This article has been accepted for publication in a future issue of this journal, but has not been

fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2019.2941906, IEEE
Transactions on Power Systems
1

Approximating Economic Dispatch

by Linearizing Transmission Losses
M. Garcia∗, R. Baldick∗

Abstract—We study a non-convex Transmission-Constrained Eco- E := [1, . . . , m] Set of edges in graph G (Transmission lines)
nomic Dispatch (TCED) problem that uses the traditional linear n, m Number of nodes and edges in graph G respectively
DC model of the transmission system together with a non-linear A ∈ Rm×n Branch-bus incidence matrix of the graph G
representation of losses. This problem is typically approximated
by the convex problem obtained by linearizing the constraints B ∈ Rm×m Diagonal matrix of edge weights (susceptances)
around some base-case state. Electricity prices and dispatch deci- H := ȦT BA Reduced weighted Laplacian matrix of graph G
sions are then chosen based on the resulting linearly-constrained S := B ȦH̊ −T = B Ȧ[ÅT B Ȧ]−1 Matrix of shift factors
economic dispatch (LCED) problem. Different LCED problems Constants
have been suggested in the literature and they are all derived D ∈ Rn Nodal demand
using one of two linearization techniques, which we call direct
and indirect linearization, respectively. An LCED problem often ρ ∈ N Angle reference bus
used in practice that uses Loss Distribution Factors (LDFs) is σ ∈ N Price reference bus
derived using indirect linearization and is termed the common F, F̄ ∈ Rm Line limits
¯
LCED problem. This paper studies the assumptions required to P, P̄ ∈ Rm Nodal generation limits
recover the optimal dispatch of the non-convex TCED problem ¯
Optimization Decision Variables
from the solution of the common LCED problem. We show that
the common LCED problem may have multiple minimizers, in P ∈ Rn Nodal generation dispatch
which case small perturbations of the base-case state may result N ∈ Rn Nodal loss allocation (Opt. B)-(Opt. E)
in large dispatch approximation error. Furthermore, even if the θ ∈ Rn Voltage angles (Opt. A)
base-case state matches a minimizer of the non-convex TCED ℓ ∈ R Total system losses (Opt. F)
problem, it is proven that there does not always exist a choice Lagrange Multipliers
of LDFs such that the optimal dispatch of the TCED problem
is also optimal for the common LCED problem. On the other π ∈ R Overall power balance constraint (Opt. B)-(Opt. E)
hand, such LDFs do exist and are identified for the special case µ̄, µ ∈ Rm Line limit constraints (Opt. B)-(Opt. E)
where no line limits are binding. β̄, ¯β ∈ Rn Generator output constraints (Opt. B)-(Opt. E)
N OMENCLATURE γ ∈¯ Rn Nodal loss allocation constraints (Opt. B)-(Opt. C)
γ ′ ∈ Rn Locally optimal Lagrange multipliers for the nodal
General Notation for a Column Vector v
loss allocation constraints (Opt. D)-(Opt. E)
v̇ Vector v with element ρ removed
Prices
v̊ Vector v with element σ removed
λ ∈ Rn Locational Marginal Prices (LMPs)
v0 Base-case value for vector v
e ∈ Rn Energy component of the LMPs
v⋆ Locally optimal value for vector v
l ∈ Rn Loss component of the LMPs
vi Element i of vector v
c ∈ Rn Congestion component of the LMPs
kvk∞ Infinity norm of vector v
Functions
kvk1 One norm of vector v
C : Rn → R Generation cost function
vT Transpose of vector v
L : Rm → Rm Loss function
General Notation for a Matrix M
Ñ : Rn−1 → Rn Nodal loss allocation function
Ṁ Matrix M with column ρ removed
∇Ñ : Rn−1→R(n−1)×n Loss sensitivity matrix valued function
M̊ Matrix M with column σ removed 2
MT Transpose of matrix M Λj :Rn × Rn−1 →Rn Loss constraint function of Opt. j
6 2
M −1 Inverse of matrix M Φj : {Rn } ×{Rm } ×R → R Lagrangian function of Opt. j
M −T Inverse transpose of matrix M Other Notation
Mij Element in row i and column j of matrix M Rd Set of d dimensional real numbers
Mi Column i of matrix M η ∈ Rn Loss Distribution Factors (LDFs)
|M | Element-wise absolute value of matrix M LF ∈ Rn Loss sensitivity vector
General Notation for a Vector Valued Function f q ∈ R Linearization offset
fi (·) Element i of the function f (·) T ∈ Rn Vector of net power injections
∇f (·) Jacobian of f (x) with respect to its arguement j ∈ {B, C, D, E} General reference to Opt. j
∇f˚(·) Jacobian ∇f (·) with column σ removed α ∈ R Scalar assumed to satisfy γ ′ = 1α
∇fi (·) Column i of the Jacobian ∇f (·) δ ∈ Rn−1 Perturbation to ideal base-case state
∇x f (·,y) Partial Jacobian of function f (x,y) with respect to x ∆P ∈ Rn Dispatch error with respect to P ⋆
Transmission System Graph ∆C ∈ R Objective value error with respect to C(P ⋆ )
G := (N, E) Undirected transmission system graph 1, 0 Vector of ones and zeros with appropriate dimension
N := [1, . . . , n] Set of nodes in graph G (Busses/generators) I Identity matrix with√appropriate dimension
i Imaginary number −1
∗ Electrical and Computer Engineering, University of Texas, Austin

0885-8950 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2019.2941906, IEEE
Transactions on Power Systems
2

I. I NTRODUCTION Reference [1] provides a summary of different linearization

techniques implemented by ISOs governed by the Federal
The majority of electricity markets in North America incorpo- Energy Regulatory Commission (FERC), highlighting vari-
rate transmission line losses into economic dispatch problems ations among ISOs in the choice of base-case state. The
used for pricing and dispatch decisions in both real-time and common LCED problem derived in [1] matches that in [3]
day-ahead markets [1]. These markets intend to solve a non- and is a linearized version of the TCED problem from [2],
convex economic dispatch problem that we will refer to as the which accommodates a general non-linear loss function. Many
Transmission-Constrained Economic Dispatch (TCED) prob- recent works have attempted to improve upon this linearly
lem. The TCED problem uses the traditional linear Direct Cur- constrained formulation in different ways. Reference [13]
rent (DC) model of the transmission system together with non- derives a linear pricing technique that does not rely on a choice
linear losses of each transmission line allocated evenly among of base-case state or LDFs. Reference [14] proposes a dispatch
the incident buses as “fictitious nodal demand” [1], [2]. The problem resulting in price components that are reference bus
TCED problem is typically approximated by first linearizing independent. Piece-wise linear representations of losses are
the loss function around some base-case state and then intro- accommodated by [15] and [16] via the load over-satisfaction
ducing loss distribution factors (LDFs). The resulting linearly relaxation, which is accurate under the assumption that prices
constrained economic dispatch (LCED) problem is termed the are positive. Other work has focused on developing more ac-
common LCED problem because it closely represents those curate linear approximations of the AC OPF problem [17–20].
used in practice as reported in [1] and [3]. For the first time, Similar linearization procedures have also been studied in the
this paper characterizes the approximation errors associated context of planning problems [21], [22]. Despite the various
with each of the three assumptions required to accurately linearly constrained formulations suggested in the literature,
recover the optimal dispatch of the non-convex TCED problem the common LCED problem from [1] and [3] remains the
from the solution of the common LCED problem. most commonly used economic dispatch problem in practice.
The economic dispatch problem that most accurately accounts Using the AC OPF problem as a benchmark, [1] illustrates
for losses also captures real/reactive power coupling as well that prices associated with the common LCED problem better
as off-nominal voltage magnitudes and is typically referred capture the marginal effect of losses as compared to economic
to as the Alternating Current Optimal Power Flow (AC OPF) dispatch problems that represent losses as being fixed. How-
problem [4]. The AC OPF problem is not used in electricity ever, using a simple 2-bus example, [1, section V-A] addi-
markets today because it is generally NP-hard [5], [6]. Instead, tionally identifies an issue that is missed by current practice.
economic dispatch problems have traditionally used the linear Specifically, using intuitive choices of base-case state and
DC model of the transmission system, which makes the sim- LDFs, the optimal dispatch of the common LCED problem
plifying assumption that there are no transmission losses [7]. may be far from feasible for the non-convex TCED problem
The resulting economic dispatch problem is termed the lossless from which it is derived. Previous work has addressed similar
DC OPF problem and assumes voltage magnitudes are fixed issues by use of post-processing methods [23], which, in our
to their nominal values, which facilitates the decoupling of context, intend to identify a dispatch that is feasible for the
real and reactive power. This lossless DC OPF problem is TCED problem by slightly altering the optimal dispatch of the
typically augmented with fixed losses allocated throughout common LCED problem. Such post-processing methods do
the network as fictitious nodal demand [8–10]. References not provide optimality guarantees and may identify a costly
[9] and [10] show that introducing a fixed loss representa- dispatch, as is illustrated in [1, section V-A]. To mitigate
tion to the lossless DC OPF problem results in price and this issue, [1] suggests a successive linearization procedure,
dispatch values that more accurately represent those from the which is similar to those from [24] and [25]. This procedure
AC OPF problem, which serves as an intuitive benchmark. is not guaranteed to converge and requires properly tuned
However, fixed loss representations do not accurately capture parameters to encourage convergence. Recognizing that the
the marginal effects of losses as operating conditions vary. issue exhibited by the 2-bus test case in [1] is difficult to
In fact, the real-power loss of a transmission line is more rectify, another contribution of this paper is to identify the
accurately approximated as the product of the line resistance source of this issue. Specifically, this issue occurs if the
and the squared real-power flowing through the line, and this common LCED problem has multiple minimizers when the
representation can capture the variation of marginal losses ideal choice of base-case state is used. (See Remark 11).
with operating conditions [11], [12]. Reference [2] derives
more accurate loss approximations from first principles and Despite the prevalence of the common LCED problem, no
establishes the proper loss allocation to distribute the losses previous work has established a set of assumptions required
of each line evenly among its incident buses. Incorporating to recover the optimal dispatch of the non-convex TCED
these non-linear representations of line losses into the lossless problem from the common LCED problem. To establish such
DC OPF problem results in the TCED problem, which is non- assumptions this paper derives the common LCED problem
convex and generally difficult to solve. To attain a tractable lin- from the non-convex economic dispatch problem adopted
early constrained formulation, the TCED problem is typically from [2] that we term the TCED problem with angles or
approximated via first order Taylor expansions taken around Optimization A (Opt. A). To illustrate the assumptions used
some base-case state. A main contribution of this paper is a to attain the common LCED problem, four intermediate prob-
clear analysis of the implications of such linearizations. lems, namely, Opt. B, C, D, and E, are also derived that are

