ETH Library
On the existence and linear
approximation of the power flow
solution in power distribution
networks
Journal Article
Author(s):
Bolognani, Saverio ; Zampieri, Sandro
Publication date:
2016-01
Permanent link:
https://doi.org/10.3929/ethz-b-000127124
Rights / license:
In Copyright - Non-Commercial Use Permitted
Originally published in:
IEEE Transactions on Power Systems 31(1), https://doi.org/10.1109/TPWRS.2015.2395452
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1
On the existence and linear approximation of the
power flow solution in power distribution networks
Saverio Bolognani and Sandro Zampieri
Abstract—We consider the problem of deriving an explicit
approximate solution of the nonlinear power equations that de-
scribe a balanced power distribution network. We give sufficient
conditions for the existence of a practical solution to the power
flow equations, and we propose an approximation that is linear in
the active and reactive power demands of the PQ buses. For this
approximation, which is valid for generic power line impedances
and grid topology, we derive a bound on the approximation error
as a function of the grid parameters. We illustrate the quality of
the approximation via simulations, we show how it can also model
the presence of voltage controlled (PV) buses, and we discuss how
it generalizes the DC power flow model to lossy networks.
Index Terms—Power systems modeling, load flow analysis,
power distribution networks, fixed point theorem.
I. INTRODUCTION
The problem of solving the power flow equations that
describe a power system, i.e. computing the steady state of
the grid (typically the bus voltages) given the state of power
generators and loads, is among the most classical tasks in
circuit and power system theory. An analytic solution of the
power flow equations is typically not available, given their
nonlinear nature. For this reason, notable effort has been
devoted to the design of numerical methods to solve systems
of power flow equations, to be used both in offline analysis of
a grid and in real time monitoring and control of the system
(see for example the review in [1]).
Specific tools have been derived for the approximate so-
lution of such equations, based on some assumptions on
the grid parameters. In particular, if the power lines are
mostly inductive, equations relating active power flows and bus
voltage angles result to be approximately linear, and decoupled
from the reactive power flow equations, resulting in the DC
power flow model (see the review in [2], and the more recent
discussion in [3]).
We focus here on a specific scenario, which is the power
distribution grid. More specifically, we are considering a
balanced medium voltage grid which is connected to the power
transmission grid in one point (the distribution substation, or
This work is supported by European Community 7th Framework Pro-
gramme under grant agreement n. 257462 HYCON2 Network of Excellence.
S. Bolognani is with the Laboratory on Information and Decision Sys-
tems, Massachusetts Institute of Technology, Cambridge (MA), USA. Email:
saverio@mit.edu.
S. Zampieri is with the Department of Information Engineering, University
of Padova, Padova (PD), Italy. Email zampi@dei.unipd.it
c
2016 IEEE. Personal use of this material is permitted. Permission from
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PCC, point of common coupling), and which hosts loads and
possibly also microgenerators.
Power distribution grids have recently been the object of an
unprecedented interest. Its operation has become more chal-
lenging since the deployment of distributed microgeneration
and the appearance of larger constant-power loads (electric
vehicles in particular). These challenges motivated the deploy-
ment of ICT (information and communications technology)
in the power distribution grid, in the form of sensing, com-
munication, and control devices, in order to operate the grid
more efficiently, safely, reliably, and within the its voltage and
power constraints. These applications have been reviewed in
[4], and include real-time feedback control [5]–[7], automatic
reconfiguration [8], [9], and load scheduling [10], [11]. In
order to design the control and optimization algorithms for
these applications, an analytic (rather than numerical) solution
of the power flow equations would be extremely convenient.
Unfortunately, because in the medium voltage grid the power
lines are not purely inductive, power flow equations include
both the active and reactive power injection/demands, and both
the voltage angles and magnitudes, in an entangled way. The
explicit DC power flow model therefore does not apply well.
The contribution of this paper is twofold. First, we give
sufficient conditions for the existence of a practical solution
of the nonlinear power flow equations in power distribution
networks (Theorem 1). Second, we derive a tractable approxi-
mate solution to the power flow problem, linear in the complex
power injections, providing a bound on the approximation
error (Corollary 2).
