Optimal Control of Electricity Production

Seminar on Optimal Control of Electricity Production
Seminar on
Optimal Control of Electricity Production
Kamrul Hasan
Supervised by
Professor Dr. Ralf Korn
Department of Mathematics, RPTU Kaiserslautern-Landau
July 07, 2023

Disclaimer
Disclaimer
The contents of the talk is based on
Göttlich S, Korn R and Lux K
Optimal control of electricity input given an uncertain demand
Mathematical Methods of Operations Research (2019)
90:301–328
All figures and tables are taken from this reference.

Introduction
Introduction
Why researchers worked actively on the modeling of energy prices?

Introduction
Introduction
Liberalization in Europe
Structural models

Introduction
Introduction
Structural models
Scheduling of electricity input and its distribution
Price decision has already been made by Provider.

Introduction
Introduction
Structural models
Actual electricity injection to satisfy the demand

Introduction
Introduction
Structural models
What are the two major challenges?

Introduction
Introduction
Structural models
What are the two major challenges?
Modeling of all ingredients and control the electricity input

Introduction

Introduction
What are the main contributions and pointed out?
A complete modeling setup

Introduction
A solved idealized stochastic control problem

Introduction
A solved idealized stochastic control problem
Pointed out some aspects for future research

The stochastic optimal control problem
Problem description
Electricity system

Problem description
Electricity system
Power inflow and actual customer’s demand location
Linear transport equation and conditions:
zt + λzx = 0, x ∈ (0, 1), t ∈ [0, T]
z(x, 0) = z0(x), z(0, t) = u(t)

Problem description
Electricity system
zt + λzx = 0, x ∈ (0, 1), t ∈ [0, T]
z(x, 0) = z0(x), z(0, t) = u(t)
Outflow should be adjusted to Yt.
u(t) = y(t + 1/λ)

Problem description
Electricity system
zt + λzx = 0, x ∈ (0, 1), t ∈ [0, T]
z(x, 0) = z0(x), z(0, t) = u(t)
Outflow should be adjusted to Yt.
u(t) = y(t + 1/λ)
min
u(t),t∈[0,T−1/λ],u∈L2
E
"Z T
1/λ
h(Ys, y(s))ds
#

Modeling of demand
Actual demand Yt at the end of the power line, i.e. at x = 1
Various indicators

Modeling of demand
Actual demand Yt at the end of the power line, i.e. at x = 1
Various indicators
Stochastic processes (Yt)t∈[0,T]

Ornstein- Uhlenbeck process (OUP)
Actual demand always fluctuates around the deterministic
process µ(t)

process µ(t)
Ornstein-Uhlenbeck process (OUP) is the natural candidate to
model the demand and the SDE follows
dYt = κ(µ(t) − Yt)dt + σdWt, Y0 = y0

process µ(t)
Ornstein-Uhlenbeck process (OUP) is the natural candidate to
model the demand and the SDE follows
dYt = κ(µ(t) − Yt)dt + σdWt, Y0 = y0
The SDE has an explicit solution given by,
Yt = y0e−κt
+ κ
Z t
0
µ(s)e−κ(t−s)
ds + σ
Z t
0
e−κ(t−s)
dWs

The special case of constant positive demand µt = µ > 0 and
Yt is normally distributed as follows,
Yt ∼ N

µ + (y0 − µ)e−κt
,
σ2
2κ
(1 − e−2κt
)

Figure: Influence of mean reversion speed κ and intensity of demand
fluctuations σ on the demand

From the figure , the demand process takes less time to return
to the µ if κ increases.

From the figure , the demand process takes less time to return
to the µ if κ increases.
κ is larger compared to σ2, the probability for the negative
value of Yt gets negligible.

Adding jump components

Brownian motion repalced by Levy process or can add a jump
martingle components

We obtain a jump diffusion process (JDP)version of the OUP
of the following form,
dYt = κ(µ(t) − Yt)dt + σdWt + γtdNt, Y0 = y0

We obtain a jump diffusion process (JDP)version of the OUP
of the following form,
The SDE has an explicit solution given by,
Yt = y0e−κt
+ κ
Z t
0
µ(s)e−κ(t−s)
ds + σ
Z t
0
e−κ(t−s)
dWs
+
Nt
X
i=1
γti e−k(t−ti)

Z t
0
γsdNs =
Nt
X
i=1
γti
The compensated Poisson integral is given by,
γt
g
dNt := γtdNt − νγ̄dt

Z t
0
γsdNs =
Nt
X
i=1
γti
The compensated Poisson integral is given by,
γt
g
dNt := γtdNt − νγ̄dt
The desired equivalent formulation of the jump diffusion
representation as,
dYt = κ(µ(t) − Yt +
γ̄ν
κ
)dt + σdWt + γt
g
dNt

Figure: Demand behavior in the presence of jumps for different mean
reversion speeds

The demand process returns faster to its mean reversion level.