0885-8950 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2019.2941906, IEEE
Transactions on Power Systems
3

not individually novel but constitute a convenient route for in [1] and [3]. Section III-C derives the intermediate indirect
the derivation. Figure 1 outlines the sequence of problems LCED problem (Opt. D) by applying indirect linearization to
derived to attain the common LCED problem or Opt. F. This the TCED problem without angles (Opt. B). Section III-C es-
derivation ultimately proves that a specific dispatch that is tablishes equivalence of the indirect LCED problem (Opt. D)
optimal for the TCED problem with angles (Opt. A) is also and the direct LCED problem (Opt. C), showing any dispatch
optimal for the common LCED problem (Opt. F) under two that is optimal for one of these problems is also optimal
key assumptions. This specific dispatch can be efficiently for the other. Section III-D then explains how to compute
recovered from the common LCED problem (Opt. F) under the LMPs from Lagrange multipliers of the indirect LCED
the additional assumption that the common LCED problem problem (Opt. D), which differ from those of the direct LCED
has a unique optimal dispatch. problem (Opt. C). We additionally emphasize that the indirect
linearization technique requires loss sensitivities to be com-
Equiv. Non-Convex Equiv. Linear Equiv. Linear puted as the solution of equations with a large Jacobian matrix
Problems Problems Problems w/ LDFs
during each market interval, e.g. every 5-15 minutes for the
Approx.
TCED w/ Indirect
Sec. III-E LDF LCED
real-time market, which may be computationally burdensome.
Angles LCED
γ ′ ≈ 1α Opt. E Section III-E introduces LDFs to the indirect LCED problem
Opt. A Opt. D
Equiv. (Opt. D) to obtain the LDF LCED problem (Opt. E). Section
Equiv. Equiv.
Sec. II-C
Sec. III-C
Sec. III-F III-E additionally explains that under Assumption 1 there do
and III-D
not generally exist LDFs that allow the base-case dispatch to
TCED w/o Direct Common
Angles
θ̇0 ≈ θ̇⋆
LCED LCED
solve the LDF LCED problem. However, under the assumption
Opt. B Approx. Opt. C Opt. F that no line limits are binding such LDFs do exist and are
Sec. III-B
identified in Section III-E. The second key assumption, termed
Fig. 1: Diagram outlining the derivation structure of the paper. Assumption 2, effectively asserts that no line limits are binding.
Vertical connections with arrows pointing in both directions Finally, Section III-F formulates the common LCED problem
indicate equivalence of the two connected problems, meaning (Opt. F) by eliminating the loss allocation variables and
any dispatch that is optimal for one is also optimal for the introducing a single loss variable in their place. The LDF
other. Horizontal connections pointing in only one direction LCED problem (Opt. E) is shown to be equivalent to the
indicate that both problems share an optimal dispatch, namely common LCED problem (Opt. F). Furthermore, the derivation
the base-case dispatch, under the appropriate assumptions. outlined in Figure 1 shows that the base-case dispatch is
optimal for the common LCED problem under Assumptions
The derivation begins in Section II by eliminating voltage 1 and 2. However, Section III-G emphasizes that the common
angles from the TCED problem with angles (Opt. A) yielding LCED problem (Opt. F) may have multiple optimal dispatches,
an ED problem that optimizes over generation dispatch and some of which may not be feasible for the TCED problem with
nodal loss allocation variables. This problem is termed the angles (Opt. A). As a result, it may be difficult to recover an
TCED problem without angles (Opt. B) and is shown to be optimal dispatch of the TCED problem with angles (Opt. A) by
equivalent to the TCED problem with angles (Opt. A), where solving the common LCED problem (Opt. F) using standard
two problems are said to be equivalent if any dispatch that is off-the-shelf optimization software. The third and final key
optimal for one is also optimal for the other. assumption, termed Assumption 3, requires the common LCED
The fundamental linearization theory in [12] uses a first problem (Opt. F) to have a unique optimal dispatch. Under
order Taylor expansion of the constraints with respect to Assumptions 1, 2, and 3 the base-case dispatch, which is
all optimization variables about the base-case state and its optimal for the TCED problem with angles (Opt. A), is
associated base-case dispatch. We call this direct linearization. the unique optimal dispatch of the common LCED problem
Section III-B derives the direct LCED problem (Opt. C) by (Opt. F), and can be easily recovered.
applying direct linearization to the TCED problem without an- Section IV provides numerical results intended to illustrate two
gles (Opt. B). The first key assumption, termed Assumption 1, key findings that point out significant dispatch approximation
requires the base-case dispatch to satisfy the Karush-Kuhn- errors may occur when certain assumptions are violated. The
Tucker (KKT) conditions and represent an optimal dispatch of first key finding states that very small perturbations to the
the TCED problem without angles (Opt. B). A result from [12] ideal base-case state can result in significantly large dispatch
can then be applied to show that the same base-case dispatch approximation error if the common LCED problem (Opt. F)
is optimal for the direct LCED problem (Opt. C). A related has multiple minimizers. This is illustrated in Section IV-A
practical question is how close does the base-case state need using an intuitive 2-bus example as well as a larger more
to be to a minimizer of the underlying non-convex problem for realistic test case. The second key finding states that significant
the approximate LCED problem to result in a good dispatch dispatch approximation error may occur when Assumption 2 is
approximation. (See Remark 4). violated, even if Assumptions 1 and 3 hold. This is illustrated
Another approach is to linearize with respect to only the net in Section IV-B using an intuitive 3-bus example that is highly
power injections using total derivatives of implicit functions. resistive and heavily congested. However, a larger more real-
We call this indirect linearization. In fact, the common LCED istic test case is used to illustrate that violating Assumption 2
problem (Opt. F) is derived using indirect linearization as typically results in insignificant dispatch approximation error.

0885-8950 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2019.2941906, IEEE
Transactions on Power Systems
4

II. T RANSMISSION C ONSTRAINED E CONOMIC D ISPATCH deriving a linearly-constrained economic dispatch problem,
we assume a linear approximation of the mid-line power
We begin by introducing notation. Lower case subscripts are flow function described in detail in [2] and represented by
used to indicate elements of matrices. For example the element H T θ̇ where H := ȦT BA is termed the reduced weighted
in the ith row and j th column of matrix M is denoted Mij . Laplacian matrix and represents the weighted Laplacian ma-
The ith column of matrix M is denoted Mi . The transpose of trix of the underlying system graph with row ρ removed. In
a matrix is denoted with a superscript T, for example M T , and the expression for H, the matrix B ∈ Rm×m is full rank
a superscript −T represents the inverse transpose of a matrix. and diagonal where the diagonal elements represent the edge
The set of n dimensional real numbers is denoted Rn . A vector weights of the underlying system graph and can be interpreted
v ∈ Rn is designated a column vector and its ith element is as transmission line susceptances. Constraint (A2) enforces
denoted vi . The element-wise absolute value of a matrix or generator output limits and constraint (A3) enforces line limits
vector is denoted |M | or |v| respectively. The identity matrix, in the form of mid-line power flow limits.
the matrix of all zeros, and the matrix of all ones are denoted
I, 0, and 1 respectively and are of appropriate dimension. Remark 1. The lossless DC OPF problem is identical to the
Furthermore, the system is modeled as an undirected graph TCED problem with angles (Opt. A) if the loss function L(·) is
G = (N, E) where N is the set of nodes (buses) and E is replaced by the zero vector 0. This lossless DC OPF problem
the set of edges (transmission lines). There are n buses and is linearly constrained, convex, and easy to solve.
m transmission lines. The branch-bus incidence matrix of the
graph G is denoted A ∈ Rm×n . Specifically, A is sparse and B. General Economic Dispatch Problem without Angles
the row representing line k connecting bus i to bus j has
Many references, e.g. [1] and [3], analyze the energy market
element i equal to 1 and element j equal to −1. In this context
with respect to an economic dispatch problem that optimizes
bus i is arbitrarily assigned to be the sending bus and power
over dispatch and loss variables. We will derive four such
flow is designated positive when flowing from bus i to bus j.
problems that optimize over the nodal dispatch vector and a
nodal loss allocation vector N ∈ Rn . These four problems,
A. Transmission-Constrained Economic Dispatch with Angles Opt. B, C, D, and E respectively, can each be expressed as the
The Transmission-Constrained Economic Dispatch (TCED) following general economic dispatch problem without angles.
problem with angles is adopted from [2]. This problem op- The price reference bus (or slack bus) is designated as bus
timizes over the nodal generation dispatch vector P ∈ Rn σ ∈ N. Throughout the paper a ring over a vector represents
and the vector of voltage angles excluding the known angle that vector with element σ removed. Similarly, a ring over a
at the bus ρ ∈ N, which will be termed the angle reference matrix represents that matrix with column σ removed, e.g. Å.
bus. Throughout the paper a dot over a vector represents min C(P ) (Opt. j)
that vector with element ρ removed. For example the vector P ∈Rn ,N ∈Rn