In the remainder of this section, we review relevant related
works. In Section II, we present the nonlinear equations that
define the power flow problem. In Section III we present our
main existence result, together with the linear approximate
solution. We illustrate such approximation via simulations
in Section IV, and we compare it with the classical DC
power flow model in Section V, where we also show how
to incorporate voltage controlled (PV) buses.
A. Related works
Conditions have been derived in order to guarantee the
existence of a solution to the power flow equations in the
scenario of a grid of nonlinear loads.
Many results are based on the degree theory [12]–[14]. In
[15], for example, exponential model is adopted for the loads,
and sufficient conditions for the existence of a solution are
derived. These conditions are however quite restrictive, and do
not include constant power (PQ) buses. In [16], on the other
Published on IEEE Transactions on Power Systems, vol. 31, no. 1, pp. 163–172, Jan. 2016.
https://doi.org/10.1109/TPWRS.2015.2395452
2
hand, the existence of a solution is proved by exploiting the
radial structure of the grid, via an iterative procedure which
is closely related to a class of iterative numerical methods
specialized for the power distribution networks [17], [18].
The existence of solutions to the power flow equations has
been also studied in order to characterize the security region of
a grid, i.e. the set of power injections and demands that yield
acceptable voltages across the network. These results include
[19], and others where however the decoupling between active
and reactive power flows is assumed [20]. Other works in
which the DC power flow assumption plays a key role are
[21] and [22], both focused on active power flows across the
grid. On the other hand, the results in [23] focus on the reactive
power flows and on the voltage magnitudes at the buses.
In [24] the implicit function theorem is used in order to
advocate the existence of a power flow solution, without
providing an approximate expression for that.
It is worth noticing that the linear approximate model that
we are presenting in this paper shares some similarities with
the method of power distribution factors [25], which allow
to express variations in the state (voltage angles) as a linear
function of active power perturbations. This method is also
typically based on the DC power flow assumptions, even if
a formulation in rectangular coordinates (therefore modeling
reactive power flows) has been proposed in [26]. Notice that,
except for the seminal works on power distribution factors
[27], [28], and the more recent results in [29], most of the
related results consists in algorithms that allow to compute
this factors only numerically, from the Jacobian of the power
flow equations.
The approximate power flow solution proposed in this paper
has been presented in a preliminary form in [6], [30], where
however no guarantees on the existence of such solution and
on the quality of the approximation were given.
II. POWER FLOW EQUATIONS
We are considering a portion of a symmetric and balanced
power distribution network, connected to the grid at one point,
delivering power to a number of buses, each one hosting loads
and possibly also microgenerators. We denote by {0, 1, . . . , n}
the set of buses, where the index 0 refers to the PCC.
We limit our study to the steady state behavior of the system,
when all voltages and currents are sinusoidal signals at the
same frequency. Each signal can therefore be represented via
a complex number y = |y|e
j∠y
whose absolute value |y|
corresponds to the signal root-mean-square value, and whose
phase ∠y corresponds to the phase of the signal with respect
to an arbitrary global reference.
In this notation, the steady state of the network is described
by the voltage v
h
∈ C and by the injected current i
h
∈ C at
each node h. We define the vectors v, i ∈ C
n+1
, with entries
v
h
and i
h
, respectively.
Each bus h of the network is characterized by a law relating
its injected current i
h
with its voltage v
h
. We model bus 0 as
a slack node, in which a voltage is imposed
v
0
= V
0
e
jθ
0
, (1)
where V
0
, θ
0
∈ R are such that V
0
≥ 0 and −π < θ
0
≤ π. We
model all the other nodes as PQ buses, in which the injected
complex power (active and reactive powers) is imposed and
does not depend on the bus voltage. This model describes
the steady state of most loads, and also the behavior of
microgenerators, that are typically connected to the grid via
power inverters [31]. According to the PQ model, we have
that, at every bus,
v
h
¯
i
h
= s
h
∀h ∈ L := {1, . . . , n}, (2)
where s
h
is the imposed complex power.
A more compact way to write these nonlinear power flow
equations is the following. Let the vectors i
L
, v
L
, s
L
be vectors
in C
n
having i
h
, v
h
, s
h
, h ∈ L as entries. Then we have
(
v
0
= V
0
e
jθ
0
s
L
= diag(
¯
i
L
)v
L
(3)
where
¯
i
L
is the vector whose entries are the complex conju-
gates of the entries of i
L
and where diag(·) denotes a diagonal
matrix having the entries of the vector as diagonal elements.
