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1
Voltage Regulation in Distribution Networks in the
Presence of Distributed Generation: A Voltage
Set-point Reconfiguration Approach
Alessandro Casavola, Member, IEEE, Giuseppe Franz`e, Member, IEEE, Daniele Menniti, Member, IEEE,
and Nicola Sorrentino, Member, IEEE,
Abstract—In this paper a control strategy based on predictive
control ideas is proposed to reconfigure on-line the on-load tap
changer (OLTC) voltage set-points in electrical Medium Voltage
(MV) power grids in the presence of Distributed Generation
(DG). The idea is that an active management of the set-
points can be effective for maintaining relevant system variables
within prescribed operative constraints in response to unexpected
adverse conditions, e.g. changing loads or generation failures.
The voltage set-points reconfiguration problem is formulated
as a constrained optimization problem by imposing that the
voltages at certain nodes have, compatibly with all prescribed
constraints and changed conditions, minimal deviations from
their nominal values. Simulation results show that the proposed
approach ensures, under certain conditions, feasible evolutions
to the overall network whenever critical events occur.
Index Terms—Predictive control, nonlinear feedback,
quadratic programming, distribution networks, distributed
generation, voltage regulation.
I. INTRODUCTION
In recent years major technological changes and restruc-
turing processes have begun to happen in power industry
due to deregulation. This will involve an increasing transition
from highly passive transmission grids towards more active
management of them, facilitating bidirectional flows of energy
and the introduction of small-scale DG units, even located at
the end-user sites.
This new scenario makes realistic the undertaken of research
initiatives aimed at investigating innovative wide-area control
and coordination strategies for ensuring safe and economical
energy dispatch under the changed regulatory rules which have
completely altered the normal practices [1]. As a consequence,
researches on DG have attracted an increasing interest and
significant advances have been achieved in recent years [2]-
[17].
Several aspects of DG have been identified and discussed
in depth in recent literature. Among these, the overvolt-
age/undervoltage problem at different nodes due to the in-
corporation of DG in the distribution network is of special
interest here and it is known to require special attention [2].
Distribution systems are generally designed to operate radi-
ally without any generation on the distribution lines or at the
customer sites. The introduction of DG units can significantly
Alessandro Casavola, Giuseppe Franz`e, Daniele Menniti and Nicola Sor-
rentino are with the Dipartimento di Elettronica, Informatica e Sistemistica
dell’Universit`a degli Studi della Calabria, Rende (CS), GA, 87036 ITALY
e-mail: {casavola,franze,menniti,sorrentino}@deis.unical.it.
affect the power flows and voltages at the customers and utility
equipments. If significant DG power is introduced, this may
deteriorate the effectiveness of the standard voltage regulation
systems in charge of handling load variations and adverse
events.
In this paper the attention is focused on the development
of a control strategy for determining the tap position of the
On-load tap changer (OLTC) under the DG presence in order
to maintain tolerable voltage values at each load connection
node.
It is worth to underline that, for an adequate analysis of
the voltage and reactive power control of distribution systems,
the equipment has to include, besides OLTCS, other control
devices (e.g. shunt capacitors, controllable DG units) and
issues as load estimation and references coordination have to
be properly addressed (see for a detailed and up-to-date survey
[18]). In this sense, this work could be seen as a starting
point because a single aspect (OLTC regulation problem) is
treated, but potentially it could open the door to more general
extensions.
OLTCs are widespread in distribution networks and are
likely to remain in service for many years to come [7].
Actually, control strategies are not able to manage a high pene-
tration of DG plants, because they are designed and optimized
on the strong assumption of unidirectional power flows from
HV to MV networks and from the Power System to the ends of
the distribution feeders. The Line Drop Compensation (LDC)
belongs to this class of regulation approaches. It imposes
a constant voltage at some remote Load Center (LC) and
determines the voltage at the HV/MV substation busbar that
compensates the voltage drops along the distribution lines [6]-
[7]. The use of a constant voltage set-point does not allow
to manage active feeders, because DG power may involve
a voltage profile which increases from the substation to the
outmost bus-bars, whereas standard compensation methods,
such as LDC, can fail when the HV/MV power flow partially
or completely reverses in consequence of DG. In recent years,
some approaches have been proposed with the primary aim of
improving the poor OLTC performance under the presence
of DG, see [5],[8],[9],[10],[18] and references therein. In
particular, [18] offers a detailed survey on voltage regulation
problems in distribution systems in presence of DG units.
There, it is analyzed the need of a voltage set-point recon-
figuration for contrasting power flow fluctuations and voltage
drops due to DG actions.
2
Here, the problem of OLTC control will be rigourously
addressed via constrained control theory. The goal is to
determine an OLTC reference voltage under changed load
conditions such that the MV bus voltages track as close as
possible the nominal values compatibly with constraints on the
MV customer bus voltages, which must be satisfied despite of
any fluctuation in active and reactive powers.