of voltage angles is denoted θ ∈ Rn and the vector of st : 1T (P − D − N ) = 0 (j1)

voltage angles excluding the angle at the angle reference bus
is denoted θ̇ ∈ Rn−1 . Similarly, a dot over a matrix represents Λj (P̊, N ; D̊) = 0 (j2)
that matrix with column ρ removed. For example, the matrix P ≤ P ≤ P̄ (j3)
Ȧ ∈ Rm×(n−1) is equivalent to A with column ρ removed. ¯
The TCED problem with angles (Opt. A) is written as follows. F ≤ S(P̊ − D̊ − N̊ ) ≤ F̄ (j4)
¯
Voltage magnitudes are assumed constant and all equal to
one per unit. The nodal demand is considered fixed and is Constraint (j1) represents power balance. Constraint (j3)
represented by D ∈ Rn . The cost of generation is represented represents generator output limits. Constraint (j4) represents
by the function C(P ) where C : Rn → R is convex. limits on the mid-line power flow on each transmission line
expressed in terms of shift factors S := B ȦH̊ −T . Note
min C(P ) (Opt. A) that the matrix H̊ is invertible under the standard assump-
P ∈Rn ,θ̇∈Rn−1 tion that the system graph is fully connected. The nodal
st : 1 T
|A| L(Ȧθ̇) + H T θ̇ = P − D (A1) loss allocation constraint (j2) incorporates a general function
2
Λj(·, ·; D̊):Rn−1 × Rn→Rn, which takes the constant parameter
P ≤ P ≤ P̄ (A2) D̊ ∈ Rn−1 as an additional argument. This general function
¯
will be defined differently for each of the next four optimiza-
F ≤ B Ȧθ̇ ≤ F̄ (A3)
¯ tion problems formulated in this paper. These four problems
will be differentiated by j taking values B, C, D, and E.
Constraint (A1) represents the power balance at each bus in
the system as presented in [2]. The vector valued loss function Remark 2. Opt. j has one more decision variable and equality
is denoted L : Rm → Rm and maps voltage angle differences constraint than Opt. A. Opt. j is also conveniently formulated
to line losses. This function is assumed to be convex on the because there is one linear overall power balance constraint
domain of the problem and is assumed to be continuously and all non-convexity is concentrated to constraint (j2).
differentiable. Half of each line’s losses are assigned to its The Lagrangian function is central to the First Order Necessary
incident buses as fictitious nodal demand. In the interest of Conditions (FONCs) for optimality. The Lagrangian function

0885-8950 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2019.2941906, IEEE
Transactions on Power Systems
5

of Opt. j is written as follows. The arguments of this function (Opt. A). Each step taken in the reformulation preserves the set
are partitioned into three categories, namely: primal variables, of feasible dispatch variables P , implying any dispatch that is
dual variables, and constant parameters. An equation number optimal for one of these problems is also optimal for the other.
is assigned to each term for future reference. Both problems additionally have the same optimal objective
value and as a result the sensitivity of the optimal objective
Φj (N, P ; π, γ, µ̄, µ, β̄, β; D) := (1) value with respect to D is the same for both problems under
¯ ¯ the assumption that the sensitivity is well defined.
C(P ) (1a)
+ π1T (−P + D + N ) (1b) 1) Loss Allocation Vector: We first introduce the nodal loss
allocation vector N ∈ Rn as a decision variable in Opt. A
+ γ T Λj (P̊, N ; D̊) (1c) along with the constraint:
T
+ β (P − P ) (1d) 1 T
¯ ¯ N= 2 |A| L(Ȧθ̇). (7)
+ β̄ T (P − P̄ ) (1e)
  2) Power Balance Constraint: By (7), we can replace the first
+ µ̄T S(P̊ − D̊ − N̊ ) − F̄ (1f) term in power balance constraint (A1) by N to obtain:
 
+ µT F − S(P̊ − D̊ − N̊ ) . (1g) N + H T θ̇ = P − D. (8)
¯ ¯
Note that H1 = 0, left multiply the left- and right-hand side
Here the Lagrangian function is defined with: π ∈ R repre-
of (8) by the full rank matrix [1, ˚
I]T and re-arrange to obtain:
senting the Lagrange multiplier of the overall power balance
constraint (j1); γ ∈ Rn representing the Lagrange multipliers 0 = 1T (P − D − N ), (9)
of the nodal loss allocation constraint (j2); (β̄, β) ∈ Rn × Rn T
H̊ θ̇ = P̊ − D̊ − N̊. (10)
¯
representing the Lagrange multipliers of the generator output
m m
constraint (j3); and (µ̄, µ) ∈ R × R representing the Since [1, ˚
I]T is full rank, constraints (9) and (10) hold if and
Lagrange multipliers of the¯ line limit constraint (j4). only if constraint (8) holds. Since H̊ is invertible, constraint
(10) can be re-arranged to:
The KKT conditions for some pair of primal variables (P, N )
require the existence of corresponding Lagrange multipliers θ̇ = H̊ −T (P̊ − D̊ − N̊ ). (11)
that jointly satisfy the primal feasibility, dual feasibility, com-
plementary slackness, and stationarity conditions [26]. Primal 3) Eliminate Voltage Angles: In summary, we first introduced
feasibility requires the primal variables to satisfy constraint constraint (7) along with variable N . We then replaced the
(j1)-(j4). Dual feasibility requires the dual variables asso- power balance constraint (A1) with equivalent constraints (11)
ciated with inequality constraints β̄, β, µ̄ and µ to be non- and (9). We now substitute the expression for θ̇ from constraint
¯
negative. Complementary slackness requires each¯ of the terms (11) into constraints (7) and (A3). The resulting optimization
(1d)-(1g) to equate to zero. The stationarity condition requires problem is equivalent to Opt. B but with additional constraint
the partial derivative of the Lagrangian function to be zero (11). Since θ̇ is otherwise unconstrained, constraint (11) can
with respect to the primal variables P and N . The stationarity be removed resulting in Opt. B.
condition is as follows, where the partial derivatives of Φj Definition 1. Let the generation dispatch P ⋆ and the nodal
with respect to Nσ , Pσ , N̊ , and P̊ are represented by (2), (3), loss allocation N ⋆ represent a local minimizer of the TCED
(4), and (5) respectively: problem without angles (Opt. B) with associated Lagrange
multipliers π ⋆, γ ⋆, µ̄⋆, µ⋆, β̄ ⋆, and β ⋆ that solve the KKT con-
0 = π + ∇Nσ Λj (P̊, N ; D̊)γ, (2) ditions for the TCED ¯problem without ¯ angles (Opt. B) under
0 = ∇Pσ C(P ) − π + β̄σ − βσ , (3) the assumption that such a local minimizer exists.
¯
0 = π1 + ∇N̊ Λj (P̊, N ; D̊)γ + S T (µ − µ̄), (4)
¯
˚ β.
0 = ∇P̊ C(P)−π1 + ∇P̊ Λj (P̊,N ; D̊)γ − S T (µ−µ̄)+ β̄−˚ (5) D. Locational Marginal Prices
¯ ¯
The Locational Marginal Prices (LMPs), denoted λ, are de-
C.Transmission-Constrained Economic Dispatch withoutAngles fined to be the partial derivative of the Lagrangian function
By choosing the function Λj appropriately Opt. j can be made with respect to the demand vector D:
equivalent to the TCED problem with angles (Opt. A) in that
λ := ∇D ΦB (N ⋆ , P ⋆ ; π ⋆ , γ ⋆ , µ̄⋆ , µ⋆ , β̄ ⋆ , β ⋆ ; D). (12)
any optimal dispatch for one of these problems is also optimal ¯ ¯
for the other. In particular, Sections II-C1-II-C3 show that Opt. where ΦB represents the Lagrangian function (1) for the
A is equivalent to the following TCED problem without angles. TCED problem without angles (Opt. B). The LMP can be
Optimization B. The TCED problem without angles is de- decomposed as λ := e+l+c, where e is the energy component
fined to be Opt. j with Λj specified
 by:  associated with Lagrange multiplier π, l is the loss component
T
ΛB (P̊, N ; D̊):=N − 12 |A| L ȦH̊ −T (P̊ − D̊ − N̊) . (6) associated with Lagrange multipliers γ, and c is the congestion
component associated with Lagrange multipliers (µ̄, µ). The
Sections II-C1 through II-C3 derive the TCED problem with- congestion and loss components of the LMP at the ¯ price
out angles (Opt. B) from the TCED problem with angles reference bus are zero, that is cσ := 0 and lσ := 0. The

0885-8950 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2019.2941906, IEEE
Transactions on Power Systems
6

remaining LMP components are explicitly written as follows: In the real-time market the base-case state is often chosen to
match the output of the state estimator. Similarly, in the day
e :=π ⋆ 1, (13) ahead market the base-case state is often chosen by predicting
˚
l := 1 H̊ −1 ȦT ∇L(Ȧθ̇⋆ ) |A| γ ⋆ , (14) the future system state based on historical data. However, other
2
T ⋆ ⋆ choices of base-case state are used in practice. For example,
c̊ :=S (µ − µ̄ ). (15)
¯ another common method constructs a base-case state from
the optimal dispatch of an alternative form of the economic
where θ̇ = H̊ (P̊ − D̊ − N̊ ⋆ ).
⋆ −T ⋆
dispatch problem [1]. A simple example of an alternative
Remark 3. The loss and congestion components of the LMP form of the economic dispatch problem is the lossless DC
are zero at the price reference bus highlighting the dependency OPF problem. Note that previous work has concluded that a
of these components of the LMP on the choice of price minimizer of the lossless DC OPF problem can be used as the
reference bus. On the other hand, the LMP is not dependent base-case to effectively approximate losses [27].
on the choice of price reference bus as stated in [2].