We model the grid power lines via their nodal admittance
matrix Y ∈ C
(n+1)×(n+1)
, which gives a linear relation
between bus voltages and currents, in the form
i = Y v. (4)
In the rest of the paper, we assume that the shunt ad-
mittances at the buses are negligible. Therefore the nodal
admittance matrix satisfies
Y 1 = 0, (5)
where 1 is the vector of all ones. Under this assumption, the
matrix Y corresponds to the weighted Laplacian of the graph
describing the grid, with edge weights equal to the admittance
of the corresponding power lines. We will show in a remark
in Section III how this assumption can be relaxed, so that the
entire analysis can be extended seamlessly to the case in which
shunt admittances of the lines are not negligible.
Considering the same partitioning of the vectors i, v as
before, we can partition the admittance matrix Y accordingly,
and rewrite (4) as
i
0
i
L
=
Y
00
Y
0L
Y
L0
Y
LL
v
0
v
L
.
where Y
LL
is invertible because, if the graph representing the
grid is a connected graph, then 1 is the only vector in the null
space of Y [32]. Using (5) we then obtain
v
L
= v
0
1 + Zi
L
(6)
where the impedance matrix Z ∈ C
n×n
is defined as
Z := Y
−1
LL
,
Objective of the power flow analysis is to determine from
these equations the voltages v
h
and the currents i
h
as functions
of V
0
, θ
0
and s
1
, . . . , s
n
, namely
v
h
=v
h
(V
0
, θ
0
, s
1
, . . . , s
n
)
i
h
=i
h
(V
0
, θ
0
, s
1
, . . . , s
n
).
3
In general, because of the nonlinear nature of the loads, we
may have no solution or more than one solution for fixed V
0
, θ
0
and s
h
, as the following simple example shows.
Example (Two-bus case). Consider the simplest grid made by
two nodes, node 0 being the slack bus (where we let θ
0
= 0),
and node 1 being a PQ bus. In this case we have that the
following equations have to be satisfied
(
v
1
¯
i
1
= s
1
v
1
= V
0
+ Z
11
i
1
(7)
Assume that Z
11
= 1, and that s
1
is real. The system of
equations (7) can then be solved analytically. In fact it can be
found that if V
2
0
+ 4s
1
< 0 there are no solutions. When on
the contrary V
2
0
+ 4s
1
> 0, there are two distinct solutions
i
1
=
−V
0
±
p
V
2
0
+ 4s
1
2
.
Notice that, if V
0
is large, then the solutions exist and, since
p
V
2
0
+ 4s
1
= V
0
p
1 + 4s
1
/V
2
0
' V
0
(1 + 2s
1
/V
2
0
) = V
0
+
2s
1
/V
0
, the current i
1
take the two values
i
+
1
' s
1
/V
0
, i
−
1
' −V
0
−
s
1
V
0
.
Therefore, when V
0
is large, one solution consists in small
currents (thus small power losses and voltage close to the
nominal voltage across all the network), while the other
consists in larger currents, larger power losses, and larger
voltage drops. Of course, the system should be controlled so
that it works at the first working point.
The intuition from this simple example is developed in the
next section, where the existence and uniqueness of a practical
solution to the power flow equations (i.e. a solution at which
the grid can practically and reliably be operated) is studied,
and an approximate power flow solution (linear in the power
terms) is proposed.
III. MAIN RESULT
Define
f := v
0
¯
i
L
− s
L
= V
0
e
jθ
0
¯
i
L
− s
L
,
so that we have
i
L
=
1
¯v
0
(
¯
f + ¯s
L
) =
e
jθ
0
V
0
(
¯
f + ¯s
L
). (8)
By putting together (8) with (3) and (6), we get
s
L
= diag(
¯
i
L
)v
L
=
e
−jθ
0
V
0
diag(f + s
L
)
e
jθ
0
V
0
1 +
e
jθ
0
V
0
Z(
¯
f + ¯s
L
)
= f + s
L
+
1
V
2
0
diag(f + s
L
)Z(
¯
f + ¯s
L
),
and therefore
f = −
1
V
2
0
diag(f + s
L
)Z(
¯
f + ¯s
L
). (9)
We can determine a ball where there exists a unique solution
f to this equation by applying the Banach fixed point theorem
[33]. In order to do so, define the function
G(f) := −
1
V
2
0
diag(f + s
L
)Z(
¯
f + ¯s
L
).
