This document discusses using fractional calculus techniques for wind and solar power generation forecasting. It introduces a simple grid network model with three generation nodes and three load dispatch points. Forecasting the generation at each node separately is important for stability, rather than just aggregating the total generation. Statistical forecasting is formulated as an error minimization problem that can be solved using fractional derivatives. This allows incorporating the non-local nature of renewable generation. The document advocates using techniques like fractional PDEs and scenario analysis for computational forecasting methods. Accuracy data for wind and solar plant-level forecasts is also presented.

1. Wind/Solar Power Generation
Forecasting
Abhik Kumar Das
del2infinity Energy Consulting

2. G3(t)
G1(t)
L1(t)
L3(t)
L2(t)
X(t), T13, C13
G1(t)-X(t), T11, C11G2(t) - Y(t), T22, C22
G3(t) - L3(t)+X(t), T32, C32
Without going complex structure of grid network let define a simple structure where we have three
variable generation grid nodes say G1, G2 and G3 for simplicity let define three Load dispatch points
L1, L2 and L3 such that L1 lies between G1 and G2, L2 lies between G2 and G3, and L3 lies between G3
and G1. The structure is made as simple as possible for the energy flow such that a generation
station can distribute its generation in its two nearest Load dispatch points. For simplification, this
analysis considers only energy flow in the network to find the stability of the network.

3. min 𝐺1 𝑡 , 𝐿3 𝑡 , 𝑇13, 𝑇32 − 𝐺3 𝑡 − 𝐿3 𝑡 ≥ 𝑋(𝑡) ≥ max 0, 𝐺1 𝑡 − 𝑇11, 𝐿3 𝑡 − 𝑇33
min 𝐺2 𝑡 , 𝐿1 𝑡 , 𝑇21 ≥ 𝑌(𝑡) ≥ max 0, 𝐺2 𝑡 − 𝑇22
𝐺3 𝑡 − 𝐿3 𝑡 + 𝐺2 𝑡 − 𝐿2 𝑡 ≥ 𝑌 𝑡 − 𝑋 𝑡 ≥ − 𝐺1 𝑡 − 𝐿1 𝑡
A(t) depends on the each value of G1, G2 and G3 but not on the sum of its values i.e. G1 + G2 + G3.
Interestingly since it is a variable generation and A(t) is not constant but to get the +ve value of A(t)
we need a prediction of G1, G2 and G3 separately but not as a sum or aggregation of those values.

4. Here the red area is actual requirement
and the area of A(t) decreases due to the
uncertainty of the generation.
Suppose schedule generation of G1,
G2 and G3 are not known separately,
then the above situation may arise:
The aggregated forecast creates instability when A(t) is not
positive.

5. Data support by CSTEP, India
The Complex Network

6. For M Generating point and N Load point of a Complex Network,
A(t) is approximately (M+N – 4) dimensional space.
If A(t) is not a connected space then it creates the instability
Hence what is the minimum temporal and spatial granularity of
Forecasting?
Minimum Temporal Granularity = 15 min already fixed
Minimum Spatial Granularity ?
Minimum Capacity of Generation?

7. Statistical Forecasting

8. Statistical Forecasting problem is an error minimization problem
min
𝑃
Ω
𝐹(𝐱, 𝑃(𝐱), 𝛻𝑃(𝐱))𝑑𝐱

9. min
𝑃
Ω
𝐹(𝐱, 𝑃(𝐱), 𝛻𝑃(𝐱))𝑑𝐱
𝛻
𝜕𝐹(𝐱, 𝑃(𝐱), 𝛻𝑃(𝐱))
𝜕 𝛻𝑃(𝐱)
=
𝜕𝐹(𝐱, 𝑃(𝐱), 𝛻𝑃(𝐱))
𝜕𝑃(𝐱)
Solve it using Euler-Lagrange:
And get an Iterative (or Dynamic) equation:
𝑃𝑡+1 𝐱 = 𝜙(𝑃𝑡 𝐱 )

10. Few Computational Techniques in Solving the
Dynamic Problem for Wind / Solar Forecasting
• Markov -> Hidden Markov
• ANN -> DNN
• PDE -> Stochastic PDE
• Single Hypothesis -> Multiple Hypothesis -> Scenario
based Analysis
• Fractional Calculus !

11. Fractional World
•
𝑑
𝑑𝑥
,
𝑑2
𝑑𝑥2 exists
• What about
𝑑0.5
𝑑𝑥0.5 or
𝑑0.9
𝑑𝑥0.9 or
𝑑1.1
𝑑𝑥1.1 ?

12. Fractional Derivative
Riemann fractional derivative (Left) is defined as
1( ) 1
( ) ( ) ( )
( )
xn
n
L x n
L
d f x d
D f x x s f s ds
ndx dx

 


 
  
  
  11( )
( ) ( ) ( )
( )
n Ln
n
x L n
x
d f x d
D f x x s f s ds
ndx dx

 


 
  
  
Riemann fractional derivative (Right) is defined as
Other forms of Fractional derivative also exist like Riesz or Caputo fractional derivative

13. • It is non-local convolution type
Why Fractional Derivative ?

14. L=0 => Riemann-Liouville Form of fractional derivative
min
𝑃
Ω
𝐹(𝐱, 𝑃(𝐱), 𝛻 𝛼,𝛽
𝑃(𝐱))𝑑𝐱
Define the error minimization problem as
And solve using Fractional Euler Lagrange as
𝛻 𝛽,𝛼
𝜕𝐹(𝐱, 𝑃(𝐱), 𝛻 𝛼,𝛽
𝑃(𝐱))
𝜕 𝛻𝑃(𝐱)
=
𝜕𝐹(𝐱, 𝑃(𝐱), 𝛻 𝛼,𝛽
𝑃(𝐱))
𝜕𝑃(𝐱)

16. 1616
Forecast Accuracy in Wind (R12) & Solar
(R1) Forecast Plant Level
Absolute
Error
Margin
Probability (%)
Wind
Probability (%)
Solar
< 15% 93.36 +/- 5 98.69 +/- 2.5
15%-25% 4.37 +/- 5 1.27+/-2.5
25%-35% 1.16 +/- 5 0.04+/- 2.5
>35% 1.11 +/- 5 0

17. Thank you !
del2infinity Energy Consulting
contact@del2infinity.xyz