February	2013

Optimal	Power	Flow	Paper	6	

Staff	paper	by	

Aaron	Schecter		

Richard	P	O’Neill		
Exploration	of	the	ACOPF	Feasible	Region	for	the	

Standard	IEEE	Test	Set	

Optimal	Power	Flow	Paper	6	

Aaron	Schecter	and	Richard	P	O’Neill		

aschec@gmail.com	richard.oneill@ferc.gov		

February	2013			

Abstract	and	Executive	Summary	

The	goal	of	our	investigation	is	to	gain	some	perspective	on	how	non‐convex	the	

feasible	region	of	the	Alternating	Current	Optimal	Power	Flow	ሺACOPFሻ	problem	is.	

First,	we	will	develop	a	metric	for	comparing	how	infeasible	different	solutions	are.	

The	set‐up	for	this	examination	will	be	to	generate	convex	combinations	of	feasible	

points,	and	determine	how	“far”	these	new	points	are	from	the	feasible	region.	We	

propose	a	metric	based	on	the	relative	two	norm.	This	value	will	be	compared	over	

various	regions	around	the	optimal	solution.	Second,	we	would	like	to	determine	

how	elastic	the	area	around	the	global	optimum	is;	in	other	words,	we	will	

determine	the	largest	range	of	values	the	optimization	variables	can	take,	given	a	

small	perturbation	from	the	global	solution.	In	doing	so,	we	will	find	the	elasticity	of	

all	optimization	variables,	which	could	give	us	insight	into	the	structure	of	the	

feasible	region.	Finally,	we	will	examine	the	two	dimensional	relationships	between	

the	optimization	variables	and	the	objective	function	value.	

Index	Terms—Optimal	Power	Flow,	Feasible	Region,	Solution	Space,	Local	

Solutions,	Elasticity	

Disclaimer:	The	views	presented	are	the	personal	views	of	the	authors	and	not	the	

Federal	Energy	Regulatory	Commission	or	any	of	its	Commissioners.	 	 	
Table	of	Contents	

Nomenclature	1	

1. Introduction	 	 	 	 	 	 	 	 	 	1	

2. Feasible	Region	Of	The	Acopf	Problem	Ac	Flow	Equations	 	 	 	2	

3. Elasticity	Near	The	Optimal	Solution	 	 	 	 	 	 	3	

4. Fixed‐Variable	Optimizationacopf	Formulation	 	 	 	 	5	

5. Conclusions	 	 	 	 	 	 	 	 	 	7	

References		 	 	 	 	 	 	 	 	 	 	7	
1

NOMENCLATURE

Indices

n, m Nodes

k Lines

g Generators

Variables

vn Voltage at Bus n

Pg Real Power at Generator g

Qg Reactive Power at Generator g

θn Voltage Angle at Bus n

θnm Voltage difference (i.e. θn − θm)

U Uniform Random Variable

x Vector of Optimization

Variables (i.e. x =



θ, V, P, Q 

)

x

(r) Vector of Optimization Variables

from Random Objective Function

x

(c) Vector of Optimization Variables

from Convex Combination

Parameters

V

min, V max Minimum and Maximum

Voltage Magnitude Vector

P

min, P max Minimum and Maximum

Real Power Capacity Vector

Qmin, Qmax Minimum and Maximum

Reactive Power Capacity Vector

θ

min

nm , θmaxnm Minimum and Maximum

Angle Difference Between Bus n and m

cgi i

th Order Cost Term for Generator g

Gnmk Electrical Conductance

from Bus n to m along Line k

Note: For our purposes, the transmission lines

are numbered 1 through |K|

Bnmk Electrical Susceptance

from Bus n to m along Line k

Sets

N Set of Nodes

G Set of Generators

K Set of Transmission Lines

S Set of Constraints

I. INTRODUCTION

The problem of finding an optimal power flow solution in

a network proves especially challenging from a computational

standpoint. The ACOPF problem is an example of a nonlinear programming problem over a non-convex feasible region.

Today’s optimization algorithms generally rely on the KarushKuhn-Tucker (KKT) conditions for optimality, but this set of

criteria assumes a convex region. Previous investigation of the

ACOPF problem suggests that the region is suitably convex

[3] for the application of the KKT conditions; however, the

region is in fact non-convex and therefore the KKT conditions

fail to guarantee global optimality. In fact, modern algorithmic

methods still do not guarantee convergence to a global optimum.

