Inverter Set-point Computation for Voltage Regulation
James Ranjith Kumar Rajasekaran
October 5, 2021
Let e and f be the vectors of real and imaginary parts of voltage phasors taken at all buses given by set Ω. With
this definition, the govening equation for the control vector u and disturbance vector w can be written as
t t−1 e
e
=
+ Bu + Dw
(1)
ft
f t−1
where
u=
d=
∆pj
∆qj
∆pj
∆qj
,
j∈C
(2)
,
j∈N
(3)
The disturbance vector w is assumed to be available from forecast and hence without control action, the real and
imaginary parts of voltage can be written as
t t−1 ê
e
=
+ Dw
(4)
f t−1
f̂ t
The primary objective is to find the control action u such that vmin ≥ vt ≥ vmax where vjt =
governing equation for u is given as
t t ê
e
=
+ Bu
ft
f̂ t
q
etj
2
2
+ fjt . The
(5)
2
2 2
2
2
+ f̂jt ≥ vmax
. It is assigned that vjt = vmax ∀j ∈ O. Let U = j êtj + f̂jt ≤ vmin
.
r
2 2
êtj + f̂jt ∀j ∈
It is assigned that vjt = vmin ∀j ∈ U. For the remaining nodes, it is assigned that vjt =
Let O =
j êtj
2
Ω \ {O ∪ U }.
2
Let (vt ) = g (et , f t ) where
2
2 · diag (et )
2 · diag (f t )
g et , f t = et
2
+ ft
(6)
Its Jacobian can be written as
∇g(et ,f t ) =
(7)
2
With the first order approximation of Taylor series, the expression for (vt ) can be linearised as follows:
t
e − êt
t t
t t
T
g e , f ≈ g ê , f̂ + ∇g(êt ,f̂ t )
f t − f̂ t
i et − êt 2
2 t 2 h
t
t
t
v
≈ ê
+ f̂
+ 2 · diag (ê ) 2 · diag f̂ t
f t − f̂ t
h
i
2
2
vt
− v̂t
≈ 2 · diag (êt ) 2 · diag f̂ t
Bu
1
(8)
(9)
(10)
Figure 1: 37 Bus System
Figure 2: Voltage magnitudes across all nodes before and after control action
To find the required control action, (10) can be written in standard form as Au = b where
h
i
A = 2 · diag (êt ) 2 · diag f̂ t
B
2
2
b = vt
− v̂t
(11)
(12)
With this definition, the cotrol action can be written as
u = AT A
−1
AT b
(13)
This control strategy is tested on 37 bus distribution network as shown in Fig. 1 where PV inverters are present
in buses 4, 15 and 35. It is considered that the PV inverters are operating MPPT mode such that the power injection
changes with the solar irradiance. To regulate the voltage, VAr compensators are available at buses 4, 15 and 35.
The set points of these control devices are calculated using (13) and the voltage magnitudes across all nodes before
and after control action are plotted in Fig. 2. The voltage magnitudes seems to regulated to the permitted limit
except a few nodes. It could be due to the propagation of error that caused due to linearisation of voltage magnitude
equations and power balance model.
2