A Novel Decomposition for Control of DC Circuits and Grid Models with
Heterogeneous Energy Sources
Shuai Wang & John Baillieul

arXiv:1709.07569v1 [] 22 Sep 2017

Abstract
The way in which electric power depends on the topology of circuits with mixed voltage and current sources is examined.
The power flowing in any steady-state DC circuit is shown to depend on a minimal set of key variables called fundamental node
voltages and fundamental edge currents. Every steady-state DC circuit can be decomposed into a voltage controlled subcircuit
and a current controlled subcircuit. In terms of such a decomposition, the I 2 R losses of a mixed source circuit are always
the sum of losses on the voltage controlled subcircuit and the current controlled subcircuit. The paper concludes by showing
that the total power flowing in a mixed source circuit can be found as critical points of the power expressed in terms of the
key voltage and current variables mentioned above. The possible relationship to topology control of electric grid operations is
discussed.

I. I NTRODUCTION
Because of the fast time constants in changing the system state, transmission line switching has been used to reduce losses
and improve grid security since the 1980s [8], [9]. More recently, with the focus on power markets, a good deal of current
research on the operation of smart grids has been focused on the co-optimization of network topology and generation in
power system operation, i.e. reducing the generation cost through changing the topology of the network whenever congestion
occurs.
It has been widely accepted that the line switching problem can be formulated as a mixed integer programming (MIP)
problem with some binary variables indicating whether the lines of the network are in or out of service [10], [11]. Formulated
in this way, the topology reconfiguration problem is NP-hard. To address this challenge, recent work has been aimed at fast
heuristic approaches to line switching. References [3], [5], [6] show the effectiveness of co-optimizing the generation and
the network topology through simulations on the IEEE 118-bus system and the WECC 179-bus system. References [7], [4]
demonstrate that topology control can be beneficial even while preserving an N-1 reliable network. The total run time of
the heuristic methods is short enough for practical use for day ahead planning, and with further development, these may
provide a useful approach to feasibility correction in optimal power flow (OPF) calculation, [14]. Despite the enthusiasm
with which the research community has pursued heuristic approaches to topology control, satisfactory grid-scale solutions
have remained elusive.
The present paper examines the loading effect of topology reconfiguration in circuits with mixed voltage and current
sources and extends our previous work on purely voltage controlled circuits, [1], and purely current controlled circuits, [2].
The aim is to provide a possible foundation for heuristic of the kind discussed above. The main results of [1] and [2] show
that in a voltage-controlled (current-controlled) circuit (see Def. 1), switching on an additional conductive line will always
increase (decrease) congestion (i.e. increase (decrease) the I 2 R losses).
Mathematically, the DC model of power flow is equivalent to a current driven network, where power injections are
equivalent to current sources; power flowing through lines is equivalent to current through edges, etc. See Table 1.
network

potential

flow

impedance

equation

grid

phase θ

power P

reactance X

P =

θ
X

circuit

voltage V

current I

resistance R

I=

V
R

Table 1: The equivalence between a current driven circuit and a transmission grid.
While there is a well established correspondence between current-controlled DC-circuits and linearized DC power flow
models, the recent increase in load shifting and demand response programs suggest that the formulation of the standard
OPF problem should be modified to take advantage of the flexibility (e.g., loads, reserve requirements, and transmission
topology) provided by the smart grid platform. Our mixed source model is better able to capture the features of power grids
in which renewables, storage, and demand response play significant roles. For example, consider the 5-bus network of Fig.
Shuai Wang is with the Division of Systems Engineering and John Baillieul is with the Departments of Mechanical Engineering; Electrical and
Computer Engineering, and the Division of Systems Engineering at Boston University, Boston, MA 02115. Corresponding author is John Baillieul (Email:
johnb@bu.edu).
The work on power systems and grid topology was supported by NSF EFRI Grant Number 1038230, and work on large scale integer optimization was
supported by DARPA Contract HR0011-16-C-0115.

1

1 with a power flow in which there is an overload of Line L25 . Traditionally, such line overloads can be alleviated either
through regulating the generators’ output or through dynamic control of the underlying network topology. The increase of
demand-side participation and the development in electrical energy storage in power markets, however, makes it also possible
to alleviate the congestion through load regulation or load shifting in time or space or both. In the simplest case, suppose
Bus 2 and Bus 5 are equipped with enough energy storage capacities that they are able to release energy during peak
times while storing energy at off-peaks. The effect of such system flexibility can be well abstracted as a “phase lock” to the
overloaded Line L25 during peak times which is equivalent to adding a voltage source to the traditional current-controlled
circuit model.
Main

DG 1

Bus 2
Line L12

~

Main Source
Bus 1
R12

Line L13

~

Bus 3

Bus 4

Line L35
Line L25
Line L45

Load 3

G2

~

L1
R25

R35
R45

L2

Load 2

~

Bus 5

R34

DG 2

Line L34

Load 1

R13

G1

L3

DG 3

(a)

G3

(b)

Fig. 1. (a) A 5-bus network with line overload at Line L25 . (b) The equivalent mixed-source circuit of (a) with the voltage source (blue) denoting the
effect of load regulating equipment at Bus 2 and Bus 5.