A Command Governor (CG) approach based on the con-
ceptual tools of predictive control methodologies will be used
for solving the problem. The CG methodology consists of
adding to a primal compensated system a nonlinear device
whose action is based on the actual reference, current state and
prescribed constraints. The aim of the CG is that of modifying,
whenever necessary, the reference in such a way that the
constraints are enforced and the primal compensated system
maintains its linear behavior. The CG action is computed on-
line by solving, at each time instant, a constrained quadratic
programming (QP) problem that usually requires low com-
putational efforts. Under linear constraints, any standard QP
solver can be used for such a task and the CG algorithm can be
implemented on any up-to-date low-cost programmable micro-
controller. Methodological studies along these lines have ap-
peared in [19], [31], [27],[32], while for CGs approached from
different perspectives see [19].
The paper is organized as follows. In Section II, the distri-
bution network model is given and the problem formulated. In
Section III, the CG scheme is discussed and its relevant prop-
erties summarized. Computer simulations are finally presented
in Section V and some conclusions end the paper.
II. DISTRIBUTION SYSTEM MODELLING
HV
T
V1V2VN1
DG1DG2DGN1
L11L12L1N1
Load 1 Load 2
VMV
~ ~ ~
Load N1
V1V2VN2
DG1DG2DGN2
L21L22L2N2
Load 1 Load 2
~ ~ ~
Load N2
V1V2VNl
DG1DG2DGNl
Ll1Ll2LlNl
Load 1 Load 2
~ ~ ~
Load Nl
Fig. 1. Distribution system network
A general description of a radial distribution network with l
distribution lines is depicted in Fig. 1. A HV/MV transformer
OLTC supplies the feeders to which small size generators
and/or loads are connected. Both loads and generators, that
make use of renewable sources (wind sun, hydro, etc), are
characterized by an high degree of uncertainty because they
follow the customer behavior and the primary source availabil-
ity. Grid generation phenomena can be modelled by means
~EMV (t)
IT1(t)
E1(t)
ITN(t)
EN(t)
Distribution Network
Fig. 2. Equivalent circuit
of a single synchronous machine directly connected to the
distribution network. The latter represents the equivalent of
one o more generators connected to the same node. The
voltage at the secondary side of HV/MV transformer with
OLTC can be modelled by an ideal voltage source ( ¯
EMV )
whose amplitude can be controlled by OLTC tap position.
As well known, the OLTC tap position is computed at each
control session twith a time period of several minutes. By
assuming the loads balanced, symmetrical generation and
a linear network behavior, one can derive the single-phase
equivalent circuit depicted in Fig. 2.
Let N=N1+...+Nlbe the network bus voltages. First of
all recall the following definitions
˙
ZLi=RLi+j XLi, i = 1,...,N,
˙
ZCi=RCi+j XCi, i = 1,...,N,
¯
Ei(t) = EDi(t) + j EIi(t), i = 1,...,N,
¯
Ii(t) = IDi(t) + j IIi(t), i = 1,...,N,
¯
EMV (t) = EMV D (t),
(1)
where ˙
ZLiare the line impedance, ˙
ZCiare the load
impedances, ¯
Ei(t)the bus voltages of Ci,¯
Ii(t)the absorbed
currents by the loads and ¯
EMV (t)denotes the HV/MV trans-
former secondary voltage.
In the sequel we assume that the distribution system evolves
in linear regimes. Such choice is motivated by two reasons.
First, the distribution network essentially has a quasi-linear
behavior as it results evident by considering the sampling
period. Then, the nonlinear load behaviors will be taken into
account via the disturbances affecting the load currents (see
(5)). Following the same reasoning, also DG units will be
treated as linear components and are represented as current
generators, namely
¯
Ji(t) = JDi(t) + j JIi(t), i = 1,...,N.
Hence, by taking into account the single phase equivalent
circuit of Fig. 2, ¯
IT i(t)are the current drawn in the buses
at each i−th node, defined as
¯
IT i(t) := ¯
Ii(t)−¯
Ji(t), i = 1,...,N. (2)
3
and, at each time session by resorting to the Kirchhoff’s laws, a
matrix description of the distribution network can be achieved
2
6
6
6
6
6
4
¯
EMV (t)
¯
E1(t+ 1)
¯
E2(t+ 1)
.
.
.¯
EN(t+ 1)
3
7
7
7
7
7
5
=2
6
4
˙
Z11 ... ˙
Z1,N+1
.
.
.....
.
.
˙
ZN+1,1... ˙
ZN+1,N+1
3
7
5
2
6
6
6
6
6
4
¯
IMV (t+ 1)
¯
IT1(t+ 1)
¯
IT2(t+ 1)
.
.
.¯
ITN(t+ 1)
3
7
7
7
7
7
5
(3)
where ¯
IMV (t)denotes the HV/MV transformer current and
˙
Zii =ϕ(˙
ZLi)(see [13] for details) are the entries of the
impedance matrix.
Finally, the load characteristic equations lead to
−¯
Ei(t+ 1) + ¯
ZCiIi(t+ 1) = 0, i = 1, . . . , N (4)
We shall, now, consider the disturbances acting on the system.
