the maximum, we are done. If the number of starts in the previous round is above the maximum, we modify the shadow cost. We make as many modifications to the shadow cost until the model reaches the maximum plant value within the start limitation per year. In our implementation typically 10- 20 rounds are needed before reaching the optimum; the whole procedure generally takes less than 1 second per simulation
50 worldPower 2010
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