Short Term River Chain Optimisation
Disclaimer
Reasonable care has been taken to ensure that the information in this paper is up to date at the time of issue. Potential users of EMarket should, however, ensure that they evaluate EMarket and this paper through an appropriate evaluation process in consultation with Energy Link. The authors are also reliant on certain information external to EMarket and Energy Link, for which no liability or responsibility can be accepted.
Introduction
This technical bulletin is intended to provide users and interested parties with a detailed explanation of how EMarket’s Short Term River Chain Optimisation (STRCO) module works. EMarket was originally designed to model medium and long-term hydro generation optimisation. This meant that hydro offers were based on medium to long-run marginal costs and did not vary in the short term. The aim of STRCO is to improve the realism of results from EMarket by ensuring that generator behaviour is consistent with the detailed day to day operation of generators in the market. STRCO is not the default for EMarket’s large hydro systems so must be enabled by the user as required.
Other Documents
This bulletin is one of a series of technical bulletins relating to Energy Link’s EMarket model. Taken together, the bulletins replace the old EMarket User Guide. The full series of bulletins covers an overview of the EMarket model, the details of the four major New Zealand hydro systems modelled in EMarket, water values and hydro offers, power flows, dispatch and nodal pricing, short term river chain optimisation and company optimisation.
Overview
Modelling in EMarket is also usually done with steps that aggregate a large number of trading periods together. Whilst changes in load through the day can be modelled, generator offers remain constant. In modelling the market behaviour over the course of a day it is necessary to determine how generators may adjust for intra-day fluctuations in demand and prices.
The benefits of determining short term behaviour extend beyond the ability to model in detail the daily behaviour of the market, since short term fluctuations in offers will have a gross effect on the overall behaviour. An example of this is the meeting of minimum flow requirements on the outflow of a large hydro system. Here the water throughput that is required for the minimum flow at night can be used for generation during the day, accumulating at the lowest reservoir which will empty over the night periods. It is difficult to determine the overall long term effect of minimum flow requirements on energy offered without understanding the short term effects in some detail.
In principle, short term modelling for hydros could include:
- all the hydrological or regulatory constraints on the system;
- the effect of generation on energy price, and price on profit;
- provision of instantaneous reserves.
In EMarket hydro systems can be enabled to optimise using the STRCO module. The optimisation horizon is one week at a resolution of 4 hours.
River Chain Hydro Modelling, the DP approach
The problem to be solved for a major hydro is the optimal pattern of generation in the short term when the configuration of storage within the system can vary considerably over a short period. Dynamic programming (DP) provides a robust algorithm for determining the basis of this behaviour.
Optimal behaviour is defined by maximising gross profit, taking into account the entire portfolio of generation assets, retail and hedge contracts, and marginal costs, of the company owning the system. The behaviour at any time is determined by the time of the day and the state variables – the most important of which represent the storage within the system. When the DP is run it becomes possible to calculate the marginal value of all the state variables, and offering behaviour at each step can be determined by optimising against these marginal values.
For example, Figure 1 above shows a hydro system with a large storage lake, A, at the head and two smaller lakes in a chain below. Three dams, a, b and c, convert the flows between the lakes into electricity at a rate of 1 MW/cumec. The storage in each lake is measured in cumec-days (CMD).
When the DP is run it should be possible to assign a marginal value to the storage at A, B and C, given the current storage levels and the time. This allows a marginal cost figure to be allocated to the generation at a, b and c. If these values are 2,520 $/CMD, 1,440 $/CMD and 480 $/CMD, respectively, then generation at a has a marginal cost of:
[(2520 – 1440) $/CMD] / [24 h/Day] × [1 MW/cumec] = $45/MWh.
Similarly the costs of generation at b and c are $40/MWh and $20/MWh respectively.
It might then be possible to offer in three bands of generation at 45, 40 and 20 $/MWh and with the maximum MW output of a, b and c.
But the marginal costs of generation do not completely determine offers for large systems. The offers have to allow for the fact that they apply over a fixed period of time, and storage or flow constraints must be obeyed over that time. For example, if generating at full output will empty one of the reservoirs before a trading period is finished, then it is not possible to offer in all generation. If the total system output can have an impact on prices, this effect needs to be taken into account and it is also necessary to know the load commitment and hedges of the company as a whole.