The idea here exploited is to take care of both the load changes
and injected active and injected/absorbed reactive powers due
to the presence of the DG units as disturbances affecting the
distribution system. It is well-known that the active/reactive
powers DG injection alters in a nonlinear fashion the loads
behavior. In order to take into account of such an aspect we
have forced the load currents to vary within a wide range.
In consequence of above considerations, the following dis-
turbances have been considered
•Module of the absorbed current from a load - The module
of the current absorbed from a load can vary within 25%
of the nominal value, i.e.
δIi=±0.25Ii, i = 1,...,N (5)
where
-δIirepresents the disturbance on the module of
absorbed current of the i-th load;
•Load power factor cos φCi-It has to belong to the
following range:
0.8≤cos φCi≤1, i = 1,...,N. (6)
Therefore, from equations (3)-(4) and definitions (1)-(2), the
following discrete-time state space description can be derived
through straightforward algebraic manipulations
xp(t+ 1) = G u(t) + GIδI(t) + GJJ(t)
y(t) = H xp(t)(7)
where
xp(t) = [ED1(t)ID1(t)ED2(t)ID2(t). . . EDN(t)IDN(t)
EI1(t)II1(t)EI2(t)II2(t)...EIN(t)IIN(t)]T
u(t) = EMV (t),
δI(t) = hδID1(t)δII1(t)δID2(t)δII2(t). . . δIDN(t)δIIN(t)iT
J(t) = [JD1(t)JI1(t)JD2(t)JI2(t)...,JDN(t)JIN(t)]T
and
y(t) = [ED1(t)ED2(t)...EDN(t)EI1(t)EI2(t)...,EIN(t)]
Therefore, the system model (7) will be rewritten according
to the following extra notations d(t) = δI(t)
J(t), Gd=
[GIGJ].
Remark - It is worth to remark that the DG action has been
considered as a disturbance, therefore the only information
we need is its amplitude depending on the available active/
reactive powers. 2
The system described by (7) is subject to the following set
of constraints:
1) Load voltages ¯
Ei(t), i = 1,2,...,N cannot be imposed
equal to the reference value En,because the voltage
drops of the corresponding distribution lines are different
each other. Therefore the following constraints must be
fulfilled at each time instant:
(1 −α)E2
n≤ | ¯
Ei(t)|2≤(1 + α)E2
n, i = 1,...,N, ∀t
(8)
Deviations of |¯
Ei(t)|i= 1,...,N within ±10%(α∈
(0 0.1]) tolerance from their nominal values Enare
reasonable in practice.
2) As it is well-known, the OLTC alters the power trans-
former turns ratio in a number of predefined steps and
in that way it changes the secondary side voltage. Each
step usually represents a change of 0.25 ÷1.25% in LV
side. Because the switching step size is small w.r.t. the
load tolerance (see (8)), one can use a continuous ap-
proximation of the OLTC dynamics [13]. Since standard
tap changers approximately offer a change of 10% of the
rated voltage, the value of ¯
EMV (t)can vary in percent
with respect to the nominal voltage En,as follows
En≤ | ¯
EMV (t)| ≤ (1 + β)En,∀t, β ∈(0 0.1]; (9)
Hence the following problem can be stated:
DG Voltage Regulation Problem (DGVRP) - At each control
session t, determine a voltage set-point such that:
•the load voltages ¯
Ei(t), i = 1,...,N satisfy the con-
straints (8);
•the HV/MV transformer secondary voltage ¯
EMV (t)sat-
isfies inequality (9).
The DGVRP will be recast as a constrained convex optimiza-
tion problem and solved by means of the CG strategy [31]. In
the next section, the main features of such an approach will
be recalled.
III. COMMAND GOVERNOR PROBLEM FORMULATION
A CG control scheme, with plant, primal controller and CG,
is depicted in Fig. 3.
--
-
CG --
-
66
Plant
Primal
controller
χp(t)χc(t)•
•
x(t) = »χc(t)
χp(t)–
r(t)g(t)u(t)y(t)≈r(t)
c(t)∈ C
d(t)∈ D
?
Fig. 3. Command Governor structure
4
Consider the following state-space description of the closed-
loop plant regulated by the primal controller
x(t+ 1) = Φx(t) + Gg(t) + Gdd(t)
y(t) = Hyx(t)
c(t) = Hcx(t) + Lg(t) + Ldd(t)
(10)
where: t∈ZZ+,x(t)∈IRnis the overall state which includes
the plant and compensator states (if a dynamic compensator is
used); g(t)∈IRm, which would be typically g(t) = r(t)if no
CG were present, is the CG action, viz. a suitably modified
version of the reference signal r(t)∈IRm;d(t)∈IRndan
exogenous disturbance satisfying d(t)∈ D,∀t∈ZZ+, with
Da specified convex and compact set such that 0nd∈ D;
y(t)∈IRmis the output, viz. a performance related signal
which is required to track r(t);c(t)∈IRncthe vector to be
constrained
c(t)∈ C,∀t∈ZZ+(11)
with Ca specified convex and compact set. It is also assumed
that
(A1)
1) Φ is a stability matrix, i.e. all eigenvalues
are in the open unit disk;
2) System (10) is offset-free, i.e.