Consider the standard 2-norm k · k on C
n
defined as
kxk :=
s
X
h
|x
h
|
2
.
Let us then define the following matrix norm
1
on C
n×n
kAk
∗
:= max
h
kA
h•
k = max
h
s
X
k
|A
hk
|
2
(10)
where the notation A
h•
stands for the h-th row of A.
The following result holds.
Theorem 1 (Existence of a practical power flow solution).
Consider the vector 2-norm k · k on C
n
, and the matrix norm
k · k
∗
defined in (10). If
V
2
0
> 4kZk
∗
ks
L
k (11)
then there exists a unique solution v
L
of the power flow
equations (3) and (6) in the form
v
L
= V
0
e
jθ
0
1 +
1
V
2
0
Z¯s
L
+
1
V
4
0
Zλ
(12)
where λ ∈ C
n
is such that
kλk ≤ 4kZk
∗
ks
L
k
2
. (13)
Proof. Let
δ :=
4kZk
∗
V
2
0
ks
L
k
2
(14)
and B := {f ∈ C
n
| kfk ≤ δ}. In order to apply the
Banach fixed point theorem, we need to show that, under the
hypotheses of the theorem,
G(f) ∈ B for all f ∈ B (15)
kG(f
0
) − G(f
00
)k ≤ kkf
0
− f
00
k for all f
0
, f
00
∈ B (16)
for a suitable constant 0 ≤ k < 1. We prove first (15). Observe
that, by using Lemma A.1 in the case p = 2, we have
kG(f)k ≤
1
V
2
0
kZk
∗
kf + s
L
k
2
≤
1
V
2
0
kZk
∗
(kfk + ks
L
k)
2
≤
1
V
2
0
kZk
∗
(δ + ks
L
k)
2
,
where we used the fact that kf k ≤ δ. Now, using the definition
(14) of δ and Lemma A.3 in the Appendix (with a = kZk
∗
/V
2
0
and b = ks
L
k) we can argue that, if (11) is true, then
kG(f)k ≤
1
V
2
0
kZk
∗
(δ + ks
L
k)
2
≤ δ.
1
The ∗ sign indicates that we are not referring to the norm induced by the
vector norm.
4
We prove now (16). It is enough to notice that, by applying
Lemma A.2 in the Appendix (with A = −Z/V
2
0
, x = f and
a = s
L
) we obtain that
kG(f
0
) − G(f
00
)k
≤
1
V
2
0
kZk
∗
(kf
0
+ f
00
k + 2ks
L
k) kf
0
− f
00
k
≤
2
V
2
0
kZk
∗
(δ + ks
L
k) kf
0
− f
00
k
= kkf
0
− f
00
k
where
k :=
2
V
2
0
kZk
∗
4kZk
∗
V
2
0
ks
L
k
2
+ ks
L
k
.
Finally, notice that by using (11) we obtain
k <
2
V
2
0
kZk
∗
V
2
0
4kZk
∗
4kZk
∗
V
2
0
V
2
0
4kZk
∗
+ 1
= 1.
Now from (15) and (16), by applying the Banach fixed point
theorem, we can argue that there exists unique solution f ∈ B
of equation (9). Then by using (6) and (8) we have that
v
L
= V
0
e
jθ
0
1 + Z
e
jθ
0
V
0
(¯s
L
+
¯
f)
= V
0
e
jθ
0
1 +
1
V
2
0
Z¯s
L
+
1
V
2
0
Z
¯
f
.
In order to prove (13) it is enough to define λ := V
2
0
¯
f.
Remark. The norm kZk
∗
can be put in direct relation with
the induced matrix 2-norm kZk, and with structural properties
of the graph that describes the power grid. Indeed
kZk
∗
= max
h
ke
T
h
Zk ≤ max
kvk=1
kZ
T
vk = kZk,
where e
h
is the h-th vector of the canonical base. It can be
shown that [
0 0
0 Z
] is one possible pseudoinverse of the weighted
Laplacian Y of the grid. Therefore we have that
kZk
∗
≤
1
σ
2
(Y )
,
where σ
2
(Y ) is the second smallest singular value of Y
(the smallest one being zero). In the special case in which
all the power lines have the same X/R ratio (i.e. their
impedances have the same angle, but different magnitudes),
then σ
2
(Y ) corresponds also to the second smallest eigenvalue
of the Laplacian, which is a well known metric for the graph
connectivity. Given this relation between kZk
∗
and kZk, the
assumption (11) in Theorem 1 is satisfied if
V
2
0
> 4
ks
L
k
σ
2
(Y )
.