One approach that does not rely on the KKT conditions is the

method proposed in [4], which establishes an algorithm based

on semi-definite programming that results in a duality gap of

zero.

In general, it can be shown that there are a large number of

physically feasible load flow solutions [5] and thus numerous

local optima. As a result, the issue of convexity becomes

important for both accuracy and reliability purposes. In this

paper, we establish some basic characteristics of these problems

using the IEEE standard test suite [1] and further contribute to

the discussion of their structure.

The first segment of our investigation focuses on points that

are nearly feasible. In other words, if we examine some convex

combination of two known feasible points, to what degree will

this new point be infeasible, if it is in fact infeasible. Because

the region is non-convex, our expectation is not to find many

feasible combinations; however, if we find only small deviations

from the feasible region, that would support the suitably convex

hypothesis.

The second area of focus is the elasticity of the optimization

variables. Rather than look only at the predefined limits on

the variables, we examine the effective limits. In other words,

what are the extreme values of each variable, such that overall

feasibility is maintained. To refine this search to a meaningful

scope, we only consider feasible points that correspond to

objective function values within one percent of the global

optimum. Therefore, our examination centers on the area around

the global optimum and we seek to establish how flat that region

of the feasible set is.

Finally, we determine the behavior of the optimization variables across their range of feasible values. In particular, we

will investigate the pairwise relationships between a single

variable’s value and the objective function value. The nature of

these relationships should lend some insight into how changes

in the power flow can affect the subsequent cost, as well as

characterize the relative convexity of the region around the

optimum. Additionally, by iterating through all of the feasible

values for each variable, we should expect to find some local

optima corresponding to those power flow profiles.

II. FEASIBLE REGION OF THE ACOPF PROBLEM

Our procedure is two stage: first, we solve for a global

optimum of the ACOPF problem. A solution is considered

globally optimal in this paper if it satisfies the first order KKT

conditions to a default tolerance of 10−6. Note that this is

a necessary but not sufficient condition for optimality. The

problem is solved in the polar formulation:

min f(x) = P

g∈G

cg2P

2

g + cg1Pg + cg0 (1)

s. t. Pn =

P

m∈N

vnvm(Gnmk cos θnm + Bnmk sin θnm) (2)

Qn =

P

m∈N

vnvm(Gnmk sin θnm − Bnmk cos θnm) (3)

P ≤ P

max (4)

P

min ≤ P (5)

Q ≤ Qmax (6)

Qmin ≤ Q (7)

V

min ≤ V ≤ Vmax (8)

We will refer to the constraint set as S = {(2),(3), ...,(8)}.

In the typical polar formulation of the ACOPF problem, we

would also include line constraints and voltage angle constraints.

However, the IEEE standard test problems are by default not
2

bounded by these constraints; the input data for branch and

angle limits are effectively infinite. Therefore, these constraints

are not included in our formulation. The original problem

(1),...,(8) is solved using the solver fmincon in the MATLAB

package MATPOWER (Version 4.1) [2].

Next, in an attempt to quantify how non-convex the feasible

region is, we consider the mid-point of two feasible points and

test it for feasibility. For each of the cases (14, 30, 57, 118, and

300 bus), we substitute the default objective function with a

randomized linear cost function. In other words, (1) is replaced

with the objective function

f(x) = P

g∈G

U cg1Pg (9)

where U is a uniform random variable, and cg1 is the

problem’s default linear cost term. All other cost terms are

ignored. We minimize this objective function over our region

S and obtain a local optimal solution that will be on the

boundary of S, by first order optimality conditions (Procedure

1, Step 1). Next, we examine all of the pairwise mid-points.

To determine feasibility, we calculate the minimum relative

`2−norm of the convex combination and a feasible point from

the set S (Procedure 1, Step 5); a result of zero indicates that

the convex combination is a feasible point. Since the point that

minimizes the Euclidean distance metric is also the minimizer

for the `2− norm, we use it for smoothness considerations. We

examine the relative magnitude in the `2−norm so that our

objective function value serves as a more meaningful metric;

it is easier to compare “how infeasible” different solutions are

when their values are scaled by the optimal values. Our process

is described in Procedure 1.