Due to the heterogeneity of the controllers (e.g. the current-control loop and voltage regulator), distribution networks
and microgrids can hardly be modeled as systems with a single class of primary energy sources, pointing out the need for
additional research on the mixed source model.
The present paper looks at whether the topology-dependent loading phenomena similar to [1] and [2] can be found in
mixed-source networks. It is organized as follows. In the next section (Section II), we review the needed background on the
topology of DC electric circuits. In particular, we revisit the calculations of total I 2 R loss for arbitrary voltage-controlled
circuits and arbitrary current-controlled circuits, respectively. In Section III, we extend the discussion from single-source
circuits (see Def. 1) to mixed-sourced circuits. It is shown that the effect of removing an edge from a mixed-source circuit
can be perfectly decomposed into two sub-effects in its voltage-controlled sub-circuit (see Def. 9) and current-controlled
sub-circuit (see Def. 9), respectively. Meanwhile, it gives a simple method to calculate the change of total I 2 R loss for
mixed-source circuit based on its reduced equivalent circuit. It is also shown that the total loss of a mixed-source circuit is
exactly the sum of total loss of its voltage-controlled sub-circuit and current-controlled sub-circuit. Section IV explores this
in terms of four different approaches to calculate the total I 2 R loss of an arbitrary mixed-source circuit. It is shown that
all of them are mathematically equivalent, pointing out a way to convert a certain type of constrained linear programming
formulation to an unconstrained non-linear programming problem. Concluding remarks and possible implications for power
networks are contained in Section V.
II. P RELIMINARIES
Definition 1: A voltage-controlled circuit is comprised purely by resistors and voltage sources. A current-controlled circuit
is comprised purely by resistors and current sources. Both voltage-controlled circuits and current-controlled circuits are called
single-source circuits. A circuit that has both current sources and voltage sources is called a mixed-source circuit.
Remark 1: As capacitors (inductors) act identically as open (short) circuits in DC steady state, the above definition can
be easily extended to general steady-state RCL circuits as was shown in [1] and [2].
Definition 2: An edge in a network graph represents a single element either a voltage source, a current source or a resistor.
A node denotes the position of connection where two or more edges meet. A cycle is any closed path.
A. Voltage-Controlled Circuit [1]
Definition 3: [1] For a voltage controlled circuit, a fundamental node basis is a maximal set of nodes among which there
exist no paths comprised purely of voltage source edges. Their voltages are called fundamental nodal voltages.
The fundamental node basis may not be unique for a voltage-controlled circuit, but its dimension is always uniquely
determined. For example, the fundamental node basis for the circuit in Fig. 3 can be nodes {1, 2}, {3, 2} or {4, 2}.
By definition, if one node, say the i-th node, is not included in the selected fundamental node basis, there must exist
one and only one fundamental node that has a pure voltage source path connecting it and the i-th node. Thus, once all

2

R1

R1
2

V

1

1

1

R1
V

2

R2

R2

2

R2

I

3

I
3

(a)

(b)

(c)

Fig. 2. (a) A voltage-controlled circuit with nodes {1, 2, 3}. (b) A current-controlled circuit with nodes {1, 2}. (c) A mixed-source circuit with nodes
{1, 2, 3}.
R1
1

E1

R3
3

2

E2

i2

R2

4

Fig. 3.

i1

i3

A simple circuit with DC-voltage sources and resistive loads.

fundamental nodal voltages are known, all other nodal voltages can be determined by adding the voltage contributions from
the voltage source edges that connect them to the fundamental nodes.
There may or may not exist a pure voltage source path between the pair of endpoints of a resistor edge. If such a path
exists, then the voltage drop between its endpoints is always fixed. We call such a resistor a Type V1 resistor. We can use
a scalar, say PV 1 , to denote the total I 2 R loss of such resistor edges.
For a resistor edge that doesn’t have a pure voltage source path between its endpoints, its I 2 R loss may vary if we change
the connectivity of another edge. We call such a resistor a Type V2 resistor. We assume, without loss of generality, that the
endpoints of each Type V2 resistor are connected to a pair of fundamental nodes, {i, j} (i, j = 1, ..., D, where D is the
cardinality of the fundamental node basis), and these are connected by a voltage source path. In general, there may be more
than one Type V2 resistor edge connected to fundamental node pair {i, j}, and we denote the total number of such resistor
edges by Li,j . Here, we assume the k-th resistor edge discussed above has value Ri,j,k (k = 1, ..., Li,j ).
It is easy to see that the sets of Type V1 resistors and Type V2 resistors are jointly exhaustive. A potential function
denoting the total I 2 R loss of all resistors on a voltage controlled circuit can then be formed by:
PV = PV 1 +

D−1
X

Li,j
D X
X
(evi + ePvi,k − evj − ePvj,k )2

Ri,j,k

i=1 j=i+1 k=1

(1)

where {ev1 , ..., evM } are the fundamental nodal voltages, and ePvi,k denotes the algebraic sum of voltages on the pure
voltage source path connecting the fundamental node i and one endpoint of the resistor Ri,j,k . Similarly, ePvj,k is the sum
of voltages along the path connecting the other endpoint of the resistor to the fundamental node j.
This can be best understood by using the circuit in Fig. 3 as an illustrative example. Clearly, there are no resistor edges
in Fig. 3 with endpoints being connected by a pure voltage source path, i.e. PV 1 = 0. The dimension of the fundamental
node basis has been shown to be 2, i.e D = 2, and we randomly choose nodes {1, 2} as the fundamental node basis. The
endpoints of three resistor edges {R1 , R2 , R3 } are either directly connected to node 1 and node 2 or indirectly connected
to node 1 by a pure voltage source path, i.e. L1,2 = 3, R1,2,1 = R1 , R1,2,2 = R2 , and R1,2,3 = R3 . For example, the left
endpoint of R2 (namely R1,2,2 ) is connected by a path of E1 and E2 to node 1, i.e. ePv1,2 = −E1 − E2 , and its right
endpoint is directly connected to node 2, i.e. ePv2,2 = 0. Denoting the voltages at node 1 and 2 as ev1 and ev2 , we know
the I 2 R loss of R2 is given by

(ev1 +ePv

1,2

−ev2 −ePv

R1,2,2

PV =

2,2

)2

. Repeating the calculation for all three resistors, we have

2
3 (e
X
v1 + ePv1,k − ev2 − ePv2,k )
k=1

R1,2,k

B. Current-Controlled Circuit [2]
Definition 4: [12] If there exists some spanning tree T for a given graph, and e denotes an edge that is not in T , then
the simple cycle consisting of e together with the path in T connecting the endpoints of e is called the fundamental cycle
defined by e. A cycle basis formed in this way is called a fundamental cycle basis.
Definition 5: Given an arbitrary DC circuit network consisting of current sources, voltage sources and resistors, its
resistance graph is formed by:

3

replacing the original position of every current source with its internal resistance: an open circuit;
replacing the original position of every voltage source with its internal admittance: a short circuit.
For a current-controlled circuit, its resistance graph can be formed by removing all source edges. A spanning tree T of
its resistance graph can be found by using depth-first search. Then a fundamental cycle basis of the resistance graph can be
formed based on T . We assume without loss of generality that there are N fundamental cycles, and the edges that define
the fundamental cycles are {e1 , e2 , ..., eN }. The original current-controlled circuit can be reconstructed from the resistance
graph by putting back all current source edges. With all current sources operating, we denote the current flowing on the
edge ei that defines the i-th fundamental cycle by Iei .
A resistor edge, if one of its endpoints is a leaf vertex, doesn’t belong to any fundamental cycle of the resistance graph. In
such case, its I 2 R loss will always be unchanged as the current flowing through the resistor is fixed. We call such resistors
Type I1 resistors. We can use a scalar, say PI1 , to denote the total loss of such resistors.
A resistor edge, if neither of its endpoints is a leaf vertex, may belong to either one or several fundamental cycles of the
resistance graph. For a resistor edge that is exclusively owned by one fundamental cycle, say the i-th fundamental cycle, we
call it a Type I2 resistor. We can denote the number of the i-th fundamental cycle’s exclusive edges by Oi (i = 1, ..., N ),
and the resistance of the k-th exclusive edge as Ri,i,k , then the total loss of such resistor edges can be computed by:
•
•

PI2 =

Oi
N X
X

(Iei + IPei ,k )2 Ri,i,k

i=1 k=1

where IPei ,k denotes the algebraic sum of current injections from current source edges and/or the Type I1 resistor edges to
the path connecting ei and the k-th exclusive edge.
For those resistor edges that are shared by two or more fundamental cycles, we call them Type I3 resistors. We can
denote the number of such resistor edges by M , the resistance of the k-th (k = 1, ..., M ) edge by Rk , the number of
fundamental cycles that are associated with the k-th edge by nk , and the edges defining these associated fundamental cycles
by {ek1 , ek2 , ..., eknk } (1 < k1 , ..., knk < N ). Then the total loss of such resistor edges can be computed by
PI3 =

nk
M X
X
(
Ieki + IPk )2 Rk
k=1 i=1

where IPk denotes the algebraic sum of current injections from current source edges and/or the Type I1 resistor edges to
the paths connecting the edges defining the associated fundamental cycles and the k-th Type I3 edge.
It is easy to see that the sets of Type I1 resistors, Type I2 resistors and Type I3 resistors are jointly exhaustive. Then a
potential function denoting the total loss of all resistors in a current-controlled circuit can be given by:
PI = PI1 + PI2 + PI3

(2)

Definition 6: [2] The adding (removing) of an edge to (from) an existing graph is called a parallel attachment (removal)
if the node set of the graph is unchanged but the number of fundamental cycles is increased (decreased) by 1 after the
operation. The adding (removing) of an edge to (from) an existing graph is called a serial attachment (removal) if the cycle
space of the graph is unchanged but the number of nodes is increased by one after the operation.
In graph theory, a serial attachment can be viewed as the subdivision of some edge. Fig. 4 shows an example of parallel
attachment and serial attachment of a resistor R2 to a circuit comprised by a voltage source V and a resistor R1 .
1

R1
2

V

1

serial
attachment
V

1

parallel
attachment

R1

V

R1

R2

R2
3

Fig. 4.

2

2

An example of parallel attachment and serial attachment.

Definition 7: [2] An electric element in the circuit is called a passive element if its current and voltage are of opposite
polarity (and therefore the element consumes power), and an active element if its current and voltage are of same polarity
(and therefore the element delivers power).
III. T HE C ASE OF M IXED -S OURCE N ETWORKS
In [1] and [2], we showed that the parallel attachment of an active current (voltage) source always increases the total loss
of a current-controlled (voltage-controlled) circuit. It is natural to ask if similar results can be generalized for an arbitrary

4

mixed-source circuit. The answer is no. [2] shows a simple counterexample which is revisited in Fig. 5. An interesting
paradoxical behavior happens in Fig. 5: the removal of an active current source causes a redistribution of the current that
results in higher total I 2 R loss of a mixed-source circuit.
0.25A
(0.25W)

1Ω
2V
(1.5W)

1Ω
2V
(1.75W)

1Ω
2Ω
1Ω

0.5A
(0.375W)

1Ω
2Ω
1Ω

(a)

0.5A
(0.406W)

(b)

Fig. 5. (a) A mixed-source network with two current sources and one voltage source. (b) A mixed-source network with one current source and one
voltage sources. (Example from [2]).

In order to explore this paradox, we have the following:
Definition 8: A source factor in a circuit is defined to be the sensitivity of the current flowing through a voltage source
(or of voltage difference between the endpoints of a current source) with respect to a change in the value of another voltage
source or current source.
By above definition, we have the following notations:
• the sensitivity of the current flowing through voltage source i with respect to a change in the value of voltage source
j is denoted as sVV
j,i ;
• the sensitivity of the current flowing through voltage source i with respect to a change in the value of current source
j is denoted as sIV
j,i ;
• the sensitivity of the voltage difference between the endpoints of current source i with respect to a change in the value
of voltage source j is denoted as sVI
j,i ;
• the sensitivity of the voltage difference between the endpoints of current source i with respect to a change in the value
of current source j is denoted as sII
j,i .
By the Generalized Notorn Theorem [13], any 4-terminal resistance graph can be reduced to an equivalent resistance
network. The equivalent network consists of 6 resistors and its graph is the complete one as shown in Fig. 6.