This condition resembles similar results that have been pro-
posed for example in [20] for the analysis of the feasibility
of the power flow problem. Interestingly, the role of similar
spectral connectivity measures of the grid has been recently
investigated also for grid synchronization and resilience prob-
lems (see the discussion in [34] and [35], respectively).
Corollary 2 (Approximate power flow solution). Consider
the vector 2-norm k · k on C
n
, and the matrix norm k · k
∗
defined in (10). Let the assumption of Theorem 1 be satisfied.
Then the solution v
L
of the power flow nonlinear equations is
approximated by
ˆv
L
:= V
0
e
jθ
0
1 +
1
V
2
0
Z¯s
L
, (17)
and the approximation error satisfies
|v
h
− ˆv
h
| ≤
4
V
3
0
kZ
h•
kkZk
∗
ks
L
k
2
, (18)
where, as before, Z
h•
is the h-th row of Z.
Proof. We have, from (12), for any bus h ∈ L,
|v
h
− ˆv
h
| =
1
V
3
0
|Z
h•
λ| ,
By using Cauchy-Schwarz inequality and the bound (13) we
obtain (18).
Theorem 1 and Corollary 2 can be interpreted as a way
to model the grid as a linear relation between the variables v
and s. In fact, power flow equations already contain the simple
linear relation (4) between the variables i and v, together with
an implicit nonlinear relation between v and s. The previous
results say that, in case s is sufficiently small, there is a way to
make the relation between v and s explicit (Theorem 1), and
to find a linear approximated relation between these variables
(Corollary 2). This interpretation will be further elaborated in
Section V, in order to extend the model to grids where voltage
regulated (PV) nodes are present.
Notice moreover that the approximate model (17) can be
manipulated, via premultiplication by the admittance matrix
Y , in order to obtain the sparse linear equation
Y ˆv =
e
jθ
0
V
0
¯s, (19)
where ˆv and s are the augmented vectors
ˆv
0
ˆv
L
and
h
−1
T
s
L
s
L
i
,
respectively. The resemblance of (19) with the DC power flow
model will be investigated in Section V, where the proposed
approximate solution is presented in polar coordinates.
Remark (Non-zero shunt admittances). The grid model (4)
in which the matrix Y is assumed to satisfy (5) is based on
the assumption of zero nodal shunt admittances. In the case
in which shunt admittances are not negligible, the proposed
analysis can be modified accordingly. In this more general
case the matrix Y is invertible, and for almost all grid
parameters of practical interest, the submatrix Y
LL
is also
invertible. Equation (6) becomes v
L
= v
0
w + Zi
L
, where
Z := Y
−1
LL
, and where w := −Y
−1
LL
Y
L0
∈ C
n
is a perturbation
of the vector 1 and corresponds to the normalized no-load
voltage profile of the grid.
The reasoning of Theorem 1 can be repeated by defining
f := V
0
e
jθ
0
W
¯
i
L
− s
L
, where the diagonal matrix W :=
diag(w) is a perturbation of the identity matrix. In this case,
f has to satisfy the equation f = G(f ) where
G(f) := −
1
V
2
0
diag(f + s
L
)W
−1
Z
¯
W
−1
(
¯
f + ¯s
L
).
![Figure 1. Bus voltages in the modified IEEE 123 test feeder [36]. The circles represent the exact solution of the nonlinear power flow equations. The dots represent the approximate solution v̂ given by (17). The solid line represent the bound (18) on the approximation error obtained by considering the 2-norm and the corresponding matrix norm ‖Z‖∗, while the dashed line represents the error bound (22), obtained by adopting the norms ‖sL‖1 and ‖Z‖∗∞.](/figures/figure-1-bus-voltages-in-the-modified-ieee-123-test-feeder-1tzps78z.webp)