H(In−Φ)−1G=Ip
One important instance of (10) satisfying (A1) consists of
linear plants under stabilizing feedback control. The offset-
free property happens to be satisfied under a suitable feedback
configuration which provides infinity dc loop gain.
Observe that the typical structure of a CG-equipped control
system consists of two nested loops. The internal loop is
designed via a whatever classic linear control method, usually
without considering the prescribed constraints, and allows the
designer to specify relevant system properties, e.g. stability,
disturbance rejection, etc, for small-signal regimes when the
constraints are supposedly not violated. The outer loop consists
of the CG unit. The latter, whenever necessary (large-signal
regimes), modifies the reference to be applied to the closed-
loop system so as to avoid constraints violation. The basic
idea is that of maintaining the closed-loop system within its
nominal linear regimes, where stability and all other closed-
loop properties are preserved.
The CG design problem is that of generating, at each time
instant t, the set-point g(t)as a function of the current state
x(t)and reference r(t)
g(t) := g(x(t), r(t)) (12)
in such a way that, under suitable conditions and regardless
of disturbances, the constraints (11) are always fulfilled along
the system trajectories generated by the application of the
modified set-points g(t)and possibly y(t)≈r(t).Moreover,
it is required that: 1) g(t)→ˆrwhenever r(t)→r, where ˆr
is either ror its best feasible approximation; and 2) the CG
has a finite settling time, viz. g(t) = ˆrfor a possibly large
but finite twhenever the reference stays constant after a finite
time.
A system equipped with a CG takes a special simplified
structure at the cost typically of performance degradation
with respect to more general approaches, e.g. constrained
Model Predictive Control (MPC). Both approaches operate
in accordance to the receding horizon strategy by exploit-
ing future system predictions, but there are some relevant
differences. In the MPC framework, the control actions are
computed by optimizing the stabilization/tracking properties
while ensuring constraints fulfillment. Differently, the CG
approach introduces a separation between these two require-
ments: stabilization is provided by the primal control and the
CG unit is only finalized to maintain the system in its linear
regimes by avoiding constraints violation. CG usage can be
however justified in large-scale industrial applications wherein
a massive amount of flops per sampling time is not allowed,
and/or one is typically only commissioned to add to existing
standard PID-like compensators peripheral units which, as
CG’s, do not change the primal compensated control system.
Specific merits of the CG approach in dealing with con-
straints are that it can handle absolute and incremental con-
straints on input and state-related variables of the plant and
that the numerical burden of the required on-line computa-
tions can be tempered according to the available computing
power, ranging from solving on-line convex multidimensional
optimization problems to consulting look-up tables. In this
respect, notice also that different sampling rates can be used
in the inner and outer loops of the CG structure if this is
advantageous from a computational point of view.
Theoretical studies along these lines appeared in [19]-[30].
In particular, CG schemes dealing with disturbances were
considered in [27], [26], with model uncertainties in [22], [27]
and with partial state information in [28]. For specific results
on CG applied to nonlinear systems see e.g. [21], [24], [25],
[29], [30].
For completeness, a brief description of the CG design
problem and its main properties is reported in the Appendix.
IV. PRIMAL CONTROLLER DESIGN AND CG CONTROL
SCHEME
The first step in the CG design consists in the synthesis of
a primal controller in such a way that the primal-controlled
distribution system is asymptotically stable and satisfies some
desirable control specifications for small-signal regimes. To
this end, let ¯ej(t) = ¯
Eri(t)−¯
Ei(t), i = 1,...,N be the
phasor representing the tracking voltage error on the i-th load
and define their real and imaginary parts as
eDi(t) = ErDi(t)−EDi(t),
eIi(t) = ErIi(t)−EIi(t), i = 1,...,N. (13)
The primal controller will be designed with respect
to the feedback control scheme of Fig. 4 where
¯
Er(t) := [ ¯
Er1(t),¯
Er2(t),..., ¯
ErN(t)] is a vector of
admissible reference values for the load voltage phasors
whose module equals g(t),and ¯
EDI (t) := [ED1(t),
EI1(t), ED2(t), EI2(t),...,EDN(t), EIN(t)].The latter is
provided by the CG by suitably modifying the nominal voltage
Enin order to enforce all the prescribed constraints (see Fig.
5). Specifically, at each time instant, the FR module generates
the couple of the reference phasors ¯
Eri(t), i = 1,...,N
5
Distribution
System
PI
Controller
EG
FR y(t)
g(t)
d(t)
EMV(t)
Er(t) e(t)
ED I
(t)
Fig. 4. Primal feedback control scheme
with amplitude g(t)and phase φias follows
¯
Eri(t) = g(t)ej φi(t), i = 1,...,N, (14)
where the phase displacements φi(t), i = 1, . . . , N are those
of the load voltages ¯
Ei(t), i = 1,...,N. This choice is
arbitrary but justified in practice because the reference and
actual voltage phasors are usually aligned under normal con-
ditions. Finally, the EG error generator device computes the
error ¯e(t) := [eD1(t)eI1(t)eD2(t)eI2(t)...eDN(t)eIN(t)]T
as indicated in (13).