Hydro System Specification
The following describes the most important features of a hydro system with respect to the modelling EMarket does in STRCO. The description has two main features:
- channels, through which water is moved, which include stations, canals or spillways, and
- nodes, which are the departure or destination points for channels.
Nodes may represent reservoirs or may simply be the point where a number of channels meet.
The system configuration is specified as follows:
| N | Number of channels |
| M | Number of nodes |
| ei | MW/cumecs for channel i = 1 to N |
| Maxi, Mini | Maximum, minimum flow though the channel, i = 1 to N |
| Ntoi, Nfromi | The nodes which are connected by the channel, i = 1 to N |
| Sj | Available storage at a node, j = 1 to M |
The state of the system at time t is specified as follows:
| sj | Current storage at a node, j = 1 to M |
The behaviour of the system at time t is specified as follows:
| Ijt | Uncontrolled inflows to nodes for time t, j = 1 to M |
| Fit | Flow (cumecs) through a channel (hydro station or spillway) at time t, i = 1 to N |
For example, a specification for the system in Figure 1 is:
N = 3; M = 4; ei = 1, 1, 1; Ntoi = 2,3,4; Nfromi = 1,2,3; Maxi = 200,300,350;
Mini = 0,0,50; Sj = 9000, 32, 120
The DP Process
Adding to the definitions given above:
MWVj,t - the marginal value of storage at node j at time t
The result of the DP will be a multi-dimensional function
MWVj,t = WVFj,t (S1t, S2t,…, SMt)
The following describes the dynamic programming analysis using a week's worth of forecast data. The dynamic program requires the following data:
- A price forecast – half hourly market prices for the week;
- Market sensitivity forecast – this would take the form of a sensitivity figure given by (change in price)/(change in generation), that estimates the amount prices will fall as more generation is put online;
- Load forecast, for company retail load only;
- Hedges for the week.
The DP starts with a finite set of seed MWV's for all reservoirs. These are assumed to be the water values at the end of the week, given for a number of storage configurations. Then, for the preceding time period, water values are calculated on the assumption that the seed water values are correct. This is done by backward projection, roughly as follows. Taking a storage configuration at time t and assuming a set of MWV's to be correct for this configuration, an optimal production schedule can be calculated. This production schedule will result in a change in storage, which is subtracted from the storage configuration at time t to give a storage configuration at time t-1. The resulting storage configuration at time t-1 is then assigned the same set of MWV's.
This is repeated back through the week to the beginning. The solution will converge, so for any seed values, the resulting MWV's for the first time period should be similar. The process can then be repeated, using the MWV's for the first period as a seed for the following iteration. The resulting MWV's, once convergence has been achieved, will be optimal for the week's operation – with the weak assumption that the following week will have roughly the same conditions. The final result says that MWV should be constant throughout the week.
Optimal Production
At each step of the DP it is necessary to calculate optimal production for a given set of MWV's. Also, when the MWV's are calculated they will be used to construct offers for the market model. As is outlined above, it is fairly straightforward to convert MWV's into marginal costs for generation, but this does not provide us with a complete solution. It is necessary to take into account all the constraints on the system and the dynamics of the market. The following method takes account of most constraints and uses a straightforward estimate of profit maximisation.
First we define total gross profit for one period:
GPt = Pt Gt + (RP - Pt) Lt + (HPt – Pt)HQt – GCt
Where Pt is energy price at time t, RP is the retail price for which energy is being sold, HPt is the hedge price and HQt is the hedge quantity. GCt is the cost of generation, which will be defined by the water values (since the marginal cost of hydro generation is virtually nil, marginal cost must be based on the MWV which is an opportunity cost giving the expected future value of water in storage.) Market price and generation are correlated giving the effect on the price of generation. Using a market sensitivity estimate calculated from the simulated offer stack within EMarket:
Pt = P0t - MStGt
and removing constant factors, the maximise gross profit problem becomes:
max {(P0t - MStGt) (Gt - Lt - HQt) - GCt }
Generation, Gt can be given as a function of the flows, Fit:
