Forecast-Error Reserve Sizing (2D)

Why the grid needs reserve

Wind and solar plants are scheduled from a day-ahead forecast, but what arrives is different. The forecast error e = actual - forecast has to be absorbed by reserve: generators or batteries that are paid in advance to stay ready (capacity price c, EUR per MW per hour) and then deliver energy when called. Energy that nobody had reserved is settled at the imbalance price p, which is usually much higher. How much reserve to buy is a balance between paying for idle capacity and paying for shortages.

Smoothing: many sites are quieter than one

Each site has its own error with standard deviation sigma_i (a percentage of its capacity), and errors of nearby sites are correlated: rho_ij = exp(-d_ij / L) within the same technology (wind with wind, solar with solar), reduced by a cross-correlation between wind and solar, where d_ij is the distance and L the correlation length (weather systems are larger than a single farm). The error of the whole fleet has

sigma_total = sqrt( sum_i sum_j rho_ij sigma_i sigma_j )

which lies between the independent case sqrt(sum sigma_i^2) (grows like the square root of the number of sites) and the fully correlated case sum sigma_i (grows linearly). Spreading plants over a wider region lowers rho and therefore the reserve needed per MW of capacity.

The newsvendor rule for reserve

For an up-reserve R the expected hourly cost is J(R) = c R + p E[(D - R)+] with the deficit D = -e. The derivative c - p P(D > R) = 0 gives the critical fractile P(D <= R*) = 1 - c / p, so for a normal error R* = m + sigma Phi^-1(1 - c / p) (m = -bias). With the default prices c / p = 6 / 90, the reserve covers the deficit in 93.3 % of hours. The expected shortage is E[(D - R)+] = (m - R) Phi(z) + sigma phi(z) with z = (m - R) / sigma. The down-reserve (surplus side) follows the same formula with its own prices.

Numerical checks on this page

(1) The fleet spread from the covariance formula is compared with a Monte Carlo of 20,000 correlated hours generated with a Cholesky factor of the correlation matrix. (2) The reserve that minimises the Monte Carlo cost curve (brute-force scan) is compared with the critical-fractile formula. (3) The closed-form expected shortage at the applied reserve is compared with the Monte Carlo average.

Honest simplifications

Forecast errors are taken as Gaussian with constant spread and the site geometry is a fixed illustrative map; real errors are skewed, depend strongly on the output level and the weather regime, and have time structure (the 24-hour sample day is a simple autoregressive realisation). Real reserve markets have several products with activation times, ramp limits and asymmetric prices, include demand forecast error and unit outages, and are designed with probabilistic or risk-based methods. The sample-day output is clipped to the site capacity while the statistics are not, which is one more reason why the day-sample graph only illustrates the numbers.

• Press Clustered sites then Dispersed sites and watch the "Smoothing gain" tile: the same fleet needs far less reserve when its sites are far apart.
• Load Expensive shortages: the critical fractile rises and the optimal up-reserve grows; the gold dot on graph 3 moves to the right.
• Load Too little reserve (manual 30 MW) and play the day: the scale tips to the red side and the city lights flicker in the hours when the error exceeds the reserve.
• Load Biased forecast: a systematic over-forecast shifts the whole error histogram and the optimal reserves shift with it.
• Switch the up-reserve to manual and drag it: the "Regret" tile is the extra expected cost compared with the optimum.
• Orbit to the Hub and reserve plant: lit gas modules are contracted reserve, brighter ones are being called.