Delft University of Technology
ENergy-Based analysis and control of
the grid: dealing with uncertainty and mARKets
Tjerk Stegink
Mahya Adibi
Claudio De Persis
Dimitri Jeltsema
Nima Monshizadeh
Arjan van der Schaft
Jacquelien Scherpen
Real-time dynamic pricing
Introduction
I
nterfacing the growing supply of renewable energy sources to the
power grid presents several challenges demanding for appropriate control strategies that guarantee stable operation. Additionally, any
control strategy for future smart grids cannot disregard economical considerations that allow producers and consumers to fairly share utilities
and costs associated with the generation and consumption of energy.
In this research we develop a novel approach for the modeling, analysis
and control of smart grids based on energy functions which also lend
themselves to the integration of dynamic pricing algorithms that allow
to consider economical factors in the control of smart grids.
Model of the power network
Based on the gradient method applied to (4), we obtain the distributed
market dynamics
power generation:
power demand:
auxiliary variable:
electricity price:
generators:
inverters:
loads:
T
MG θ̈G + AG θ̇G = −DG Γsin(D θ) + uG
AI θ̇I = −DI Γsin(D T θ) + uI
ALθ̇L = −DLΓsin(D T θ) + PL
(1)
where θ = col(θG , θI , θL), D = blockcol(DG , DI , DL).
G
I
L
L
L
Figure 1: A power network consisting of generators, inverters and loads.
Control areas
Another model of the power system is obtained when dividing the grid
into large control areas, each having a power supply and demand:
(2)
M θ̈ + Aθ̇ = −DΓsin(D T θ) + u − PL
Optimal power dispatch
W
e consider two optimal power dispatch problems. The first one
focuses only on the power producers and the second one on both
producers as well as consumers.
Minimizing generation cost
Each generator and inverter is associated to a quadratic cost function.
X
X
Ci (ui ) =
qi ui2
min. generation cost:
min
supply-demand matching:
s.t.
i
X
(3)
i
ui =
i
X
PL,j
Extensions:
• Congestion (of e.g. the transmission lines or nodal power production)
• Power transmission costs
T
he classical complex phasor representation of sinusoidal voltages and
currents generalizes to arbitrary waveforms leading to the notion of
the time-varying power triangle. Using the Hilbert Transform H{·},
the analytical signal representation of the voltage u(t) and current i(t) is
formulated as
√ jα(t)
√ jβ(t)
u(t) = u(t) + jH{u(t)} = U(t) 2e
, i(t) = i(t) + jH{i(t)} = I (t) 2e
,
√
with j := −1, U(t) := |u(t)|, α(t) := arg{u(t)}, etc..
1
S(t) := u(t)i ∗(t) = P(t) + jQ(t),
2
1
1
with P(t) := 2 u(t)i(t) + H{u(t)}H{i(t)} and Q(t) := 2 H{u(t)}i(t) − u(t)H{i(t)} .
The instantaneous power then takes the form:
p(t) = u(t)i(t) = P(t) 1 + cos 2α(t) + Q(t)sin 2α(t) ,
where P(t) = U(t)I (t)cos ϕ(t) , Q(t) = U(t)I (t)sin ϕ(t) , and ϕ(t) := α(t) − β(t).
Budeanu's reactive and distortion power revised
Budeanu's reactive power can be related to energy oscillations, but only
in an average sense. Indeed, consider distorted waveforms:
X √
X √
u(t) =
Uk 2cos(kωt + αk ) , i(t) =
Ik 2cos(kωt + βk ).
k
k
Then, the active power PA and Budeanu's reactive power QB are
obtained from averaging P(t) and Q(t) over a period:
Z T
X
1
PA :=
P(t)dt =
Uk Ik cos(ϕk ),
T 0
k
Z T
X
1
QB :=
Q(t)dt =
Uk Ik sin(ϕk ),
T 0
k
with ϕk := αk −βk . Let DP (t) := P(t)−PA, DQ (t) := Q(t)−QB , IP (t) := I (t)cos ϕ(t) ,
and IQ (t) := I (t)sin ϕ(t) , then Budeanu's distortion power DB is decomposed into two parts:
( 2
2
2
2
D
:=
kUk
kI
k
−
P
P
PU
A,
2
2
2
2
2
2
DB := S − PA − QB = DPU + DQU , with
DQ2 U := kUk2kIQ k2 − QB2 ,
We associate U(PL) to the total utility of the power consumers.
X
X
max. social welfare:
max U(PL) − C (u) =
Ui (PL,i ) −
Ci (ui )
u
s.t.
i
X
ui =
X
i
PL,i
i
(4)
i
Distributed controllers
T
Definition of power under nonsinusoidal conditions
j
Social welfare problem
supply-demand matching:
(6)
The nonsinusoidal equivalent of the classical complex power is defined by
the time-varying complex power
I
u
−∇C (u) + λ − ω
∇U(PL) − λ + ω
−D T λ
Dv − u + PL
Time-varying complex power
G
control areas:
=
=
=
=
where τu , τL, τv , τλ > 0 are time-scales. Interconnecting (2) with (6) gives an
asymptotically stable closed-system w.r.t. the optimal points of (4)
in addition to zero frequency deviation.
C
onsider a simplified model of a power network containing three types
of elements, which are generators, inverters and frequency dependent loads:
τu u̇
τLṖL
τv v̇
τλλ̇
wo distributed controller designs are proposed to tackle each of the
optimal power dispatch problems.
where DPU and DQU are measures of fluctuation around the active power
and Budeanu's reactive power, respectively, relative to the voltage.
i(t)
P(t)
1Ω
G
Internal-model-based controller
Based on model (1) and optimization problem (3), consider the controller
X
δ̇i = −
aij (δi − δj ) − qi−1ωi
(5)
j
ui = qi−1δi .
Zero frequency deviation is achieved with an optimal power generation
identified by (3), despite the uncertainty of a constant unknown PL.
Extensions: A variant of (5) can also deal with a time-varying sinusoidal power demand which have known amplitude and frequency but
unknown offset and phase.
u(t)
2H
ωφ(t)
u(t)=10√2cos(t)+5√2cos(5t)
PA
D P (t)
time
Q(t)
S(t)
QB
DQ (t)
Q(t)
φ(t)
time
P(t)
Figure 2: A series RL circuit supplied by a nonsinusoidal voltage source.
Here ωϕ(t) = ϕ̇(t) denotes the instantaneous frequency.