B. Direct Linearization
III. L INEARLY C ONSTRAINED E CONOMIC D ISPATCH
Reference [12] suggests directly linearizing the nodal loss al-
The economic dispatch problems formulated in the previous location constraint ΛB (P̊, N ; D̊) = 0 with respect to decision
section are non-convex and may be difficult to solve. For this variables P̊ and N using a first order Taylor expansion. The
reason it is typical for ISOs to use approximations that result in gradient of the Left-Hand Side (LHS) with respect to P̊ is
a convex problem. Today ISOs use linearization techniques to − 21 H̊ −1 ȦT ∇L(Ȧθ̇0 )|A| and the gradient of the LHS with
simplify non-convex economic dispatch problems into convex respect to N̊ is I + 12 H̊ −1 ȦT ∇L(Ȧθ̇0 )|A|. With this in mind,
linearly-constrained problems. In this section we present a the direct LCED problem is defined as follows.
sequence of four linearly-constrained economic dispatch prob- Optimization C. The direct LCED problem is defined to be
lems, respectively, Opt. C, D, E, and F, ultimately resulting in Opt. j with Λj specified by:
the common LCED problem (Opt. F) used in [1] and [3].  
Presenting the sequence of problems highlights the nature of T
ΛC (P̊, N ; D̊):=N − 21 |A| L ȦH̊ −T (P̊ 0 −D̊−N̊ 0)
the approximations used in practice.
Section III-A describes linearization about a base-case state, − 21 |A|T ∇LT (Ȧθ̇0 )ȦH̊ −T ((P̊ − P̊ 0 )−(N̊ −N̊ 0)). (16)
which is fundamental to all of the linearization approaches.
Section III-B derives the direct LCED problem (Opt. C), which Note that with the specification of ΛC as in (16), Opt. C has
serves as an approximation to the TCED problem without linear constraints. With cost function C(·) assumed convex,
angles (Opt. B) as explained by a result from [12]. Section Opt. C is a convex problem. If the cost function C(·) is linear,
III-C introduces the indirect LCED problem (Opt. D) and then Opt. C is a linear program.
proves that it is equivalent to Opt. C. Section III-D then The following assumption provides conditions under which
explains how to compute the loss component of the LMP from an optimal dispatch of the TCED problem without angles
Lagrange multipliers of the indirect LCED problem (Opt. D). (Opt. B) is also optimal for the direct LCED problem (Opt. C).
LDFs are then introduced to the indirect LCED problem in
Assumption 1. The base-case dispatch and nodal loss alloca-
Section III-E to obtain the LDF LCED problem (Opt. E). This
tion vectors (P 0,N 0) represent a local minimizer of the TCED
problem is shown to be a good approximation of the indirect
problem without angles (Opt. B) and solve the KKT conditions
LCED problem under the additional assumption that no line
for the TCED problem without angles (Opt. B) along with some
limits are binding. The common LCED problem (Opt. F) is
Lagrange multipliers π 0, γ 0, µ̄0, µ0, β̄ 0, and β 0.
derived in Section III-F and is shown to be equivalent to the ¯ ¯
LDF LCED problem (Opt. E). Section III-G then introduces We will employ this assumption for the remainder of this
a uniqueness assumption required to efficiently recover an section. Notice that the LMPs are well defined under As-
optimal dispatch of the TCED problem with angles (Opt. A). sumption 1 and are expressed as in Section II-D. Under
Assumption 1 reference [12] proves that (P 0 , N 0 ) along with
Lagrange multipliers π 0 , γ 0 , µ̄0 , µ0 , β̄ 0 , β 0 satisfy the KKT
A. Linearization about Base-Case State conditions for the direct LCED problem ¯ ¯(Opt. C), implying
0
The linearization procedure will use a first order approx- that the base-case dispatch P is optimal for the direct LCED
imation around a base-case state θ̇0 . Associated with the problem, which is a convex program with linear constraints as
base-case state are the base-case nodal loss allocation vector mentioned above. This result is easy to verify by noting that
N 0 := 21 |A|T L(Ȧθ̇0 ) and the base-case generation dispatch these values satisfy the stationarity condition and the primal
feasibility condition for both problems. Additionally note that
vector P 0 := H T θ̇0 + N 0 + D. Note that by construction the
the dual feasibility condition and complementary slackness
base-case values θ̇0 and P 0 satisfy all equality constraints in
conditions are identical for both problems.
Opt. A and the base-case values N 0 and P 0 satisfy all equality
constraints in Opt. B.

0885-8950 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2019.2941906, IEEE
Transactions on Power Systems
7

Remark 4. A base-case state θ̇0 satisfying Assumption 1 Proof: Differentiating all rows of (17) excluding the price
is termed an ideal base-case state. However, in practice reference bus around the point T̊ 0 yields the following:
Assumption 1 will only be approximately true and the base- ˚(T̊ 0)H̊ −1ȦT ∇L(Ȧθ̇0)|Å|.
˚(T̊ 0)= 1 H̊ −1ȦT ∇L(Ȧθ̇0)|Å|− 1 ∇Ñ
case values (P 0 , N 0 ) will fall in the vicinity of a minimizer ∇Ñ 2 2
of the TCED problem without angles (Opt. B). Section IV Algebraic manipulation yields the following and the result then
explains the conditions under which this is a reasonable follows by the assumption on invertibility:
approximation. Specifically, Section IV-A studies test cases
˚(T̊ 0 )(I+ 1 H̊ −1ȦT ∇L(Ȧθ̇0 )|Å|)= 1 H̊ −1ȦT ∇L(Ȧθ̇0 )|Å|.
∇Ñ
where this approximation does not work well and Section IV-B 2 2
studies test cases where this approximation does work well.
Differentiating the row of (17) corresponding to the price
reference bus yields the expression for ∇Ñσ (T̊ 0 ).
C. Indirect Linearization
D. Equivalent Loss Component of LMP
In this section we take another approach, similar to that of [1]
and [3] by deriving loss sensitivities with respect to nodal The expression for the loss component of the LMP in terms
net power injections about the base-case state. To do this of the Lagrange multipliers of the indirect LCED problem
we must first define the nodal loss allocation as an implicit (Opt. D) evaluated at the base-case point (P 0, N 0) is as follows:
vector valued function of the net power injections at the non- ˚
l = ∇D̊ ΛD (P̊ 0 , N 0 ; D̊)γ ′ = ∇Ñ(P̊ 0 −D̊)γ ′ , (19)
price reference buses Ñ : Rn−1 → Rn . The vector of net
power injections is denoted T ∈ Rn and is interpreted as the where lσ = 0 and (P 0 , N 0 ) along with Lagrange multipliers
generation dispatch vector less the demand vector P − D. The π 0 , γ ′ , µ̄0 , µ0 , β̄ 0 , β 0 satisfy the KKT conditions for the
indirect LCED ¯ problem ¯ (Opt. D). Of course this expression
function Ñ is defined implicitly by the relationship between N
and P̊ − D̊ in the constraint ΛB (P̊, N ; D̊) = 0 from the TCED is slightly different than the expression given for the loss
problem without angles (Opt. B), and therefore satisfies: component of the LMP from (14). This is because the La-
grange multipliers γ ′ corresponding to the indirect LCED
˚(T̊ )) .
 
Ñ (T̊ ) := 1 |A|T L ȦH̊ −T (T̊ − Ñ
2 (17) problem (Opt. D) do not match the Lagrange multipliers γ 0
corresponding to the direct LCED problem (Opt. C). For the
The loss sensitivity matrix is denoted ∇Ñ : Rn−1 →R(n−1)×n remainder of the paper γ ′ will denote the optimal Lagrange
and is defined to be the Jacobian of Ñ with respect to multiplier for the constraint ΛD (P̊, N ; D̊) = 0 of the indirect
its argument. The nodal loss allocation vector can then be LCED problem (Opt. D).
approximated using a simple first order Taylor expansion of the
function Ñ (T̊ ) evaluated at the base-case net power injections The following theorem derives a relationship between the
T̊ 0 := P̊ 0 −D̊. The indirect LCED problem is as follows. Lagrange multipliers of the direct LCED problem (Opt. C) and
the indirect LCED problem (Opt. D). By Theorem 2 below,
Optimization D. The indirect LCED problem is defined to be the KKT point from Assumption 1, with γ 0 replaced by γ ′ ,
Opt. j with Λj specified by: solves the KKT conditions for the indirect LCED problem
ΛD (P̊, N ; D̊):=N−Ñ (P̊ 0−D̊)−∇Ñ (P̊ 0−D̊)T (P̊ − P̊ 0 ).(18) (Opt. D). As a result, the base-case dispatch P 0 is optimal
for the indirect LCED problem (Opt. D). In addition, the
We derive the loss sensitivity matrix ∇Ñ in Theorem 1 below. expressions of the loss component of the LMP from (14) can
Theorem 1 allows us to prove equivalence of linear constraints be obtained from the expression (19) by substituting γ ′ and
ΛC (P̊, N ; D̊) = 0 and ΛD (P̊, N ; D̊) = 0. Specifically, these ∇Ñ(P̊ 0 − D̊) using the expressions from Theorems 1 and 2.
constraints are equivalent in that one constraint holds if and Theorem 2. Let (P 0 , N 0 ) along with Lagrange multipliers
only if the other holds. This can be proven algebraically by π 0, γ 0, µ̄0, µ0, β̄ 0, β 0 satisfy the KKT conditions for the direct
substituting the expression for the loss sensitivity matrix, ∇Ñ , LCED problem ¯ (Opt. ¯ C). Then (P 0, N 0 ) along with Lagrange
from Theorem 1 into the expression for ΛD (P̊, N ; D̊) from multipliers π , γ , µ̄0 , µ0 , β̄ 0 , β 0 satisfy the KKT conditions
0 ′
(18). As a result, the direct LCED problem (Opt. C) and the for the indirect LCED problem ¯ ¯(Opt. D) where γ ′ = γ 0 and
σ σ
indirect LCED problem (Opt. D) are equivalent in that any
optimal dispatch of one of these problems is also optimal γ̊ ′=(I+ 21 H̊ −1ȦT∇L(Ȧθ̇0 )|Å|)γ̊ 0+ 21 H̊ −1ȦT∇L(Ȧθ̇0 )|Aσ|γσ0 .
for the other. Also notice that computing the loss sensitivity
matrix, ∇Ñ , requires the inversion of a large Jacobian matrix. Proof: Notice that the primal feasibility, dual feasibility, and
complementary slackness conditions do not include γ and are
Theorem 1. Assuming the matrix I+ 12 H̊ −1ȦT ∇L(Ȧθ̇0)|Å| is identical for Opt. C and Opt. D. Therefore, by hypothesis these
invertible, the columns of the loss sensitivity matrix evaluated conditions also hold for Opt. D. It remains to show that the
at T̊ 0 corresponding to the non-price reference buses are stationarity conditions (2)-(5) hold for Opt. D. Notice that the
expressed as: stationarity conditions for Opt. C and Opt. D differ only by
˚ T̊ 0)= 1 H̊ −1ȦT∇L(Ȧθ̇0 )|Å|(I+ 1 H̊ −1ȦT∇L(Ȧθ̇0)|Å|)−1.
∇Ñ( their loss terms and so we need only show equivalence of the
2 2
partial derivatives of the loss terms for both problems.
The column of the loss sensitivity matrix evaluated at T̊ 0 1) Stationarity Condition w.r.t. Nσ : For Opt. D we have
corresponding to the price reference bus is expressed as: ∇Nσ ΛD (P̊ 0 , N 0 ; D̊)γ ′ = γσ′ . Replacing γσ′ with its given ex-
˚(T̊ 0 ))H̊ −1 ȦT ∇L(Ȧθ̇0 )|A |.
∇Ñ (T̊ 0 ) = 1 (I − ∇Ñ pression yields γσ0 =∇Nσ ΛC (P̊ 0 , N 0 ; D̊)γ 0, matching Opt. C.
σ 2 σ