The demand process returns faster to its mean reversion level.
The amplitude is lower.

Choice of the objective function

The objective function given by,
OF(Ys, y(s)) =
Z T
1
λ
E[(Ys − y(s))2
]ds

The objective function given by,
OF(Ys, y(s)) =
Z T
1
λ
E[(Ys − y(s))2
]ds
Contoller’s information about demand Ys, s ≤ t.
Optimal output y(t + 1
λ)

Stochastic optimal control (SOC) problem for JDP
The complete constrained stochastic optimal control (SOC)
problem as follows,
minu(t),t∈[0,T−1/λ],u∈L2 OF(Ys, y(s))

problem as follows,
zt + λzx = 0, x ∈ (0, 1), t ∈ [0, T]
z(x, 0) = z0(x), z(0, t) = u(t)

problem as follows,
zt + λzx = 0, x ∈ (0, 1), t ∈ [0, T]
z(x, 0) = z0(x), z(0, t) = u(t)
γt = 0 for obtaining an OUP- type demand

Optimal control strategies: different information levels
Figure: Updates algorithoms CM1-CM3 with transportation time 1
λ = 6

CM1 Setting without demand updates
Have to decide in advance about injected power
u(t) is assumed to be F0- predictable.
Transportation time is ∆t := 1/λ = 6

CM2 Setting with regular demand updates
Current demand can only be updated at prespecified points.
The forecasted value is adapted optimally with the updated
information.

CM2 Setting with regular demand updates
Current demand can only be updated at prespecified points.
The forecasted value is adapted optimally with the updated
information.
In the short run, upcomming demand can be higher or lower.
Quadratic deviation of CM2 Quadratic deviation of CM1

CM3 Idealized setting
The current demand information is available.
A time dealy because of instantaneouly updated information.

CM3 Idealized setting
The current demand information is available.
A time dealy because of instantaneouly updated information.
The smallest quadratic deviation

Proposition 3.1
Let(Ω,F,P)be a complete probability space, G be sub-σ-algebra of
F. Let further X, Z both be real-valued and square integrable
random variables on Ω, where in addition Z is G-measurable.
Then,the conditional expectation Ẑ := E(X|G) is the minimizer of
the mean-square distance from X,
msd(X, Z) := E((X − Z)2
)

Theorem 3.2
Let us assume the demand is a jump diffusion process. Then, given
a time homogenous jump height process with existing second
moment and γ̄ = E(γt). Then, the optimal control,

Theorem 3.2
Let us assume the demand is a jump diffusion process. Then, given
a time homogenous jump height process with existing second
moment and γ̄ = E(γt). Then, the optimal control,
1 for u(t) being F0 measurable in the CM1 as,
u∗
(t; t0) = e−κ(t+1/λ)
y0 + κ
Z t+1/λ
0
exp(−κ(t + 1/λ − s))
µ(s)ds +
γ̄ν
κ
(1 − e−κ(t+1/λ)
)

Theorem 3.2
2 for u(t) being Ft̂i
measurable in the CM2 as,
u∗
(t; t̂i) = eκ(t+1/λ−t̂i )
Yt̂i
+ κ
Z t+1/λ
t̂i
e−κ(t+1/λ−s)
µ(s)ds
+
γ̄ν
κ
(1 − e−κ(t+1/λ−t̂i )
)

Theorem 3.2
3 and for u(t) being Ft measurable in the CM3 as,
u∗
(t) = e−κ/λ
Yt + k
Z t+1/λ
t
e−κ(t+1/λ−s)
µ(s)ds
+
γ̄ν
κ
(1 − e−κ/λ
)

Relation between the different approaches
Theorem 3.3
The optimal control in the idealized setting CM3 is the limit of the
optimal control with updates at the discrete times t̂i = i∆tup if
the time between the updates ∆tup tends to zero,
lim
∆tup→0
u∗
(t) − u∗
(t; t̂i) = 0, P − a.s.