Ra

3

Re

Rb

2

1

Rc

Fig. 6.

Rf

Rd

4

The equivalent reistance network for an arbitrary 4-terminal resistance network.

The red box (and green box) in Fig. 6 can then be filled with a voltage source or current source in order to study some
useful properties of the source factor. Basically, we have following results (whose proof is a basic calculation based on Fig.
6 and thus is omitted here):
VV
sVV
j,i = si,j
VI
sIV
j,i = −si,j

sII
j,i

=

(3)

sII
i,j .

Lemma 1: The parallel attachment of a resistor edge to endpoint pair {m, n} in a mixed-source circuit will always decrese
the voltage difference between {m, n}, and will keep the voltage polarity of {m, n} unchanged.
Proof: Thevenin’s theorem states that any linear circuit with voltage and current sources and resistances can be replaced at
terminals m-n by an equivalent voltage source Vmn in series connection with an equivalent resistor Rmn . Vmn is the voltage
obtained at terminals m-n before we add the new resistor edge. Denoting the new voltage obtained at terminals m-n after
0
0
we add the new resistor edge as Vmn , it is easy to prove that Vmn must be smaller than Vmn , and the voltage polarity of
{m, n} must be unchanged.

Definition 9: For a given mixed-source circuit, CM , its voltage-controlled sub-circuit, CV , is created by replacing all
current source edges with open circuits in CM ; and its current-controlled sub-circuit, CI , is created by replacing all voltage

5

source edges with short circuits in CM . It is easy to see that CM , CV , and CI have the same set of resistance edges. To
prevent confusion, we denote the i-th resistance edge by RiM in CM , RiV in CV , and RiI in CI , respectively, and denote
the current flowing on the i-th resistance edge by IiM in CM , IiV in CV , and IiI in CI , respectively.
For example, the circuits in Fig. 2(a) and Fig. 2(b) are the voltage-controlled sub-circuit and current-controlled sub-circuit
of the circuit in Fig. 2(c), respectively.
Proposition 1: The change of total losses, ∆P , resulting from the parallel removal (parallel attachment) of a resistance
link RjM from a mixed-source circuit is given by ∆P = ∆PV + ∆PI , where ∆PV denotes the change of losses resulting
from removing (adding) the link RjV from its voltage-controlled sub-circuit, and ∆PI denotes the change of losses resulting
from removing (adding) the link RjI from its current-controlled sub-circuit.
Proof: It is easy to prove that the parallel attachment and parallel removal have exactly the opposite effect on the total loss
of a circuit. The parallel removal part of the proposition is thus logically equivalent to the parallel attachment part of the
proposition. Hence we just need to prove the parallel removal part of the proposition.
We assume without loss of generality that there are k current sources {I1 , ..., Ik }, and l voltage sources {V1 , ..., Vl }, in
the circuit. Suppose we are going to remove the j-th resistor edge Rj , and its endpoint pair is {m, n}.
By the principle of energy conservation, the change of total I 2 R loss must be equivalent to the change of total sources’
energy output
k
l
X
X
Ii ∆Vi +
Vi ∆Ii
i=1

i=1

where ∆Vi denotes the change of voltage difference between the endpoints of the i-th current source, and ∆Ii denotes the
change of current flowing through the i-th voltage source. Here, in order to calculate ∆Vi and ∆Ii , we replace the j-th
resistor edge by a passive current source with value Imn , i.e. the current flowing on the resistor edge before its removal.
Clearly, such a replacement increases the number of source edges by 1 and deceases the number of resistor edges by 1, but
it has no effect on the rest of the circuit. The voltage difference between the endpoints of the new current source edge must
be equivalent to Vmn , i.e. the the voltage difference between node pair {m, n} before the removal.
By the superposition principle, it is easy to prove that Vmn is a linear combination of {I1 , ..., Ik , Imn } and {V1 , ..., Vl },
i.e.
 II

 VI 
s1,mn
s1,mn


.

  ..  
 . 
Vmn = I1 · · · Ik Imn 
 + V1 · · · Vl  .. 
 sII

k,mn
sVI
l,mn
II
smn,mn
II
II
VI
VI
II
where {sII
1,mn , · · · , sk,mn , smn,mn } and {s1,mn , · · · , sl,mn } are source factors. To be more specific, su,w (u, w =
1, ..., k, mn) is the sensitivity of the voltage difference between the w-th current source’s endpoints with respect to a
change in the value of the u-th current source. sVI
u,w (u = 1, ..., l and w = 1, ..., k, mn) is the sensitivity of the voltage
difference between the w-th current source’s endpoints with respect to a change in the value of the u-th voltage source.
Similarly, we have
 II 
 VI 
s1,i
s1,i
..  



 . 
 .. 
Vi = I1 · · · Ik Imn 
 + V1 · · · Vl  . 
 sII

k,i
sVI
II
l,i
smn,i

where Vi (i = 1, ..., k)denotes the voltage difference between the endpoints of the i-th current source, and
 IV 
 VV 
s1,i
s1,i


.

  ..  
 . 
Ii = I1 · · · Ik Imn 
 + V1 · · · Vl  .. 
 sIV

k,i
sVV
l,i
sIV
mn,i
where Ii (i = 1, ..., l)denotes the current flowing through the i-th voltage source.
Clearly, the removal of the the j-th resistor edge Rj and the removal of the current source Imn have exactly the same
effect on the total energy output of the k current sources and l voltage sources. In addition, the removal of the current source
Imn doesn’t further change the resistance graph. Thus, we have


 II 
smn,1
∆V1
 .. 
 .. 
 .  = −Imn  . 
∆Vk

sII
mn,k

6


 IV 

smn,1
∆I1
 .. 
 .. 
 .  = −Imn  . 
sIV
mn,l
 II 
 IV 
smn,1
smn,1
k
l
X
X

 .  
 . 
Ii ∆Vi +
Vi ∆Ii = −Imn { I1 · · · Ik  ..  + V1 · · · Vl  .. }.
∆Il

i=1

i=1

sII
mn,k

sIV
mn,l

VI
IV
II
By (3), we have sII
mn,i = si,mn (i = 1, ..., k, mn) and smn,i = −si,mn (i = 1, ..., l) which further gives us
 VI 
 II 
s1,mn
s1,mn
k
l
X
X
 . 