0885-8950 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2019.2941906, IEEE
Transactions on Power Systems
8

2) Stationarity Condition w.r.t. N̊ : For Opt. D we have ∇P̊ ΛD (P̊, N ; D̊)γ ′ = −∇Ñ (P̊ 0− D̊)γ ′ . A sufficient condition
∇N̊ ΛD (P̊ 0 , N 0 ; D̊)γ ′ = γ̊ ′ . Substituting the given expression for the base-case dispatch to be optimal for the LDF LCED
for γ̊ ′ results in ∇N̊ ΛC (P̊ 0 , N 0 ; D̊)γ 0 , matching Opt. C. problem (Opt. E) requires that γ ′ be proportional to the vector
3) Stationarity Condition w.r.t. P̊ : For Opt. D we have of ones as embodied in Assumption 2 below. We will employ
Assumption 2 for the remainder of this section and we will
˚(P̊ 0− D̊)γ̊ ′−∇Ñ (P̊ 0−D̊)γ ′ .
∇P̊ ΛD (P̊ 0 , N 0 ; D̊)γ ′ =−∇Ñ consider some test cases for which it is also satisfied; however,
σ σ
˚ it is important to note that this assumption is not generally true.
Substituting the expression for ∇Ñ (P̊ 0−D̊) from Theorem 1,
substituting the given expressions for γ̊ ′ and γσ′ , and rearrang- Assumption 2. The optimal Lagrange multipliers γ ′ associ-
ing the expression algebraically results in the following: ated with the indirect LCED problem (Opt. D) are uniform.
E.g. there exists some α ∈ R such that γ ′ = 1α.
∇P̊ ΛC (P̊ 0 , N 0 ; D̊)γ 0 = − 21 H̊ −1 ȦT ∇L(Ȧθ̇0 )|Å|γ̊ 0
To understand Assumption 2 it is useful to analyze the
− 1 H̊ −1 ȦT ∇L(Ȧθ˙0 )|A |γ 0 ,
2 σ σ
stationarity condition for the indirect LCED problem (Opt. D)
again matching Opt. C. with respect to N as defined by (2) and (4), which is as
follows:
E. Loss Distribution Factor LCED Problem
π 0 + γσ′ = 0 and 1π 0 + γ̊ ′ + S T (µ0 − µ̄0 ) = 0
To reduce the number of optimization variables many ref- ¯
This shows that the Lagrange multipliers γ ′ = −1π 0 − c are
erences, including [1] and [3], introduce Loss Distribution
indeed uniform if and only if the congestion component of
Factors (LDFs) in the form of a vector η ∈ Rn where 1T η = 1
the LMP c is zero as defined in Section II-D, in which case
and typically η ∈ Rn+ . Each LDF represents the fraction of total
γ ′ = −1π 0 . In turn, the congestion component of the LMP
system losses allocated to node i. Recognizing that the total
is zero if no line limits from constraint (j4) are binding at
system losses are represented by the function 1T Ñ (P̊ − D̊),
the base-case values (P 0 , N 0 ), leading to the statement in
the LDF LCED problem is defined as follows.
Remark 6. This is because non-binding line limits require that
Optimization E. The LDF LCED problem is defined to be µ0 = µ̄0 = 0 by the complementary slackness condition.
Opt. j with Λj specified by: ¯
  Remark 6. Assumption 2 holds if no line limits from con-
ΛE(P̊,N; D̊)=N−η 1TÑ (P̊ 0−D̊)+1T∇Ñ (P̊ 0−D̊)T(P̊ −P̊ 0) .(20) straint (j4) are binding at the base-case values (P 0 , N 0 ).
The following theorem states that the base-case dispatch
References [1] and [3] only provide intuitive choices of LDFs P 0 is optimal for the LDF LCED problem (Opt. E) under
that are not proven optimal. In fact, there may not exist LDFs Assumptions 1 and 2.
that allow the base-case dispatch to be optimal for the LDF
LCED problem (Opt. E). To see this notice that the constraint Theorem 3. Let (P 0 , N 0 ) along with Lagrange multipliers
ΛE (P̊,N; D̊) = 0 reduces to N 0 = η1T N 0 when evaluated π 0 , γ ′ , µ̄0 , µ0 , β̄ 0 , β 0 satisfy the KKT conditions for the
indirect LCED ¯ problem ¯ (Opt. D) where γ ′ = 1α. Then
at the base-case values (P 0 , N 0 ). For this constraint to be
0 0
satisfied we must have the following choice of LDFs: (P , N ) along with the same Lagrange multipliers also satisfy
the KKT conditions for the LDF LCED problem (Opt. E).
1
η := 1T N 0
N 0. (21)
Proof: Notice that the primal feasibility, dual feasibility and
Remark 5. References [1] and [3] suggest the use of LDFs complementary slackness conditions are identical for Opt. D
from (21). However, these references do not prove if or when and Opt. E and by hypothesis these conditions therefore hold
such LDFs are optimal. Section IV-B illustrates that the LDFs for Opt. E. It remains to show that the stationarity conditions
from (21) are not always optimal but typically result in very (2)-(5) hold for Opt. E. Notice that the stationarity conditions
small approximation error. for Opt. D and Opt. E differ only by their loss terms and so
Unfortunately, the LDFs in (21) do not generally allow the we need only show equivalence of the partial derivatives of
stationarity condition for the LDF LCED problem (Opt. E) the loss terms for both problems. First, the partial derivatives
to be satisfied at the KKT point from Assumption 1. To see with respect to N and Pσ are equivalent for both problems:
this notice that the stationarity conditions for the LDF LCED
∇N Λj (P̊ 0 , N 0 ; D̊)1α = 1α and ∇Pσ Λj (P̊ 0 , N 0 ; D̊)1α = 0.
problem (Opt. E) and the indirect LCED problem (Opt. D)
differ only by their loss terms. In order for these problems Furthermore, the partial derivative with respect to P̊ is equiv-
to be equivalent, the partial derivative of the loss term in the alent for both problems. This is shown as follows:
Lagrangian function from Opt. E with respect to P̊ must be
equivalent to that of Opt. D. This partial derivative is written ∇P̊ ΛE (P̊ 0 , N 0 ; D̊)1α = −∇Ñ (P̊ 0−D̊)1( 1T1N 0 N 0 )T 1α,
as follows: = − ∇Ñ (P̊ 0− D̊)1α.
∇P̊ ΛE (P̊, N ; D̊)γ ′ =−∇Ñ (P̊ 0−D̊)1η T γ ′ ,
=−∇Ñ (P̊ 0−D̊)1(1T1N 0 N 0)Tγ ′ . (22) F. Common LCED Problem
In general this partial derivative does not match that of the The commonly formulated economic dispatch problem repre-
indirect LCED problem (Opt. D), which is expressed as sents the total system losses as a single decision variable ℓ ∈ R