Because one of the requirements is to compensate the effect
of constant disturbances, the following primal feedback PI
control law has been considered
¯
EMV (t) = ¯
EMV (t−1) +
N
X
i=1
[kiki]»eDi(t)
eIi(t)–(15)
Finally, when a CG device is inserted in the previously
described setup, the complete control scheme becomes that
depicted in Fig. 5.
Primalcontrolled
Distribution System
CG FR EG
x(t)
En
ED I (t)
d(t)
e(t)
Er(t)
g(t)
c(t)
y(t)
Fig. 5. CG primal-controlled distribution system control scheme
V. SIMULATIONS
HV/MV
T
V1V2V3
V4V5V6
DG1DG2DG3
L11L12L13
L21L22L23
C
VMV
~ ~ ~
1C2C3
C4C5C6
Fig. 6. MV italian distribution network
Let us consider in Fig. 6 the radial power system network
obtained by suitably simplifying a typical part of the MV
italian distribution network fully described in [14]. In this
example, for the sake of simplicity, all variables will be
considered in pu units. The OLTC tap position is computed
at each control session twith a time period of 15 minutes.
The nominal set-point of the load voltage modules has been
selected as En= 1 pu and we have imposed β= 0.04 in (9).
Therefore at each time session the following constraint must
be fulfilled
1≤ | ¯
EMV (t)| ≤ 1.04,∀t. (16)
Moreover, the load voltage constraints (8) on |¯
Ei(t)|(quadratic
constraints) have been (conservatively) replaced by constraints
on the real parts EDi(linear constraints). The involved approx-
imations are usually acceptable because in typical distribution
networks the line reactances XLiand the line resistances
RLihave negligible values w.r.t. the load voltage modules
|¯
Ei(t)|.In fact, because the phase displacement of the phasor
¯
Ei(t)is almost zero [13], the imaginary part EIi(t)of ¯
Ei(t)
can be approximate as EDi(t)∼
=10 EIi(t).Then, it can be
assumed w.l.o.g. that ¯
Ei(t)∼
=EDi(t), i = 1,...,6,∀tand
the constraint (8) can be replaced by
(1 −α)En≤EDi(t)≤(1 + α)En,
α∈(0 0.1], i = 1,...,6,∀t.
Here, an admissible variation in percent (α= 0.04) has been
chosen.
A primal controller has been selected along the lines de-
scribed in the previous Section IV. For the power network
here considered, it results that parameters k1∈[0,0.5],
k2∈[0,0.5] k3= 0.01, k4=k1, k5=k2and k6=k3ensure
that the compensated system is asymptotically stable and the
tracking error specification (8) is satisfied with α= 0.1. In
particular, the linear compensator (15), with k1=k4= 0.01,
k2=k5= 0.305 and k3=k6= 0.01,was selected so as to
obtain a satisfactory transient response.
In the simulations, we have supposed that the load power
factors are held constant at each time instant to cos φCi=
0.9, i = 1,...,6.Moreover, it has been assumed that the GD
units on the distribution line L1have the following properties:
•GD1injects active power: PGD1= 0.1;
•GD2injects active power: PGD2= 0.1;
•GD3injects active power: PGD3= 0.15 and absorbs
reactive power: QGD3= 0.0726,
at a constant power factor cos φGD = 0.9.The disturbance
δI(t)on the loads Ci, i = 1,...,6has been generated as an
uniformly distributed sequence of random values satisfying the
bounds defined in (5).
Finally, it has been assumed that the distribution system is
equipped with a 16 tap positions OLTC with a voltage range
of variation ±0.25% within the voltage constraint (16).
A. No Faulty case
The control objective in these simulations consists of impos-
ing that at each time session the load voltages ¯
Ei, i = 1,...,6
deviate less than ±4% w.r.t. their nominal value Endespite
of any possibly occurring disturbance sequence d(t).
Fig. 7 shows the behavior of the compensated linear system
without the action of the CG unit. It is worth to note that
6
the constraints on the first feeder L1are fulfilled thanks to
the effect of the DG units on the load variations, while on
the distribution line L2,where the DG is absent, voltage
constraints are violated (see Fig. 7). The latter is due to the
fact that ¯
EMV reduces the voltage at the secondary substation
busbar and this pull down all the voltages on the feeder [16].
Moreover, Fig. 7 shows also the behavior of the bus voltage
¯
EMV (t)due to the action of the OLTC.