Procedure 1: Test for Convexity

1. Let (P) denote the optimization problem

minf(x) = P

g∈G

U cg1Pg (10)

s. t. x ∈ S, U ∼ Uniform(0, 2)

2. Solve (P) 100 times

3. Let x

(r)

i

be the ith solution

4. For each pair of solutions i 6= j

x

(c)

ij = 0.5x

(r)

i + 0.5x

(r)

j

(11)

5. Solve the following ∀i, j pairs, i 6= j

min g(x) = P

g∈G

(P

(c)

g,ij−Pg)

2

|P ∗| +

P

g∈G

(Q

(c)

g,ij−Qg)

2

|Q∗|

(12)

+

P

n∈N

(v

(c)

n,ij−vn)

2

|v

∗|

s.t. x ∈ S

6. Report the values of g(x

∗

) from (12)

To determine the convexity of the region around the global solution, we set U so that U is drawn from the Uniform(0, 2) distribution. The results of this procedure are summarized in Table

1. After doing so, we restrict the region to U ∼Uniform(0.99,

1.01). By doing so, we should restrict our randomized local

solutions to a much tighter region of the feasible set. The results

of this change are summarized in Table 2.

Figure 1 illustrates the value we are trying to minimize. In

two dimensions, we would simply try to minimize the Euclidean

distance between the midpoint of two feasible points (f1(x

∗

)

and f2(x

∗

)) and the feasible set S.

Figure 1. A two-dimensional illustration of the infeasibility metric we seek to

minimize

A. Convex Combination Feasibility

For our convex combination procedure, we report the mean

and median values for each case, and we record the number of

feasible points (objective function value of zero). To give some

notion of how variable the differences are, we also report the

maximum and minimum values of the objective function.

As we can see from Table I, none of the convex combinations

are feasible points when the random variable is drawn from

the Uniform(0, 2) distribution; additionally, there is no obvious

correlation between size of the problem and “how non-convex”

the constraint set is. In fact, the 300 bus case has the lowest

values of our infeasibility metric; this implies that, relative to

the variable values, the convex combinations of 300 bus case

solutions are closest to known feasible points.

Table I

SUMMARY OF MIDPOINT FEASIBILITY TEST RESULTS, WITH RANDOM

VARIABLE U ∈ [0, 2]

Case Mean Median Feasible Points Minimum Maximum

14 0.542 0.512 0 0.29 1.595

30 0.47 0.455 0 0.336 0.823

57 0.66 0.643 0 0.341 1.178

118 1.39 1.335 0 0.648 2.704

300 0.082 0.082 0 0.062 0.105

Next, we restrict the objective function’s variation by introducing a uniform random variable with a smaller range and

repeat Procedure 1. Our results for the Uniform(0.99, 1.01)

randomized objective function are as follows in Table II:

Table II

SUMMARY OF MIDPOINT FEASIBILITY TEST RESULTS, WITH RANDOM

VARIABLE U ∈ [0.99, 1.01]

Case Mean Median Feasible Points Minimum Maximum

14 0.551 0.522 0 0.292 1.535

30 0.468 0.453 0 0.329 0.806

57 0.686 0.673 0 0.424 1.071

118 1.469 1.466 0 0.501 2.473

300 0.077 0.078 0 0.050 0.103

Restricting the region of points does not have much of

an impact on the mean and median `2−norm. However, the

minimum and maximum difference values do decrease. These
3

results suggest that on average, the midpoint between any two

feasible points will be as infeasible as either point, regardless

of our choice of points. This conclusion may give us some

insight into the shape of S; if the non-convexities of the feasible

region were sharp, then we would see wider variation in the

infeasibility metric as we moved the points closer together.

However, if we produce similar values for different choices

of random variables, then the notion of a flat feasible region

becomes more tractable.

Aside from the midpoint, we look at a more general case of

convex combinations:

x

(c) = αx1 + (1 − α)x2, α ∈ [0, 1] (13)

Generally speaking, as α approaches 0 (or 1), the convex

combination of variables will become more similar to one of the

two feasible points. Thus, we investigate how our infeasibility

metric changes over different values of α. For this procedure,

we repeat Procedure 1 with U ∼ Uniform(0.99, 1.01), but

replace (11) with (13), and test at the following points: α =

0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45. This test is

performed for the 14 and 30 bus case; we report the following

results in Table III:

Table III

RESULTS FOR INFEASIBILITY TEST AT VARIOUS ALPHA VALUES; MEAN

RESULTS REPORTED FOR THE 14 AND 30 BUS CASES

Mean Infeasibility

α 14 Bus 30 Bus

0.05 0.690 0.574

0.10 0.660 0.553

0.15 0.634 0.533

0.20 0.612 0.517

0.25 0.593 0.502

0.30 0.578 0.491

0.35 0.566 0.481

0.40 0.557 0.474

0.45 0.553 0.470

0.50 0.542 0.468

The results show that the convex combination that varies the

least in our `2−norm is the midpoint. In fact, as the convex

combination favors one point over the other, the combination

actually varies more from the feasible set. This relationship is

illustrated by both the 14 and 30 bus case, with a nearly identical

shape to the plots (Figure 2).

III. ELASTICITY NEAR THE OPTIMAL SOLUTION

Elasticity is a measure of the magnitude of change in a

variable as a result of change in another variable. We will denote

the elasticity in variable y with respect to variable x as:

Ey,x =







∆y/y

∆x/x





 (14)

For the ACOPF problem, we are interested in how greatly

the optimization variables can change while maintaining an

objective function value that is within 1% of the global optimum

(Procedure 2, Step 3). Because the feasible region S is a nonconvex set, we are interested in the elasticity as a measure of

Figure 2. A plot of Alpha level againts mean infeasibility, 14 bus case (top)

and 30 bus case (bottom)

0.1 0.2 0.3 0.4 0.5

0.56 0.58 0.60 0.62 0.64 0.66 0.68

Alpha

Mean Infeasibility

0.1 0.2 0.3 0.4 0.5

0.48 0.50 0.52 0.54 0.56

Alpha

Mean Infeasibility

how flat the area around the global optimum of problem (1)

is; if in fact the area is highly elastic (flat), then it is possible

that the non-convexities of S are not severe enough to force our

algorithms into local optima.

To measure the elasticity, we calculate the absolute difference

between the minimum and maximum values (Procedure 2, Step

4), divided by the magnitude of the optimal value, then divided

by the allowed percentage change in the optimal objective

function value (Procedure 2, Step 5). These problems were all

solved in MATLAB using the non-linear solver fmincon, with

tolerances for costs and constraint violations set at 10−6. These

problems were solved for every bus and every generator. The

data used were the IEEE 14, 30, 57, 118, and 300 bus cases.

Our process is described in Procedure 2.
4

Procedure 2: Compute Elasticity

For each case:

1. Solve for the optimal solution of (1)

x

∗ =



θ

∗

, V ∗, P∗, Q∗



(15)

2. Find the corresponding objective function value

f(x

∗

) = P

g∈G

cg2P

∗2

g + cg1P

∗

g + cg0 (16)

3. Add new constraint to S

f(x) ≤ 1.01f(x

∗

) (17)

S

0 = S ∪ (17)

4. Solve the new optimization problems

min / max vn (18)

subject to x ∈ S’

min / max Pg (19)

subject to x ∈ S’

min / max Qg (20)

subject to x ∈ S’

5. Compute the elasticities

Evn,f =

v

max

n −v

min

n

v

∗

n

/(0.01) (21)

EPg,f =

P

max

g −P

min

g

P ∗

g

/(0.01) (22)

EQg,f =

|Qmax

g −Qming

|

|Q∗

g

|

/(0.01) (23)

A. Test Ranges

We report the optimal value, maximum, and minimum value

of each optimization variable for the 14 (Table IV) and 30

(Table V) bus problems. These results correspond to step 4

of Procedure 2 ((18), (19), and (20)). It should be noted that

the default limits for voltage magnitude in the 14 bus case are

[0.94, 1.06], while the limits for the 30 bus case are [0.95,

1.05]. Individual generator power limits were also left at their

default values. These limits were not changed for the sake of

easy replication. Interestingly, we see that while many of these

variables have a wide range, the voltage magnitudes and real

power generally do not span all feasible values, while reactive

power can fluctuate across its entire spectrum. The likely cause

is that reactive power costs are not included in the standard

objective function, which is to minimize the cost of real power.

B. Elasticities

We report the elasticity of our optimization variables for the

14 and 30 bus cases, as calculated in Step 5 of Procedure 2

((21), (22), and (23)) in Table VI.

Summarized in Table VII are the mean and median elasticities

for each of the cases. The median was considered due to the

high frequency of extreme values, which skew the mean. In the

14 bus case, there are particularly glaring discrepencies between

the mean and median power (active and reactive) elasticities.