 .  
Ii ∆Vi +
Vi ∆Ii = −Imn { I1 · · · Ik  ..  − V1 · · · Vl  .. }.
i=1
i=1
sVI
sII
l,mn
k,mn

By replacing all voltage source edges with short circuits (i.e. setting the value of all voltage sources to zero), we get the
current-controlled sub-circuit, CI . It is easy to prove that ∆PI , the change of I 2 R loss resulting from removing Rj from
the current-controlled sub-circuit, is given by
 II 
s1,mn

 . 
∆PI = −Imn I1 · · · Ik  ..  .
sII
k,mn
Similarly, we have

∆PV = Imn V1

···

 VI 
s1,mn
 . 
Vl  ..  .
sVI
l,mn

This ends the proof.

Proposition 2: The change of total losses, ∆P , resulting from the serial removal (serial attachment) of a resistance link
RjC from a mixed-source circuit is given by ∆P = ∆PV + ∆PI , where ∆PV denotes the change of losses resulting from
removing (adding) the link RjV from its voltage-controlled sub-circuit, and ∆PI denotes the change of losses resulting from
removing (adding) the link RjI from its current-controlled sub-circuit.
Proof: As in Proposition 1, the serial removal part of the proposition is logically equivalent to the serial attachment part of
the proposition. So we just need to prove the serial removal part of the proposition.
Again, we assume there are k current sources {I1 , ..., Ik }, and l voltage sources {V1 , ..., Vl } in the circuit. Suppose we
are going to remove the j-th resistor edge Rj , and its endpoint pair is {m, n}. Since we are doing a serial removal, it will
merge node m and node n together. Electrically, it is also equivalent to the parallel attachment of a zero resistance edge
to {m, n}. Proposition 1 states that the change of total I 2 R loss resulting from the parallel attachment of a zero resistance
edge to {m, n} is given by ∆PV + ∆PI .

We now state a sequence of corollaries that follow fairly directly from Proposition 1 and 2.
Corollary 1: The change of total I 2 R loss, ∆PV , resulting from the parallel removal of a resistor edge with endpoint
0
pair {m, n} from a voltage-controlled circuit is given by ∆PV = Imn Vmn , where Imn denotes the current flowing on the
0
edge before its removal, and Vmn denotes the voltage difference between node pair {m, n} after its removal.
Proof: At the end of the proof for Proposition 1, we know that ∆PV , the change of I 2 R loss resulting from removing Rj
from the voltage-controlled sub-circuit, is given by
 VI 
s1,mn

 . 
∆PV = Imn V1 · · · Vl  ..  .
sVI
l,mn
Since there are no current sources in a voltage-controlled circuit, we have
 VI 
s1,mn


0


Vmn = V1 · · · Vl  ...  .
sVI
l,mn
0

Thus ∆PV = Imn Vmn .

Corollary 2: The change of total I 2 R loss, ∆PI , resulting from the parallel removal of a resistor edge with endpoint pair
0
{m, n} from a current-controlled circuit is given by ∆PI = −Imn Vmn , where Imn denotes the current flowing on the edge
0
before its removal, and Vmn denotes the voltage difference between node pair {m, n} after its removal.

7

Proof: At the end of the proof for Proposition 1, we know that ∆PI , the change of I 2 R loss resulting from removing Rj
from the current-controlled sub-circuit, is given by
 II 
s1,mn

 . 
∆PI = −Imn I1 · · · Ik  ..  .
sII
k,mn
Since there are no voltage sources in a current-controlled circuit, we have
 II 
s1,mn


0


Vmn = I1 · · · Ik  ...  .
sII
k,mn
0

Thus ∆PI = −Imn Vmn .

Corollary 3: The change of total I 2 R loss, ∆PV , resulting from the serial removal of a resistor edge with endpoint pair
0
{m, n} from a voltage-controlled circuit is given by ∆PV = −Imn Vmn , where Vmn denotes the voltage difference between
0
node pair {m, n} before its removal, and Imn denotes the current flowing through the short circuit edge connecting the
node pair {m, n} after the removal.
Proof: Since we are doing a serial removal, it will merge node m and node n together. Electrically, it is equivalent to the
parallel attachment of a zero resistance edge to {m, n}. Corollary 1 shows that the change of total I 2 R loss, ∆PV , resulting
0
from the parallel removal of a zero resistance edge from {m, n} is Imn Vmn . In other words, it means the parallel attachment
0

of a zero resistance edge to {m, n} will change the total loss by −Imn Vmn .
Corollary 4: The change of total I 2 R loss, ∆PI , resulting from the serial removal of a resistor edge with endpoint pair
0
{m, n} from a current-controlled circuit is given by ∆PI = Imn Vmn , where Vmn denotes the voltage difference between
0
node pair {m, n} before its removal, and Imn denotes the current flowing through the short circuit edge connecting the
node pair {m, n} after the removal.
Proof: Again, the serial removal of a resistor edge from {m, n} is equivalent to the parallel attachment of a zero resistance
edge to {m, n}. Corollary 2 shows that the change of total I 2 R loss, ∆PI , resulting from the parallel removal of a zero
0
resistance edge from {m, n} is −Imn Vmn . In other words, it means the parallel attachment of a zero resistance edge to
0
{m, n} will change the total loss by Imn Vmn .