6 12 18 24
0.95
1
1.05
ED4
6 12 18 24
0.9
0.95
1
1.05
1.1
ED5
Time (Hours)
6 12 18 24
0.9
0.95
1
1.05
ED6
Time (Hours)
2 4 6 8 10 12 14 16 18 20 22 24
0.99
0.995
1
1.005
1.01
1.015
1.02
1.025
1.03
1,035
1,04
Time (Hours)
EMW
Fig. 7. Constraints without CG: (Up) Loads C4, C5, C6on the MV
distribution line L2(Down) MV Bus voltage
Fig. 8 depicts the system evolutions resulting when the CG
device is used with δ= 10−6,and Ψ = 1.Under these
choices, the constraint horizon k0= 25 has been computed
via the method of [31].
As a result, the previous undesirable behavior is avoided and
all variables are always constrained inside their bounds. This
is obtained by the CG action by modifying the nominal bus
voltage set-point En= 1 into its best feasible approximation
g(t), depicted in Fig. 9. It is important to note that the CG
device modifies the bus voltage set-point when it recognizes
that a constraint violation will occur. Moreover it has to
be remarked that the maximum value does not exceed the
saturation constraint (16).
Fig. 10 depicts the tap position switching signals T P (t)
6 12 18 24
0.95
1
1.05
ED4
6 12 18 24
0.95
1
1.05
ED5
Time (Hours)
6 12 18 24
0.95
1
1.05
ED6
Time (Hours)
2 4 6 8 10 12 14 16 18 20 22 24
0.99
0.995
1
1.005
1.01
1.015
1.02
1.025
1.03
1.035
1.04
Time (Hours)
EMW
Fig. 8. Constraints with CG: (Up) Loads C4, C5, C6on the MV distribution
line L2(Down) MV Bus voltage
with and without the CG action, i.e. the number of tap position
changes at each hour. It is worth to point out that, even if
the CG strategy imposes an higher number of tap positions
switchings, the constraints are never violated (see Figs. 7-8)
which is the main aim of the strategy. However, a reduction
in the number of switchings could be achieved by suitably
modifying the CG index as indicated in (29).
Finally in order to take care of unavoidable errors/uncertainties
in the process of measure/estimation of the distribution system
parameters, a robustness test has been performed. In particular,
we have verified that the CG strategy does work in presence of
a3% perturbation w.r.t the line impedance ˙
ZLnominal values
reported in [14]. In the case which larger inaccuracies arise,
a robust CG approach can be used by resorting to a polytopic
description of the uncertainty as described in [27].
B. Faulty case
The previous situation is considered here with in addition
the following faulty scenario:
A voltage drop of 5% lasting 15 minutes occurs on the HV
network at t= 12 h. Such an undesirable phenomenon could
be caused by several reasons, e.g. heavily load conditions
7
2 4 6 8 10 12 14 16 18 20 22 24
1
1.005
1.01
1.015
1.02
1.025
1.03
1.035
1.04
Time (Hours)
g(t)
Fig. 9. Output of CG unit
1 2 4 6 8 10 12 14 16 18 20 22 24
0
1
2
3
4
Time (hours)
TP(t)
1 2 4 6 8 10 12 14 16 18 20 22 24
0
1
2
3
4
Time (hours)
TP(t)
Fig. 10. Number of tap position switchings per hour: (Up) without CG
(Down) with CG
or transmission line failure. This implies that the network
distribution lines are supplied at only 95% of the voltage
(En) w.r.t. the no faulty condition.
For the sake of brevity, we will show in the following
only the fault effect on the first loads (C1and C4) of both
feeders. This choice is motivated by the fact that the impact
of adverse phenomena mainly affect the loads at the end of
the distribution lines.
Figs. 11 shows the dynamics of the compensated linear
system without the action of the CG unit. Under the proposed
scenario, violations of the prescribed actuator and load voltage
constraints result for both feeders. In fact, during the faults
time interval, starting at time t= 12 h, the voltage constraint
(9) is active and, as a consequence, all voltages on the feeders
are pulled down.
On the contrary, in Fig. 12, where the responses of the
system under the CG action are reported, no constraints
violation is observed. This means that the viability of the
system during the pre-post fault transient is ensured despite
of the presence of the three DG units on the feeder L1and
the fault occurrence.
This has been achieved by modifying the OLTC reference
voltage from its nominal value in the one reported in Fig. 13.
Here, it is remarkable to point out that the CG output g(t)
is further modified w.r.t. the no-faulty case. In fact when the
fault event occurs (grey zone) the bus voltage reference is
suitably adjusted for dealing with the new operating scenario:
essentially, the CG reconfigures the OLTC reference voltage.
From the simulations, the ability of CG to manage the fault
occurrence or other adverse conditions is evident. It is worth
commenting that the CG is not informed of the fault occur-
rence and its behavior hinges on its intrinsic reconfiguration
capability.