0885-8950 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2019.2941906, IEEE
Transactions on Power Systems
9

as opposed to representing the loss allocation to each node sensitivity matrix LF and the offset constant q to be constant
as individual decision variables. This effectively eliminates parameters independent of the demand vector D. This may
n − 1 decision variables and n − 1 constraints, making the be counter-intuitive because LF and q are indeed (implicitly)
problem much easier to solve. Using notation similar to [1], defined in terms of the demand vector D.
the common LCED problem (Opt. F) is written as follows:

min C(P ) (Opt. F) G. Recovering a Locally Optimal Dispatch

P ∈Rn ,ℓ∈R
Recall that, by Assumption 1, the base-case values (P 0 , N 0 )
st : 1T (P − D − ηℓ) = 0 (F1) represent a local minimizer of the TCED problem without
ℓ = LF T (P − D) + q (F2) angles (Opt. B). Thus far we have shown that the base-
case generation dispatch P 0 is locally optimal for the TCED
P ≤ P ≤ P̄ (F3) problem with angles (Opt. A) and is globally optimal for the
¯
F ≤ S(P̊ − D̊ − η̊ℓ) ≤ F̄ (F4) common LCED problem (Opt. F) under Assumptions 1 and 2.
¯ However, as mentioned in the introduction, the common LCED
This formulation is similar to the general economic dispatch problem (Opt. F) may have multiple optimal dispatch vectors,
problem without angles (Opt. j). In fact, the objective function some of which may not be optimal or even feasible for
and the generation output limit constraint are identical to the TCED problem with angles (Opt. A). This is an issue
those of Opt. j. The power balance constraint (F1) and because standard off-the-shelf optimization software used to
line limit constraint (F4) represent constraints (j1) and (j4) solve the common LCED problem (Opt. F) only identifies one
after replacing the nodal loss allocation vector N with the of potentially multiple optimal dispatch vectors. An additional
expression ηℓ. Constraint (F2) represents the total system assumption is required to guarantee the identified dispatch
losses where q ∈ R is an offset constant and LF ∈ Rn matches the base-case dispatch P 0 .
represents the sensitivity of the total system losses with respect Assumption 3. The common LCED problem (Opt. F) has a
to the net power injections. Notice this formulation enforces unique minimizer (P, ℓ).
transmission line limits despite the use of Assumption 2, which
effectively assumes no line limits are binding. Remark 8. The common LCED problem (Opt. F) is said to
have a unique optimal dispatch if Assumption 3 holds true.
References [1] and [3] observe that different approximations Similarly, the common LCED problem (Opt. F) is said to
of LF can be used and they derive three different versions of have multiple minimizers or multiple optimal dispatch vectors
LF using total derivatives and implicit functions. With respect if Assumption 3 does not hold true.
to our formulation the loss sensitivity vector is defined based
on the loss sensitivity matrix as LF ˚ = ∇Ñ (P̊ 0−D̊)1 and Assumption 3 ensures that the common LCED problem has
LFσ = 0. Our formulation also uses a constant offset term of a unique optimal dispatch. Under Assumptions 1, 2, and 3
q = 1T Ñ (P̊ 0−D̊) + 1T ∇Ñ (P̊ 0−D̊)T (D̊ − P̊ 0 ). Using these the unique optimal dispatch for the common LCED problem
definitions of LF and q the loss constraint (F2) is as follows: (Opt. F) is indeed locally optimal for the TCED problem with
angles (Opt. A) and represents the base-case dispatch P 0 .
ℓ =1T Ñ (P̊ 0−D̊)+1T∇Ñ (P̊ 0−D̊)T(P̊ −P̊ 0). (23) Under these three assumptions a locally optimal dispatch for
the TCED problem with angles (Opt. A) can be identified by
Using this proposed definition of the loss sensitivity vector solving the common LCED problem (Opt. F) using standard
LF and offset constant q, it should now be apparent that the off-the-shelf optimization software. The numerical results sec-
common LCED problem (Opt. F) is equivalent to the LDF tion investigates errors associated with relaxing each of these
LCED problem (Opt. E) in that any optimal dispatch of one three assumptions.
problem is also optimal for the other. This is because both
problems have the same feasible set of dispatch variables P . IV. N UMERICAL R ESULTS
In fact, a feasible solution of the common LCED problem
This section provides numerical results intended to illus-
(P, ℓ) can be constructed from any feasible solution of the
trate two key findings that point out significant dispatch
LDF LCED problem (P, N ) where ℓ = 1T N . Similarly, a
approximation errors may occur when certain assumptions
feasible solution of the LDF LCED problem (P, N ) can be
are violated. Section IV-A illustrates the first key finding
constructed from any feasible solution of the common LCED
that small perturbations to the ideal base-case state can
problem (P, ℓ) where N = ηℓ.
result in significantly large dispatch approximation error if
Remark 7. Although we do not prove this explicitly, the Assumption 3 fails to hold. Associated errors are illustrated
base-case values P 0 and ℓ0 = 1T N 0 indeed satisfy the KKT using an intuitive 2-bus test case as well as a realistically
conditions of the common LCED problem (Opt. F) along with large test case with 2383 buses, neither of which enforce
Lagrange multipliers π 0 , α, µ̄0 , µ0 , β̄ 0 , β 0 under Assumptions line limits so that Assumption 2 holds. Section IV-B then
1 and 2. Furthermore, in agreement ¯ ¯ (19) the loss price
with illustrates the second key finding that significant dispatch
can now be expressed as l = LF α. It is interesting to note that approximation error may occur when Assumption 2 fails to
this expression of the loss price can also be derived from the hold even if Assumptions 1 and 3 hold. Associated errors are
common LCED problem (Opt. F) when considering the loss illustrated using an intuitive 3-bus test case as well as the same

0885-8950 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2019.2941906, IEEE
Transactions on Power Systems
10

2383 bus test case as in Section IV-A but with line limits
enforced. In this specific example, introducing transmission 100
line limits to the 2383 bus test case causes the common LCED
problem (Opt. F) to have a unique optimal dispatch and we 95
show that there is no dispatch approximation error despite
90
the presence of transmission congestion. This suggests that
realistic systems may not experience significant approximation 85
error when Assumption 2 fails to hold. Section IV-B continues
to select different base-case states that introduce dispatch 80
approximation error into the 2383 bus test case when enforcing 75
line limits. Since the uniqueness Assumption 3 holds for this
specific test case, small perturbations to the ideal base-case 70
state result in little dispatch approximation error.
65
In this section the k th diagonal element of B is Bkk = x1k
and the k th element of the vector valued loss function is 60
Lk (Θ) = xrk2 Θ2k where the impedance of line k ∈ E is rk +ixk
k 55
and Θ ∈ Rm represents the voltage angle difference across
each transmission line Ȧθ̇. The LDFs η are chosen as in (21). 0 10 20 30 40 50 60
A computer with a 2.0 GHz processor is used.
Fig. 3: Illustration of the feasible set of dispatch vectors P for
various ED problems associated with the 2-bus test case.
A. Multiple Minimizers of the Common LCED Problem
This subsection studies two test cases that satisfy Assumptions descent direction. The unique optimal generation dispatch
1 and 2; however, the common LCED problem (Opt. F) has vector is P ⋆ = [28.125, 78.125]Tp.u., is represented by the
multiple minimizers for these two test cases, violating As- star in the figure, and is intuitively the feasible generation
sumption 3. In this context small perturbations to the ideal base dispatch vector that is furthest downstream in the descent
case-state are shown to result in a common LCED problem direction. The associated optimal vector of voltage angles is
(Opt. F) with a unique optimal generation dispatch; however, θ̇⋆ = θ2⋆ = −0.25 radians.
this unique optimal generation dispatch differs significantly The black dashed line represents the set of all feasible genera-
from the desired generation dispatch P ⋆ that solves the TCED tion dispatch vectors for the common LCED problem (Opt. F)
problem with angles (Opt. A), resulting in large generation when using the ideal base-case state θ̇0 = θ̇⋆ . Notice that the
dispatch approximation error. This is shown using an intuitive descent direction is perpendicular to the black dashed line and
2-bus test case as well as a large test case with 2383 buses. as a result all feasible generation dispatch vectors are optimal.
In this subsection neither test case enforces line limits so that Since the common LCED problem (Opt. F) has multiple
Assumption 2 holds. (See Remark 6). minimizers, Assumption 3 does not hold. Additionally, only
1) 2-Bus Test Case: A one-line diagram of the 2-bus test case one of the infinitely many minimizers of Opt. F is feasible for
is provided in Figure 2 along with various parameters of the the TCED problem with angles (Opt. A).
test case. Notice that the transmission line is highly resistive In practice the ideal base-case state is typically estimated.
and has no transmission limit. All system demand is located at To emulate this, linearize around the base-case state θ̇⋆ + δ
bus 2 and is fixed to D2 = 100p.u. The demand is co-located where δ = 0.07 radians represents a small perturbation. The
with expensive generation with cost function C2 (P2 )=P2 . resulting linear feasible set of generation dispatch vectors is
The generation at bus 2 is unlimited so that 0 ≤P2 ≤∞. represented by the dashed gray line. Furthermore, the common
Inexpensive generation is located remotely at bus 1 as the LCED problem (Opt. F) now has a unique optimal generation
cost of this generator is C1 (P1 )=0.6P1 . The inexpensive dispatch vector P ≈ [60, 55.559]Tp.u. that is represented by
generation is limited as 0 ≤ P1 ≤ 60p.u. The total system cost the black circle and also significantly differs from the desired
C(P ) is the sum of the individual generator costs. Bus 1 is generation dispatch vector P ⋆ . The dispatch approximation
designated the angle reference bus and the price reference bus error vector is ∆P = P − P ⋆ ≈ [31.875, −22.566]Tp.u.
so that σ = ρ = 1. The resulting system state is θ̇ = θ2 .
Perturbing the ideal base-case state in the other direction
D2 P2 Fig. 2: 2-bus system one-line diagram. The de- also results in large dispatch approximation error. Specifically,
Bus 2 mand is fixed at D2 = 100p.u. The generators have linearizing around the base-case state θ̇⋆ − δ results in the
cost functions C1 (P1 )=0.6P1 and C2 (P2 ) = P2 . linear feasible set of generation dispatch vectors represented
Bus 1 The line has per-unit impedance of 0.01+i0.01 by the solid gray line. In this case the common LCED problem
P1 and has no limit so that F̄1=−F1 =∞. (Opt. F) has a unique optimal generation dispatch vector
¯ P ≈ [0, 92.242]Tp.u. that is represented by the black diamond
Figure 3 plots the set of all feasible generation dispatch and also significantly differs from the desired generation
vectors for the TCED problem with angles (Opt. A) as a black dispatch vector P ⋆ . The dispatch approximation error vector
curve. The gray arrows in this figure represent the objective is ∆P = P − P ⋆ ≈ [−28.125, 14.117]Tp.u.