Corollary 5: The change of total loss, ∆PI , resulting from adding a resistance edge to a current-controlled network is
equivalent to the change of total loss resulting from adding the resistance edge to its associated Norton equivalent circuit.
The change of total loss, ∆PV , resulting from adding a resistance edge to a voltage-controlled network is equivalent to the
change of total loss resulting from adding the resistance edge to its associated Thevenin equivalent circuit.
Proof: Based on Corollary 1 through Corollary 4, we know that in a single-source network the change of total loss resulting
from the attachment (removal) of a resistor edge is completely determined by
•
•

the current flowing through the link after (before) the change, and
the voltage difference between the endpoints of the edge before (after) the change.

In other words, the change of total loss is independent of the changes of edges other than the one to be removed or added.
This good property enables us to use the Thevenin equivalent circuit or the Norton equivalent circuit to predict the change
of total loss in a single-source circuit. Since we know the attachment of a link has opposite effect on the loss of a currentcontrolled circuit and a voltage-controlled circuit, we’d better use Norton equivalent circuit to replace the current-controlled
network and use Thevenin equivalent circuit to replace the voltage-controlled network. Of course, using Thevenin equivalent
circuit to replace a current-controlled network is theoretically acceptable, but it requires an additional step to get the right
answer, i.e. flipping the sign of the change of total loss. So is using Norton equivalent circuit to replace a voltage-controlled
network.

Corollary 6: The change of total loss, ∆P , resulting from adding a resistance edge to node pair {m, n} in a mixed-source
network is equivalent to the change of total loss, ∆Peq , resulting from adding the resistance edge to the equivalent circuit at
terminal m-n in Fig. 7 where Ieq is the current source in the Norton equivalent circuit of the current-controlled sub-circuit,
Veq is the voltage source in the Thevenin equivalent circuit of the voltage-controlled sub-circuit, and Req is the equivalent
resistance.
n

m

Veq

Fig. 7.

The equivalent circuit for a mixed-source network.

Req

I eq

8

Proof: Corollary 5 tells that we can use Norton equivalent circuit to replace the current-controlled sub-circuit and use
Thevenin equivalent circuit to replace the voltage-controlled sub-circuit when calculating the change of total loss in each
sub-circuit for a mixed-source one. Moreover, Proposition 1 and 2 show that the change of total loss in a mixed-source circuit
can be completely decomposed into two parts, one for its voltage-controlled sub-circuit and one for its current-controlled
sub-circuit 4. Thus, we just need to “merge” together the Norton equivalent circuit of the current-controlled sub-circuit and
Thevenin equivalent circuit of the voltage-controlled sub-circuit for the purposed of calculating the change of total loss for
a mixed-source circuit. The idea is visualized in Fig. 7.

Remark 2: Results similar to Corollary 1 through Corollary 4 hold for the removal of multiple resistors together, although
the proof becomes more involved.
Remark 3: A resistor in the circuit is always a passive element as it consumes power. So Imn and Vmn are always of
0
opposite polarity. By Lemma 1, we know Vmn and Vmn are always of same polarity. Thus by Corollary 1 and Corollary
2, we know the parallel removal of a resistor edge from a voltage-controlled circuit will always decrease total I 2 R loss,
and the parallel removal of a resistor edge from a current-controlled circuit will always increase total I 2 R loss. It is worth
1
) connection
mentioning that this is much deeper than the formulas for serial (R = R1 + R2 ) and parallel (R = 1/R1 +1/R
2
of resistors, as the removal of a resistor not only change the resistance of the graph but also redistribute the currents in the
graph. Both the above results are consistent with [1] and [2].
Proposition 3: The change of total losses, ∆P , resulting from the parallel removal (parallel attachment) of a current
source Ij from (to) a mixed-source circuit is given by ∆P = ∆PI , where ∆PI denotes the change of losses resulting from
removing (adding) the link Ij from its current-controlled sub-circuit.
Proof: As in Proposition 1, we just need to prove the parallel removal part of the proposition.
Again, we assume there are k current sources {I1 , ..., Ik }, and l voltage sources {V1 , ..., Vl }, in the circuit. Suppose we
are going to remove the last current source Ik , and its endpoint pair is {m, n}.
By the principle of energy conservation, the change of total I 2 R loss must be equivalent to the change of total sources’
energy output
k−1
l
X
X
Ii ∆Vi − Ik Vk +
Vi ∆Ii
i=1

i=1

where ∆Vi denotes the change of voltage difference between the endpoints of the i-th current source, Vk denotes the voltage
difference between the endpoints of the k-th current source before the removal, and ∆Ii denotes the change of current
flowing through the i-th voltage source.
By the definition of resistance graph, we know putting back all sources will just create some short circuits and open circuits
which essentially don’t change the resistance graph. In addition, the sensitivity of the current flowing through an edge (or
of voltage difference between the endpoints of an edge) is completely determined by the resistance graph, i.e. constant
resistance graph means constant source factors. Thus by the superposition principle, we know Vk is a linear combination of
{I1 , ..., Ik } and {V1 , ..., Vl }, i.e.
 II 
 VI 
s1,k
s1,k

 .  
 . 
Vk = I1 · · · Ik  ..  + V1 · · · Vl  .. 
sII
sVI
k,k
l,k
II
VI
VI
where {sII
1,k , · · · , sk,k } and {s1,k , · · · , sl,k } are source factors. Following a procedure similar to Proposition 1, we have


 II 
sk,1
∆V1
 .. 
 .. 
 .  = −Ik  . 