2 4 6 8 10 12 14 16 18 20 22 24
0.95
1
1.05
ED1
2 4 6 8 10 12 14 16 18 20 22 24
0.95
1
1.05
Time (Hours)
ED4
2 4 6 8 10 12 14 16 18 20 22 24
0.94
0.96
0.98
1
1.02
1.04
Time (Hours)
EMW
Fig. 11. Constraints without CG: (Up) Loads C1, C4on the MV distribution
lines L1and L2(Down) MV Bus voltage
8
2 4 6 8 10 12 14 16 18 20 22 24
0.95
1
1.05
ED1
2 4 6 8 10 12 14 16 18 20 22 24
0.95
1
1.05
Time (Hours)
ED4
2 4 6 8 10 12 14 16 18 20 22 24
0.99
1
1.01
1.02
1.03
1.04
1.05
Time (Hours)
EMW
Fig. 12. Constraints with CG: (Up) Loads C1, C4on the MV distribution
lines L1and L2(Down) MV Bus voltage
2 4 6 8 10 12 14 16 18 20 22 24
1
1.005
1.01
1.015
1.02
1.025
1.03
1.035
1.04
1.045
Time (Hours)
g(t)
Fig. 13. Output of CG unit
VI. CONCLUSIONS
The paper has addressed the voltage regulation problem in
distribution network in the presence of distributed generation
by suitably modifying the On Load Tape Changer voltage
set-point. Network failures as well as the voltage regulation
effects on the distribution system reliability are analyzed.
A Command Governor approach is used to determine the
OLTC voltage reference that allows to satisfy operating con-
straints despite of adverse conditions that may occur in real
life operations. The simulation results referring to a typical
distribution network show the potential benefits on system
reliability introduced by the proposed approach under both
normal operating conditions and in the presence of undesirable
phenomena.
Future developments can be rely on the introduction of non-
linear constraints, such as thermal capacity line and capability
chart limits, and on the coordination of different devices in
the distribution network. The latter means that the actions
due to the presence of tap-changers, shunt-compensators and
distributed generators can be jointly taken into account by
formulating a multi-objective optimization CG problem.
APPENDIX
In this appendix we summarize the CG design method-
ology and give a brief account of its main properties. For
all theoretical and mathematical details concerning the mo-
tivations, developments and proofs and for alternative CG
design philosophies in dealing with disturbances and model
uncertainties we refer the interested readers to the existing
exhaustive literature [19]-[30].
We start by showing that the CG design problem (12) can
be solved by considering only the disturbance-free evolutions
of (10) under constant command sequences. Such evolutions
will be referred to as the “virtual” system evolutions. It will
be also shown that the CG action (12) is built by switching,
on the basis of current state and reference signal, amongst
admissible virtual constant sequences.
By linearity, one is allowed to separate the effects of
initial conditions and input from those of disturbances, e.g.
x(t) = ¯x(t) + ˜x(t), where ¯xis the disturbance-free component
(depending only on initial state and input) and ˜xdepending
only on disturbances. The same can be done for yand c, viz.
y(t) = ¯y(t) + ˜y(t)and c(t) = ¯c(t) + ˜c(t). Let us denote the
disturbance-free steady-state solutions of (10), for a constant
command g(t)≡w, as follows
¯xw:= (In−Φ)−1Gw
¯yw:= Hy(In−Φ)−1Gw (17)
¯cw:= Hc(In−Φ)−1Gw +Lw
Consider the following set recursions (see Fig.14 (Left))
C0:= C ∼ LdD,Ck:= Ck−1∼HcΦk−1GdD(18)
C∞:=
∞
\
k=0
Ck
where the symbol ∼denotes the geometrical difference be-
tween sets, that is A ∼ E := {a∈ A :a+e∈ A,∀e∈ E }.
It is shown in [23] that the sets Ckare the largest restrictions
of Csuch that ¯c(t)∈ Ckimplies c(t+i)∈ C,any d(t+i)∈
D,∀i∈ {0,1, ..., k −1}. In words, if the disturbance-free con-
strained vector ¯c(t)is contained in Ckat a certain time instant
t, then one is ensured that no constraint violations can occur
for the next ktime instants due to disturbances. In particular,
¯c(t)∈ C∞,∀t∈ZZ+, implies that c(t)∈ C,∀t∈ZZ+. If C∞
9
-
6
¯x
w
1
.
.
.
.
.
.
.
.
.
.
..
.
.................
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
:
:
j
CC0
C∞
:C1
Ck
6
-
¯x
w
K
δ
-
Wδ
Cδ
z
s¯cw
w
¯xw
C∞
Fig. 14. (Left) Nested set family Ck. (Right) Sets Cδ,Wδand V(x). The
set ¯cwdenotes the equilibrium points corresponding to constant commands
w∈ Wδ.
is empty the problem has no solution. If nonempty, all Ck’s,
∀k∈ZZ+, are non-empty, convex, compact and satisfy the
nesting condition Ck⊂ Ck−1,∀k∈ZZ+. Thus, one is allowed
to consider the disturbance-free system evolutions only and
adopt a “worst case” approach. Next, consider, for a small
enough δ > 0, the sets:
Cδ:= C∞∼ Bδ
Wδ:= {w∈IRm: ¯cw∈ Cδ}(19)
where Bδis the ball of radius δcentered at the origin (see
Fig.14 (Right)). In particular, Wδ, which we will assume
hereafter non-empty, is the closed and convex set of all com-
mands whose corresponding equilibrium vector ¯cwsatisfies
the constraints with margin δ. Again, emptiness of Wδmeans
that the problem has no solution.