0885-8950 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2019.2941906, IEEE
Transactions on Power Systems
11

Remark 9. Similarly large approximation error remains for TABLE I: Results for test case 2383wp without line limits
an arbitrarily small perturbation δ > 0. Alternatively, similarly enforced and with the ideal base-case state θ̇0 = θ̇⋆ . The error
large approximation error remains after adding slight curvature quantities are denoted with ∆ and represent the difference
to the objective descent direction that might arise from a small between identified optimal quantities of Opt. A and Opt. F.
positive quadratic coefficient in the cost function. Initial Guess for the k∆P k1 k∆P k∞ |∆C| Solver
Remark 10. The lossless DC OPF problem yields a unique Interior Point Algorithm (MW) (MW) ($) Time (s)
optimal generation dispatch vector of P = [60, 40]T p.u. for the P = P ⋆ and ℓ = 1T N ⋆ 1.73 0.88 0.00 5.52
2-bus test case. In fact, the lossless DC OPF problem and the Typical Operating Point 21.65 10.70 0.00 5.33
TCED problem with angles (Opt. A) both have unique optimal Minimizer of lossless DC OPF 308.77 152.77 0.00 7.61
generation dispatch vectors; however, the common LCED
problem (Opt. F) has multiple optimal generation dispatch
vectors when the base-case state matches its ideal value θ̇⋆ . To demonstrate that Opt. F has multiple minimizers, two
alternative initial guesses were supplied as shown in the second
Remark 11. Reference [1, section V-A] makes a similar
and third rows of table I. The first alternative was constructed
observation regarding a different 2-bus test case. In their ex-
from the typical operating point provided by the test case
ample the common LCED problem (Opt. F) also has multiple
description. The second alternative was constructed from the
minimizers when the ideal base-case state is chosen. Rather
minimizer of the lossless DC OPF problem, which was solved
than perturbing the base-case state, as is done in this paper,
using the DC OPF function available in MATPOWER [30].
they perturb the bids of the generators. Keeping the base-case
When using these alternative initial guesses the algorithm
state fixed, they show that a small perturbation in generator
converges to dispatch values that do not match P ⋆ but do
bids can result in significant dispatch approximation error. This
attain the same optimal objective value C(P ⋆ ). Furthermore,
observation is consistent with the observations made here.
these identified dispatch values are not feasible for the TCED
2) Test Case 2383wp without Line Limits Enforced: Now problem with angles (Opt. A).
consider the test case 2383wp from the NESTA archive, which Similar to the simple 2-bus example, a small perturbation
represents the Polish power system during the 1999-2000 of the base-case state results in significantly large dispatch
winter evening peak [28]. Line limits are not enforced to approximation error. To illustrate this we perturb the ideal
ensure that Assumption 2 holds. The identified minimizer of base-case state θ̇⋆ by a perturbation vector δ ∈ Rn−1 that
the TCED problem with angles (Opt. A) is found using the was sampled from a normal distribution with zero mean
interior point algorithm provided by the MATLAB function and a diagonal covariance matrix of 10−10 × I. The spe-
FMINCON and takes 17.10sec. to converge. The optimal cific sample drawn from this distribution has the properties
objective value of this problem is C(P ⋆ )=1865459.40$. kδk∞ = 3.6 × 10−5 radians and kδk1 = 0.018 radians. The
Let’s first analyze the common LCED problem (Opt. F) with resulting common LCED problem appears to have a unique
the base-case state chosen to match the identified minimizer of optimal dispatch because the same dispatch is identified when
the TCED problem with angles (Opt. A) θ̇0 = θ̇⋆ in order to using any initial guess for the interior point algorithm. The
satisfy Assumption 1. In this case the loss sensitivity matrix dispatch approximation error is k∆P k1 = 314.70 MW and
∇Ñ(P̊ 0 − D̊) takes 26.13sec. to compute due to the inverse k∆P k∞ = 156.19 MW. This dispatch approximation error is
computation. There exist multiple optimal dispatch vectors significant compared to the small perturbation δ.
P for the common LCED problem (Opt. F), some of which
are not feasible for the TCED problem with angles (Opt. A).
B. LDF Approximation Error with Congestion
To illustrate this we use an interior point algorithm provided
by Artelys KNITRO software to solve the common LCED This subsection studies test cases that satisfy Assumptions
problem (Opt. F) with different supplied initial guesses [29]. 1 and 3; however, transmission line congestion causes As-
Table I compares the identified minimizers of both problems sumption 2 to be violated. A highly resistive 3-bus test case
where ∆P ∈ Rn represents the difference between P ⋆ and with significant transmission congestion is used to illustrate
the identified optimal dispatch of the common LCED problem the potential dispatch approximation errors associated with
(Opt. F) and ∆C represents the difference between C(P ⋆ ) and Assumption 2. Although associated errors are large for this
the optimal objective value of the common LCED problem extreme 3-bus test case, they are typically very small in
(Opt. F). practice as is illustrated by the 2383 bus test case with line
Turning to the first row of table I, the main body of the paper limits enforced. In fact, this test case exhibits no error despite
proves that P = P ⋆ and ℓ = 1T N ⋆ is a minimizer of the having transmission congestion. To illustrate error associated
common LCED problem (Opt. F). When supplying this as with Assumption 1 alternative choices of base-case state are
the initial guess, the algorithm immediately converges to a investigated that introduce dispatch approximation error into
dispatch that nearly matches P ⋆ . Notice that the associated the 2383 bus test case.
errors, ∆P , are very small relative to the total dispatch 1) 3-Bus Test Case: Figure 4 provides the details of the 3-bus
generation kP ⋆ k1 = 49231.67MW and the optimal objective test case in a one-line diagram. The generation dispatch values
values of both problems are nearly identical. However, ∆P is P have no upper limit but are restricted to be non-negative,
not identically zero due to insignificant computational error. e.g. P = 0. The system is highly resistive as the impedance
¯

0885-8950 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2019.2941906, IEEE
Transactions on Power Systems
12

of each line is rk + ixk = 0.01 + i0.01 in units of p.u. The allow for a user supplied initial guess [31]. Table II provides
cost function is given by C(P )=C1 (P1 )+C2 (P2 )+C3 (P3 ). numerical results for each of the four choices of base-case
Bus 1 is assigned to be the slack bus and the reference bus. state. Notice that the solver times are much faster and the
optimal generation dispatch vector P ⋆ is exactly recovered
P2 D2 =100p.u. Fig. 4: 3-bus system one-line diagram. when setting the base-case state to its ideal value θ̇0 = θ̇⋆ .
Bus 2 The line limits F̄ = −F are such that Note that the loss sensitivity matrix ∇Ñ(P̊ 0 − D̊) requires
¯ approximately 28 seconds of computation time for each base-
e1

Lin

F̄2 = F̄3 =∞ and F̄1 =11p.u. The gener-

e2
Lin

Line 3 ators have cost functions C1 (P1 )=P12 , case state.

P1 Bus 1 Bus 3 P3 C2 (P2 )=100P22 and C3 (P3 )=0.01P32 . Since the common LCED problem (Opt. F) now has a unique
optimal dispatch when using the ideal base-case state, small
perturbations of the base-case state should result in small
The minimizer of the TCED problem with angles (Opt. A)
approximation errors. This is illustrated by using the same
is P ⋆ ≈ [82.38, 6.81, 123.61]T and θ̇⋆ ≈ [−0.690, 0.110]T
perturbed base-case state θ̇⋆ + δ as in Section IV-A2. Notice
in units of p.u. and radians respectively. The ideal base-case
that the dispatch approximation error is very small as com-
state is chosen θ̇0 = θ̇⋆ and thus the base-case dispatch is
pared to the error witnessed in Section IV-A2. In fact, we
P 0 = P ⋆ . The line limit for line 1 is binding and the con-
should expect the dispatch approximation error to disappear
gestion component of the LMP is non-zero, taking the value
as the perturbation becomes smaller, e.g. kδk → 0.
c=[0, 137.26, −137.26]T in units of dollars per p.u. Notice
that Section III-E draws no conclusion regarding whether or TABLE II: Results for test case 2383wp with line limits en-
not P 0 is optimal for the LDF LCED problem (Opt. E). forced. The error quantities are denoted with ∆ and represent
As expected the base-case point P 0 and ℓ0 := 1T N 0 is the difference between identified optimal quantities of Opt. A
feasible for the common LCED problem (Opt. F); however, and Opt. F.
this point is not optimal. An alternative unique optimal Choice of kθ̇ ⋆− θ̇ 0 k1 k∆P k1 k∆P k∞ |∆C| Solver
dispatch P of Opt. F is identified that is not feasible for Base-Case State, θ̇ 0 (radians) (MW) (MW) ($) Time (s)
the TCED problem with angles (Opt. A). The optimal ob- Local Min. of Opt. A, θ̇ ⋆ 0.00 0.00 0.00 0.00 1.26
jective value for the common LCED problem (Opt. F) is Small Perturbation, θ̇ ⋆ + δ 0.02 0.08 0.04 0.95 1.33
$11512.63, which is $62.36 lower than the optimal objective Min. of Lossless DC OPF 34.15 73.95 29.67 1061.11 1.10
value for the TCED problem with angles (Opt. A). The Typical Operating Point 257.68 501.37 306.58 24007.79 1.37
optimal dispatch for the common LCED problem (Opt. F)
is P =[86.96, 6.17, 122.41]Tp.u. and the dispatch approxima-
tion error is P ⋆−P =[−4.58, 0.64, 1.20]T. This highly resistive As discussed in Section III-A, the ideal base-case state θ̇⋆
network exhibits small error, suggesting this approximation is is not known in practice. We now consider two alternative
typically accurate. choices of base-case state θ̇0 . First, is the minimizer of the
lossless DC OPF problem. The second alternative is deter-
2) Test Case 2383wp with Line Limits Enforced: Consider the mined by solving power flow equations at the typical operating
same test case as Section IV-A but with line limits enforced. point provided by the test case description. The power flow
The identified minimizer of the TCED problem with angles equations and the lossless DC OPF problem are solved using
(Opt. A) is found using the interior point algorithm provided MATPOWER. Table II shows that the generation dispatch
by FMINCON and takes 18.81sec. to converge. The optimal error is small relative to the total generation, which satisfies
objective value of this problem is C(P ⋆ )=1890940.57$. There kP ⋆ k1 = 25108.74MW. Furthermore, the minimizer of the
are 4 congested lines with binding limits, which is a small lossless DC OPF problem serves as a better base-case state in
amount relative to the 2896 total transmission lines. terms of the generation dispatch error.
When using the ideal base-case state θ̇0 = θ̇⋆ the common
LCED problem (Opt. F) appears to have a unique minimizer
because the interior point algorithm converges to the same V. C ONCLUSIONS
point with any choice of initial guess. Notice that test case For the first time, this paper characterizes the generation
2383wp has the unusual property that its common LCED dispatch approximation errors associated with each of the
problem (Opt. F) has a unique minimizer when enforcing line three assumptions required to recover the optimal dispatch
limits but has multiple minimizers when line limits are not of the TCED problem from the common LCED problem.
enforced. This property is not typical as many other test cases Four intermediate problems are used to derive the common
in the NESTA archive do not have this property. LCED problem and identify the three key assumptions. Nu-
We consider four different base-case states when formulating merical results illustrate dispatch approximation error caused
the common LCED problem (Opt. F). Each of the four by choosing a non-ideal base-case state, using LDFs un-
versions of this problem appear to have a unique minimizer der congested conditions, and the common LCED problem
because the interior point algorithm converges to the same having multiple minimizers. The first key finding states that
point for any choice of initial guess. Since there is a unique the common LCED problem may have multiple minimizers
minimizer, we are able to take advantage of the much faster when using the ideal base-case state, in which case small
dual simplex algorithm provided by MOSEK, which does not perturbations of the base-case state may result in significantly