∆Vk−1
sII
k,k−1


 IV 
sk,1
∆I1
 .. 
 .. 
 .  = −Ik  .  .
∆Il

sIV
k,l

II
IV
VI
By (3), we have sII
k,i = si,k (i = 1, ..., k) and sk,i = −si,k (i = 1, ..., l) which gives us
 II 
 II 
s1,k
s1,k
k−1
l
X
X

 .  
 . 
Ii ∆Vi − Ik Vk +
Vi ∆Ii = −Ik { I1 · · · Ik−1  ..  + I1 · · · Ik  .. }.
i=1
i=1
sII
sII
k−1,k
k,k

As we can see, the above result is independent of the voltage sources in the mixed-source circuit. By setting the value of
all voltage sources to zero, we get the current-controlled sub-circuit. It is easy to prove that ∆PI , the change of I 2 R loss
resulting from removing Ik from the current-controlled sub-circuit, is equivalent to above result.


9

Using the same idea, we can get a similar proposition for the serial removal (serial attachment) of a voltage source from
a mixed-source circuit.
Proposition 4: The change of total losses, ∆P , resulting from the serial removal (serial attachment) of a voltage source
Vj from (to) a mixed-source circuit is given by ∆P = ∆PV , where ∆PV denotes the change of losses resulting from
removing (adding) the link Vj from its voltage-controlled sub-circuit.
Proof: As in Proposition 1, we just need to prove the parallel removal part of the proposition.
Again, we assume there are k current sources {I1 , ..., Ik }, and l voltage sources {V1 , ..., Vl }, in the circuit. Suppose we
are going to remove the last current source Vl , and its endpoint pair is {m, n}.
By the principle of energy conservation, the change of total I 2 R loss must be equivalent to the change of total sources’
energy output
k
l−1
X
X
Ii ∆Vi +
Vi ∆Ii − Vl Il
i=1

i=1

where ∆Vi denotes the change of voltage difference between the endpoints of the i-th current source, Il denotes the current
flowing through the l-th voltage source before the removal, and ∆Ii denotes the change of current flowing through the i-th
voltage source.
Again, by the superposition principle, we know Il is a linear combination of {I1 , ..., Ik } and {V1 , ..., Vl }, i.e.
 VV 
 IV 
s1,l
s1,l

 .  
 . 
Il = I1 · · · Ik  ..  + V1 · · · Vl  .. 
sIV
sVV
k,l
l,l
VV
VV
IV
where {sIV
1,l , · · · , sk,l } and {s1,l , · · · , sl,l } are source factors. Following a procedure similar to Proposition 1, we have


 VI 
sl,1
∆V1
 .. 
 .. 
 .  = −Vl  . 

∆Vk

sVI
l,k


 VV 
sl,1
∆I1
 .. 
 .. 
 .  = −Vl  .  .
∆Il−1
sVV
l,l−1


VI
IV
VV
By (3), we have sVV
l,i = si,l (i = 1, ..., l) and sl,i = −si,l (i = 1, ..., k) which gives us
 VV 
s1,l
l−1
k
X
X


 . 
Vi ∆Ii − Vl Il = −Vl V1 · · · Vl−1  ..  − Vl V1
Ii ∆Vi +
i=1
i=1
sVV
l−1,l

···

 VV 
s1,l
 . 
Vl  ..  .
sVV
l,l

As we can see, the above result is independent of the current sources in the mixed-source circuit. By setting the value of
all current sources to zero, we get the voltage-controlled sub-circuit. It is easy to prove that ∆PV , the change of I 2 R loss
resulting from removing Vl from the voltage-controlled sub-circuit, is equivalent to above result.

By the definition of resistance graph, we know a mixed-source network can be created by the parallel attachment of all
current sources and the serial attachment of all voltages to the resistance graph. Thus, combining the result of Proposition
3 and Proposition 4, we have the following:
Proposition 5: (Superposition of I 2 R loss) The total I 2 R loss, P , of a mixed-source circuit is given by P = PI + PV ,
where PI denotes the total I 2 R loss of its current-controlled sub-circuit, and PV denotes the total I 2 R loss of its voltagecontrolled sub-circuit.
Corollary 7: The change of total I 2 R loss, ∆PI , resulting from the parallel removal of a current source I with endpoint
0
0
pair {m, n} from a current-controlled circuit is given by ∆PI = −I(Vm,n + Vm,n ), where Vm,n and Vm,n denote the voltage
differences between node pair {m, n} before and after its removal, respectively.
Proof: Again, we assume there are k current sources {I1 , ..., Ik } in the circuit. Suppose we are going to remove the last
current source Ik , and its endpoint pair is {m, n}. At the end of the proof of Proposition 3, we know
 II 
 II 
s1,k
s1,k

 . 

 . 
∆PI = −Ik I1 · · · Ik−1  ..  − Ik I1 · · · Ik  ..  .
sII
sII
k−1,k
k,k

10

Also we know



Vm,n = I1


sII
1,k


Ik  ... 
sII
k,k

···

and



0
Vm,n = I1


sII
1,k


Ik−1  ...  .
sII
k−1,k

···

0

Thus ∆PI = −I(Vm,n + Vm,n ).

Corollary 8: The change of total I 2 R loss, ∆PV , resulting from the serial removal of a voltage source V with endpoint
0
pair {m, n} from a voltage-controlled circuit is given by ∆PV = −V(Im,n + Im,n ), where Im,n denotes the current that
0
flowed through the voltage source before the removal, and Imn denotes the current flowing through the short circuit edge
connecting the node pair {m, n} after the removal.
Proof: Again, we assume there are l voltage sources {V1 , ..., Vl }, in the circuit. Suppose we are going to remove the last
current source Vl , and its endpoint pair is {m, n}. At the end of the proof of Proposition 4, we know
 VV 
 VV 
s1,l
s1,l

 . 

