• Introduction
• Methodology
• Results
• Conclusion

• Power systems analysis : balance between generation and demand
• Models differ with time horizon
• Input data differ with time horizon
• Focus on weather-based variables : consistency required
  • Temperature / Consumption
    
  • Wind / Wind power
    
  • Solar radiation / Solar power
    
  • (Water inflows / Hydro-power)

• Short-term (up to 1 month)
  • weather forecasts give consistency
• Mid-term / Long-term (>1 month)
  • historical data
  • independent time-series models
  • connected time-series models with temperature as a reference (using linear correlation)

• Post-processing ensemble forecasts :
  • Schaake shuffle
  • Ensemble Copula Coupling
  • Bayesian model averaging
• Weather generators :
  • univariate/multivariate time series models
  • Markov chains (rain)
  • GCM coupled with downscaling approach

Adapt Ensemble Copula Coupling

Ensemble forecasts replaced by historical data

Two-steps :

• Estimate and simulate from marginal densities
  
• Model dependence using ranks (discrete copula)

Dimensions :
- a=1,…,A : index of variables
- k=1,…,K : number of geographical locations
- t=1,…,T : number of time steps per reference series
- s=1,…,S : number of reference series

Our application :
- A = 3 variables : temperature, wind and solar power load factors
- K = 1 geographic location : France
- T = 365 time steps per historical scenario : daily data
- S = 53 reference series (1958-2010)

Step 1 : Marginal densities
For each time step t, each variable a and each geographical location k , estimate marginal density (using neighbouring values in time) using non-parametric density estimation

Let v the size of the window to set the estimation sample : S(2v+1) data

Denote the sample by \(y=(y_1,…,y_{(S(2v+1))})\)

The non-parametric density estimation is (\(K\) is a gaussian kernel and \(h\) is optimized): \(\hat{f}(y_i)=\frac{1}{S.(2v+1).h}\sum_{l=1}^{S(2v+1)}K(\frac{y_i-y_l}{j})\)

Simulation step
- From each marginal density, simulate S values

Step 2 : Inter-variable and Time dependence

• For each variable \(a\), each time step \(t\) and each geographic location \(k\), order the \(S\) historical values and keep the vector of ranks
  
• Put all vectors in a multidimensional array to link variables and geographic locations for each time step

Simulation step

• Final outputs are obtained by ranking simulated values and join variables, time-steps and geographical locations using rank vectors
• The number of simulations is constrained by the size S of the sample
• However the procedure can be executed several times

January : low temperature - low wind power

July : high temperature - low wind power

July : high temperature - high solar power

Pros and Cons

• simple method
  
• non-linear correlations
  
• require large dataset

To go further

• adapt to local data : less historical data, inconsistent time granularity
  
• translate ranks in time

R. Schefzik, Ensemble Copula Coupling, 2011.