Numerical results for the SOC

Deterministic demand
Yt = 2 + sin(0.5πt)
0.5 ≤ t ≤ 5, ∆x = 0.5, λ = 2 and T = 5
Optimal control is close to the deterministic demand and
mean demand respectively.

Figure: Optimal control and available power in a deterministic and mild
stochastic demand setting

Stochastic demand
Slightly modify the control problem for the update setting

Stochastic demand
A partition of intervals to subintervals

Stochastic demand
A partition of intervals to subintervals
The following sequence of optimization problems determined
by the sub-intervals,
minu(t),t∈[t̂i,t̂i+1]
R min{t̂i+1+1/λ,T}
t̂i +1/λ
E[(Yt − y(t))2|Ft̂i
]dt
zt + λzx = 0, z(0, t) = u(t),
z(x, t̂i) = zold(x, t̂i), x ∈ (0, 1), t ∈ [t̂i , min{t̂i+1 + 1/λ, T}]

Parameter PS1 PS2 PS3
Transport velocity λ 4 PS1 PS1
Time horizon T 1 PS1 PS1
Mean demand level µ(t) 2 + 3. sin(2πt) PS1 PS1
Speed of mean reversion κ 1 3 3
Intensity of demand fluc-
tuations
σ 2 PS1 PS1
Initial demand y0 1 PS1 PS1
Jump height γt = γ 0 PS1 1
Jump intensity ν 5 PS1 PS1
Table: Parameter setting

OUP- type demand : Parameter setting PS1 and PS2
Figure: Available power in x = 1, mean realization and confidence levels
of demand
Less speed and more concentrated around the mean

Figure: Available power in x = 1, mean realization and confidence levels
of demand
Less speed and more concentrated around the mean
Difficult of demand forecasts
More fluctations around the latter

Figure: Numerical results for PS1 based on the update algorithm

Figure: Numerical results for PS1 based on the update algorithm
The output of CM2 outperforms than CM1

Cumulative root mean squared error (cumRMSE)
cumRMSE(y(t)) :=
R T
∆t
p
E[(Yt − y(t))2]

cumRMSE(y(t)) :=
R T
∆t
p
E[(Yt − y(t))2]
The numerical cumRMSE of CM1 and CM2 can be found in
the Table,
CM1 CM2 relative reduction
PS1 0.9325 0.7526 19.29%
PS2 0.6434 0.6002 6.71%
Table: Comparison of cumRMSEs with and without updates for
PS1 and PS2

cumRMSE(y(t)) :=
R T
∆t
p
E[(Yt − y(t))2]
The numerical cumRMSE of CM1 and CM2 can be found in
the Table,
CM1 CM2 relative reduction
PS1 0.9325 0.7526 19.29%
PS2 0.6434 0.6002 6.71%
Table: Comparison of cumRMSEs with and without updates for
PS1 and PS2
CM2 is considered to be better.
The relative reduction is more pronounced for lower κ.

JDP-type demand: Parameter setting PS3
Figure: Numerical results for PS3 without updates

Figure: Numerical results for PS3 without updates
Upward trend of the JDP-type demand because of γ = 1
Non-zero jump height leads to increase the amplitude of the
C.I.

Figure: Numerical results for PS3 based on update algorithm

Figure: Numerical results for PS3 based on update algorithm
Updates help to enhance the performance
Better capture the upward trend due to fixed positive γ

Theoretical and numerical point of view
Time instances between updates cumRMSE
41 0.3885
5 0.1924
3 0.1278
2 0.0814
1 2.6701 e-06
Table: Convergence of numerical solution based on CM2 against
numerical implementation of theoretical solution for CM3

Theoretical and numerical point of view
Time instances between updates cumRMSE
41 0.3885
5 0.1924
3 0.1278
2 0.0814
1 2.6701 e-06
Table: Convergence of numerical solution based on CM2 against
numerical implementation of theoretical solution for CM3
Cumulative RMSE decreases with decreasing ∆tup
Update setting converges to the idealized setting

Conclusion
What we have learned today?

Conclusion
OUP and JDP version of the OUP

Conclusion
Three control methods

Conclusion
Demand process is observed by using different values of
parameters.

Conclusion
parameters.
The amplitude of the Confidence Interval

Conclusion
parameters.
The amplitude of the Confidence Interval
Update setting is observed by theoretical and numerical point
of view.

References
Reference
Göttlich S, Korn R and Lux K
Optimal control of electricity input given an uncertain demand
Mathematical Methods of Operations Research (2019)
90:301–328

End
Thank You!

Optimal Control of Electricity Production

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