 . 
∆PV = −Vl V1 · · · Vl−1  ..  − Vl V1 · · · Vl  ..  .
sVV
sVV
l−1,l
l,l
Also we know

Imn = V1

 VV 
s1,l
 . 
Vl  .. 
sVV
l,l

···

and



0
Imn = V1

···


sVV
1,l


Vl−1  ...  .
sVV
l−1,l

0


Thus ∆PV = −V(Im,n + Im,n ).
Remark 4: Results similar to Corollary 7 through Corollary 8 hold for the removal of multiple sources together, although
the proof becomes more involved.
Remark 5: Proposition 3 and Corollary 7 together explain the reason for the paradoxical behavior in Fig. 5. After we
create the current-controlled sub-circuit of Fig. 5(a), we can compute the voltage difference between the node pair from
which the 0.25A current source is removed. The voltage difference between this node pair is 0V and 0.125V before and
after the removal, respectively. So the total loss will decrease by 0.25(0 + 0.125) = 0.03125W.
Remark 6: (Orthogonality between the effects of voltage sources and current sources) Suppose there are n resistance
edges {R1 , ..., Rn } in the mixed-source circuit, and we denote the currents flowing on the i-th resistance edge as IiM in the
mixed-source circuit, IiV in its voltage-controlled sub-circuit, and IiI in its current-controlled sub-circuit, respectively. The
diagonal matrix of edge resistances is
Λ = diag{R1 , ..., Rn }
Write the vector of currents flowing through each edge as I~M = [I1M , ..., InM ]T in the mixed-source circuit, I~V =
[I1V , ..., InV ]T in the voltage-controlled sub-circuit, and I~I = [I1I , ..., InI ]I in the voltage-controlled sub-circuit, respectively.
Then by the superposition principle, we must have
T
I~M
ΛI~M = (I~VT + I~IT )Λ(I~V + I~I )
= I~T ΛI~V + I~T ΛI~I + 2I~T ΛI~I .
V

I

Combining the above result with Proposition 5, we must have
I~VT ΛI~I = 0.

V

11

IV. F OUR E QUIVALENT L OSS C OMPUTING M ETHODS
Based on the discussion in this paper, we have four different methods to calculate the total I 2 R loss of an arbitrary
mixed-source circuit. To describe and compare them, we assume there are k current sources {I1 , ..., Ik }, l voltage sources
{V1 , ..., Vl }, and t resistors {R1 , ..., Rt } in the circuit.
(a) The first way is the most traditional one. We calculate either the total power output of all voltage sources and current
sources or the total I 2 R loss of all resistors which is given by
k
X
i=1

Ii Vi +

l
X

Vi Ii

(or

i=1

t
X

PR i )

i=1

(4)

s.t. Kirchhof f voltage laws
Kirchhof f current laws
where Vi denotes the voltage difference between the endpoints of the i-th current source before the removal, Ii denotes
the current flowing through the i-th voltage source before the removal, and PRi denotes the I 2 R loss of the i-th resistor.
Ii and Vi are the controlling currents and voltages respectively, they remain constant under the considered topology
reconfiguration.
(b) The second way is based on the following idea: a mixed-source circuit can be created by first constructing the currentcontrolled sub-circuit and then putting back all voltage sources. In [2], we show that the partial derivative of the cost
function PI described by (2) in the direction Iei is exactly double the algebraic sum of all voltages on the fundamental
cycle defined by ei . This means we are including the constraints associated with Kirchhoff’s voltage law and the voltage
sources Vi relative to fundamental cycles in which they appear. The operating point is thus given by minimizing cost
function PI with linear constraints denoting the effect of voltage sources.
min PI
s.t. some linear constraints denoting the ef f ect of voltage sources are satisf ied

(5)

(c) The third way is quite similar to the second way: a mixed-source circuit can be created by first constructing the voltagecontrolled sub-circuit and then putting back all current sources. Following an idea similar to (b), the total loss can be
given by minimizing cost function PV described in (1) with linear constraints denoting the effect of current sources.
min PV
s.t. some linear constraints denoting the ef f ect of current sources are satisf ied

(6)

(d) The fourth way is based on Proposition 5 and is given by
min PV + min PI

(7)

where PV is the potential function (described by (1)) of the voltage-controlled sub-circuit, and PI is the potential
function (described by (2)) of the current-controlled sub-circuit.
Although the four methods are of different mathematical forms, they must give us the same result. Specifically, methods
(b), (c), and (d) are closely related to each other. In mixed source networks, the operating values of voltages and currents
are determined as critical values of PV (as a quadratic function of fundamental node variables) and PI (as a quadratic
function of fundamental cycle variables) where it is assumed that all voltage sources Vi and current sources Ii are present.
This is the approach of methods (b) and (c). This approach can be carried out by solving for critical points of PI (with
respect to the fundamental cycles variables) subject to the Kirchhoff voltage constraints that are obtained by adding the
Vi to the resistance graph. Similarly, one can solve for the critical points of PV (with respect to the fundamental node
variables) subject to the Kirchhoff current constraints by including the Ii to the resistance graph. Method (d) utilizes the
novel decomposition of mixed-source circuit in Proposition 5 and integrates (b) and (c) together. By calculating the loss of
the voltage-controlled sub-circuit PV and the loss of the current-controlled sub-circuit PI individually, method (d) solves
for the loss of a mixed-source circuit in an unconstrained quadratic programming form.
V. C ONCLUSION
Our previous work, [1] and [2], proved that in a single source network, although the detailed change of loss is impossible
to predict without solving the Kirchhoff’s equations, the sign of the overall change in I 2 R loss is always certain. Such
predictability with respect to total loss is not present in a mixed-source network, and this may result in interesting paradoxes
such as fewer sources producing more power. While the demonstrated uncertainty to changes in a mixed-source circuit
together with the well recognized complexity in line switching suggest that active control of grid topology in a mixedsource model is a formidable problem, our results nevertheless offer a clean decomposition for the mixed-source circuit
that completely separates the effect of current sources and voltage sources on network total loss. As the world’s power
grids increasingly embrace novel energy sources and new classes of assets associated with storage and demand response,

12

our ongoing research seeks to use the decomposition concepts we have presented for developing new approaches to
resource allocation that appropriately balance generation scheduling, grid topology configuration, and recruitment of demand
response.
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