Then, our approach in selecting at each time tthe CG action
g(t)in (12) will consists of restricting the choice amongst
all vectors of a suitable x(t)-depending subset of Wδ, each
vector of which, if constantly applied as a command to the
system from the time instant tonwards, gives rise to system
evolutions which do not produce constraint violations. Notice
that the choice g(t)∈ Wδ, only ensures constraint fulfillment
in steady-state, viz. ¯cg(t)∈ Cδ, but nothing can be said about
the transient from x(t)to xg(t). If many choices exist, the
vector that best approximates r(t)is selected. Such a command
is applied, a new state is measured and the procedure is
repeated at next time instant t+ 1 on the basis of the new
state x(t+ 1).
In this respect, we consider the following family of constant
virtual command sequences
gw(·) = g(k)≡w∈ Wδ,∀k∈ZZ+(20)
along with the following quadratic selection index
J(r(t), w) := kw−r(t)k2
Ψ(21)
where kxk2
Ψ:= x′Ψx,Ψ=Ψ′>0. Moreover, define the set
V(x)as
V(x) := {w∈ Wδ: ¯c(k, x, gw(·)) ∈ Ck,∀k∈ZZ+}(22)
where
¯c(k, x, gw(·)) := Hc Φkx+
k−1
X
i=0
Φk−i−1Gw!+Lw (23)
is to be understood as the disturbance-free virtual evolution
at the virtual time k(opposite to the real time t) of cfrom
the initial condition xat time 0under the constant command
gw(·)≡w. Notice that, because of time-invariance, the above
machinery (20)-(23) can be used at each time instant tfrom
the state x=x(t)in order to make predictions on the future
system evolutions under a constant command sequence applied
from tonwards. As a consequence, V(x(t)) ⊂ Wδwhich,
if non-empty, represents the set of all vectors in Wδwhich,
if applied as constant commands to the system in the state
x(t)at time t, give rise to system evolutions that satisfy the
constraints for all future time instants.
Then, provided that V(x(t)) is nonempty, closed and convex
for every t∈ZZ+the following minimizer uniquely exists
w(t) = arg min
w∈V(x(t)) J(r(t), w)(24)
and represents the best approximation of r(t)which, if con-
stantly applied from tonwards to the system, would never
produce constraints violation. Thus, we adopt the following
CG selection rule
g(t) = w(t)(25)
and repeat the same procedure at next time t+ 1. Obviously
w(t)∈ V(x(t+ 1)), viz. it is an admissible, although not
necessarily optimal, solution for (24) at the next time instant
t+ 1.
It has been shown in [27] that the following properties
hold true for the above described CG.
Main Result - Let assumptions (A1) be fulfilled and
Wδbe nonempty. Consider system (10) along with the CG
selection rule (24)-(25), and let V(x(0)) be non-empty. Then:
1) The minimizer in (25) uniquely exists at each t∈ZZ+
and can be obtained by solving a convex constrained
optimization problem, viz. V(x(0)) non-empty implies
V(x(t)) non-empty along the trajectories generated by
the CG command (25);
2) The set V(x),∀x∈IRn, is finitely determined, viz. there
exists an integer k0such that if ¯c(k , x, g(·)w)∈ Ck,
k∈ {0,1,...k0}, then ¯c(k, x, g(·)w)⊂ Ck∀k∈ZZ+.
Such a constraint horizon k0can be determined off-line;
3) The constraints are fulfilled for all t∈ZZ+;
4) The overall system is asymptotically stable; in particular,
whenever r(t)≡r,g(t)monotonically converges in fi-
nite time to either ror its best admissible approximation
ˆr
g(t)→ˆr:= arg min
w∈Wδkw−rk2
Ψ(26)
Consequently, by the offset free condition ((A1).2),
lim
t→+∞¯y(t) = ˆr. (27)
where ¯yis the disturbance-free component of y.
2
Notice that all above results of hold true for an arbitrary time-
varying reference signal r(t)and not only for a constant set-
points, except for the last part of item 4). Moreover, it is also
worth pointing out that the above results hold true under full
10
state information. The problem of partial state information has
been considered in [28]. Rigorous results, as the ones stated
in the previous Main Result, can be recovered also in the case
of incomplete state information by modifying the CG strategy
(25) as follows
g(t) := arg min
w∈V( ˆx(t)) ||w−r(t)||2
Ψ(28)
where ˆx(t)represents the state estimate given by a Luenberger
observer and considering the state reconstruction error and
measurement noise as additional disturbances acting on the
system.
Notice finally that the selection index (21) can easily be
modified as follows
J(r(t), w(t−1), w) := kw−r(t)k2
Ψ+kw−w(t−1)k2
Φ(29)
in order to penalize large set-point variations. This general-
ization can be instrumental to incorporate and trade-off in the
approach the minimization of the tap changes w.r.t. tracking
performance.
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