0885-8950 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2019.2941906, IEEE
Transactions on Power Systems
13

large dispatch approximation error. This type of approximation [19] Z. Yang, H. Zhong, A. Bose, T. Zheng, Q. Xia, and C. Kang, “A
error is problematic in realistic systems and is illustrated using linearized OPF model with reactive power and voltage magnitude: a
pathway to improve the MW-only DC OPF,” IEEE Transactions on
a simple 2-bus example as well as a larger more realistic test Power Systems, vol. 33, no. 2, pp. 1734–1745, 2018.
case with 2383 buses, neither of which enforce line limits. [20] Y.-H. Moon, B.-K. Choi, H.-S. Ryu, J.-S. Jung, and H.-M. Park, “Slack-
The second key finding states that under congested conditions bus independent penalty factor for spot pricing under deregulation,” in
there may not exist LDFs that allow the optimal dispatch Power Engineering Society Winter Meeting, 2000. IEEE, vol. 2. IEEE,
2000, pp. 1017–1021.
of the TCED problem to also be optimal for the common
[21] H. Zhang, G. T. Heydt, V. Vittal, and J. Quintero, “An improved network
LCED problem, even if the ideal base-case state is used. model for transmission expansion planning considering reactive power
This is illustrated using an extreme 3-bus example that is and network losses,” IEEE Transactions on Power Systems, vol. 28,
highly resistive. However, the larger 2383 bus test case with no. 3, pp. 3471–3479, 2013.
line limits enforced is then used to illustrate that associated [22] T. Akbari and M. T. Bina, “A linearized formulation of AC multi-year
transmission expansion planning: A mixed-integer linear programming
approximation error is typically insignificant. approach,” Electric Power Systems Research, vol. 114, pp. 93–100, 2014.
[23] M. Wang, H. B. Gooi, S. Chen, and S. Lu, “A mixed integer quadratic
R EFERENCES programming for dynamic economic dispatch with valve point effect,”
IEEE Transactions on Power Systems, vol. 29, no. 5, pp. 2097–2106,
[1] B. Eldridge, R. ONeill, and A. Castillo, “An improved method for the 2014.
DCOPF with losses,” IEEE Transactions on Power Systems, vol. 33,
no. 4, pp. 3779–3788, 2018. [24] B. F. Hobbs, G. Drayton, E. B. Fisher, and W. Lise, “Improved
transmission representations in oligopolistic market models: quadratic
[2] M. Garcia, S. Siddiqi, and R. Baldick, “A general economic dispatch losses, phase shifters, and DC lines,” IEEE Transactions on Power
problem with marginal losses,” American Control Conference (ACC), Systems, vol. 23, no. 3, pp. 1018–1029, 2008.
2019. Also available on arXiv:1811.06572.
[25] A. Castillo, P. Lipka, J.-P. Watson, S. S. Oren, and R. P. O’Neill, “A
[3] E. Litvinov, T. Zheng, G. Rosenwald, and P. Shamsollahi, “Marginal loss successive linear programming approach to solving the IV-ACOPF,”
modeling in LMP calculation,” IEEE Transactions on Power Systems, IEEE Transactions on Power Systems, vol. 31, no. 4, pp. 2752–2763,
vol. 19, no. 2, pp. 880–888, 2004. 2016.
[4] M. B. Cain, R. P. Oneill, and A. Castillo, “History of optimal power [26] R. Baldick, Applied optimization: Formulation and algorithms for engi-
flow and formulations,” Federal Energy Regulatory Commission, pp. 1– neering systems. Cambridge University Press, 2006.
36, 2012.
[27] J. L. M. Ramos, A. G. Exposito, F. Moron, and S. Becerra, “On the use
[5] K. Lehmann, A. Grastien, and P. Van Hentenryck, “AC-feasibility on tree of loss penalty factors for generation scheduling,” in 2003 IEEE Power
networks is NP-hard,” IEEE Transactions on Power Systems, vol. 31, Engineering Society General Meeting (IEEE Cat. No. 03CH37491),
no. 1, pp. 798–801, 2016. vol. 2. IEEE, 2003, pp. 926–931.
[6] D. Bienstock and A. Verma, “Strong NP-hardness of AC power flows [28] C. Coffrin, D. Gordon, and P. Scott, “NESTA, the NICTA energy system
feasibility,” arXiv preprint arXiv:1512.07315, 2015. test case archive,” arXiv preprint arXiv:1411.0359, 2014.
[7] J. D. Glover, M. S. Sarma, and T. Overbye, Power System Analysis & [29] R. H. Byrd, J. Nocedal, and R. A. Waltz, “Knitro: An integrated pack-
Design, SI Version. Cengage Learning, 2012. age for nonlinear optimization,” in Large-scale nonlinear optimization.
[8] B. Stott, J. Jardim, and O. Alsaç, “DC power flow revisited,” IEEE Springer, 2006, pp. 35–59.
Transactions on Power Systems, vol. 24, no. 3, pp. 1290–1300, 2009. [30] R. D. Zimmerman, C. E. Murillo-Sánchez, and R. J. Thomas, “Mat-
[9] F. Li and R. Bo, “DCOPF-based LMP simulation: algorithm, comparison power: Steady-state operations, planning, and analysis tools for power
with ACOPF, and sensitivity,” IEEE Transactions on Power Systems, systems research and education,” IEEE Transactions on power systems,
vol. 22, no. 4, pp. 1475–1485, 2007. vol. 26, no. 1, pp. 12–19, 2011.
[10] T. J. Overbye, X. Cheng, and Y. Sun, “A comparison of the AC and [31] M. ApS, The MOSEK optimization toolbox for MAT-
DC power flow models for lmp calculations,” in 37th Annual Hawaii LAB manual. Version 8.1., 2017. [Online]. Available:
International Conference on System Sciences, 2004. Proceedings of the. http://docs.mosek.com/8.1/toolbox/index.html
IEEE, 2004, pp. 9–pp.
[11] F. C. Schweppe, M. C. Caramanis, R. D. Tabors, and R. E. Bohn, Spot
pricing of electricity. Springer Science & Business Media, 2013.
[12] W. Hogan, “Financial transmission right formulations,” Report, Center
for Business and Government, John F. Kennedy School of Government,
Harvard University, Cambridge, MA, 2002.
[13] Z. Yang, A. Bose, H. Zhong, N. Zhang, J. Lin, Q. Xia, and C. Kang,
“LMP revisited: A linear model for the loss-embedded LMP,” IEEE
Transactions on Power Systems, vol. 32, no. 5, pp. 4080–4090, 2017.
[14] F. Li, “Fully reference-independent LMP decomposition using reference-
independent loss factors,” Electric Power Systems Research, vol. 81,
no. 11, pp. 1995–2004, 2011.
[15] R. Palma-Benhke, A. Philpott, A. Jofré, and M. Cortés-Carmona, “Mod-
elling network constrained economic dispatch problems,” Optimization
and Engineering, vol. 14, no. 3, pp. 417–430, 2013.
[16] A. Philpott, “Experiments with load flow pricing models,” in Proceed-
ings of CRNEC Policy Conference, Centre for Research in Network
Economics and Communications, Univ. Auckland, New Zealand, 1999.
[17] C. Coffrin and P. Van Hentenryck, “A linear-programming approxima-
tion of AC power flows,” INFORMS Journal on Computing, vol. 26,
no. 4, pp. 718–734, 2014.
[18] C. Coffrin, P. Van Hentenryck, and R. Bent, “Approximating line losses
and apparent power in AC power flow linearizations,” in Power and
Energy Society General Meeting, 2012 IEEE. IEEE, 2012, pp. 1–8.

0885-8950 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.