As we can see from Table VII, each variable has an elasticity

much higher than 1. Reactive power is especially elastic, and is

consistently more elastic than the other optimization variables.

In the 57 bus case, however, reactive power is not significantly

Table IV

THE RESULTS FOR THE TEST OF VARIABLE RANGES FOR THE 14 BUS

CASE; MINIMUM, MAXIMUM, AND OPTIMAL VALUES ARE REPORTED

Variable Minimum Optimal Maximum

v1 0.969 1.06 1.06

v2 0.945 1.041 1.048

v3 0.94 1.016 1.047

v4 0.94 1.014 1.026

v5 0.94 1.016 1.025

v6 0.959 1.06 1.06

v7 0.950 1.046 1.052

v8 0.94 1.06 1.06

v9 0.951 1.044 1.051

v10 0.951 1.039 1.045

v11 0.958 1.046 1.049

v12 0.945 1.045 1.045

v13 0.942 1.04 1.041

v14 0.94 1.024 1.029

P1 156.308 194.327 231.296

P2 19.074 36.719 54.366

P3 1.99E-06 28.737 68.350

P6 2.00E-08 0.0138 48.316

P8 2.00E-06 8.491 59.668

Q1 2.00E-08 0.0234 10

Q2 -29.628 23.681 50

Q3 2.00E-06 24.126 40

Q6 -6 11.529 24

Q8 -6 8.274 24

more elastic than the other variables. In the 118 bus case,

the median range of reactive power is more than 600% the

magnitude of the optimal solution. Despite these large results

for the 118 bus case, we observe that there is not a consistent

trend between size of the problem and elasticity of the variables,

similar to our findings regarding our infeasibility metric. The 57

bus case is the least elastic across all three types of variables,

and the 300 bus case is the second least elastic. We can thus

hypothesize that the size of the problem is not the driver of

variable elasticities, and therefore does not directly affect the

shape of the feasible region.

IV. FIXED-VARIABLE OPTIMIZATION

To determine the isolated affect of one variable on the overall

objective function value, multiple optimization problems can

be solved at various levels, and the changes in cost can be

tracked. For the ACOPF problem, we fixed a single variable

value (Procedure 3, Line 3), and performed the optimization

using that variable as an additional constraint (Procedure 3, Line

5). As an output of this procedure, we will obtain a graphical

representation of the relationship between various levels of the

particular variable and objective function values. This process

was completed for the IEEE 14, 30, and 57 bus cases. For the

purposes of space, we will include a deeper look at the 14 bus

case only. Our process is decribed more fully in Procedure 3:
5

Table V

THE RESULTS FOR THE TEST OF VARIABLE RANGES FOR THE 30 BUS

CASE; MINIMUM, MAXIMUM, AND OPTIMAL VALUES ARE REPORTED

Variable Minimum Optimum Maximum

v1 0.951 0.983 1.05

v2 0.951 0.979 1.059

v3 0.961 0.977 1.041

v4 0.961 0.977 1.039

v5 0.955 0.971 1.042

v6 0.961 0.973 1.034

v7 0.950 0.963 1.028

v8 0.95 0.961 1.022

v9 0.955 0.991 1.039

v10 0.951 1.000 1.043

v11 0.955 0.991 1.039

v12 0.954 1.018 1.05

v13 0.95 1.066 1.1

v14 0.95 1.007 1.045

v15 0.951 1.01 1.05

v16 0.954 1.003 1.04

v17 0.95 0.996 1.037

v18 0.95 0.994 1.035

v19 0.95 0.988 1.03

v20 0.95 0.99 1.033

v21 0.95 1.009 1.05

v22 0.951 1.016 1.059

v23 0.95 1.026 1.1

v24 0.95 1.017 1.05

v25 0.968 1.044 1.05

v26 0.95 1.027 1.033

v27 0.983 1.069 1.069

v28 0.963 0.982 1.037

v29 0.962 1.05 1.05

v30 0.95 1.039 1.039

P1 25.624 41.51 61.989

P2 38.768 55.362 79.893

P22 12.114 22.733 34.043

P27 10.058 39.948 55

P23 0.925 16.3 30

P13 0.93 16.206 33.79

Q1 -20 -5.469 81.401

Q2 -20 1.539 60

Q22 -15 33.905 62.5

Q27 -15 31.639 46.347

Q23 -10 7.06 40

Q13 -15 36.445 44.7

Procedure 3: Fixed-Variable Optimization

For each case:

1. Select an optimization variable Pg

(or Qg, vn)

2. Determine the range, RPg = P

max

g − P

min

g

,

of feasible values for the variable

3. Let Pg = P

min

g + ciRPg

for i = 1, ..., 100 (24)

where ci = i/100

4. Add new constraint to S

S

00 = S ∪ (24) (25)

5. Solve the optimization problem (1)

under constraint set S

00

6. Plot objective function value

againts values of Pg

A. Graphical Representations of the 14 Bus Problem

1) Well Behaved Graphs: We first report on the general class

of variables that are “well-behaved”; in other words, they display

Table VI

RESULTS OF ELASTICITY CALCULATION; REPORTED ARE ELASTICITY

VALUES FOR ALL OPTIMIZATION VARIABLES FROM THE 14 BUS (LEFT) AND

30 BUS (RIGHT) CASES

Variable Elasticity

v1 8.57

v2 9.86

v3 10.54

v4 8.46

v5 8.39

v6 9.50

v7 9.73

v8 11.32

v9 9.52

v10 9.08

v11 8.75

v12 9.62

v13 9.54

v14 8.64

P1 38.59

P2 96.12

P3 237.85

P6 350625.54

P8 702.71

Q1 42733.22

Q2 336.26

Q3 165.80

Q6 260.22

Q8 362.59

Variable Elasticity

v1 10.11

v2 11.07

v3 8.16

v4 7.97

v5 8.91

v6 7.45

v7 8.10

v8 7.50

v9 8.51

v10 9.19

v11 8.51

v12 9.40

v13 14.07

v14 9.42

v15 9.80

v16 8.64

v17 8.77

v18 8.59

v19 8.13

v20 8.35

v21 9.91

v22 10.63

v23 14.61

v24 9.83

v25 7.81

v26 8.08

v27 8.07

v28 7.60

v29 8.39

v30 8.57

P1 87.61

P2 74.28

P22 96.47

P27 112.50

P23 178.38

P13 202.77

Q1 1854.24

Q2 5199.19

Q22 228.58

Q27 193.92

Q23 708.18

Q13 163.81

Table VII

SUMMARY STATISTICS FOR OPTIMIZATION VARIABLE ELASTICITIES

Test Case Voltage Power Reactive Power

14 Median 9.51 237.85 336.26

Mean 9.40 70340.16 8771.62

30 Median 8.58 104.48 468.38

Mean 9.12 125.33 1391.32

57 Median 2.88 102.49 243.15

Mean 3.79 101.87 360.35

118 Median 11.02 499.33 676.37

Mean 10.88 57053.86 4228.16

300 Median 7.80 137.19 229.32

Mean 7.46 10712.25 3403.45

smooth relationships between their values and cost and are

generally convex. On the whole, the majority of optimization

variables fall into this category; their values with respect to

the objective function are parabolic with minima at the global

optimum. It should be noted that for visualization purposes,

infeasible points (or points that failed to lead to a convergent

solution) are designated with objective function value of 1.5f

∗

,
6

which serves as effectively infinite. For the inset graphs, we

observe a “bucket” shape; this effect is caused by the values

falling outside the feasible region. A “well-behaved” variable

generally had a shape similar to Figure 3.

Figure 3. Real Power at Generator 2 plotted against Objective Function Value.

The inset of the graph represents a plot over the full range of feasible values;

the larger graph is a “zoomed-in” region focused on the 1% neighborhood of

the optimal objective function value.

36.45 36.5 36.55 36.6 36.65 36.7 36.75 36.8 36.85 36.9 36.95

8081.525

8081.53

8081.535

8081.54

8081.545

8081.55

P

F value

Objective Function Values for 14 Bus Case

Against Generator 2 P Levels

-100 0 100 200

0.8

1

1.2

1.4

x 10

4

There are interesting observations to be made about the

power, reactive power, and voltage profiles. The real power,

since it is the driver of cost in the objective function, seems to

have a very strong correlation with cost. Since our cost terms

are quadratic, it makes sense that over the full range of values

real power has a parabolic relationship to cost.

Reactive power, on the other hand, is rather flat. Conversely

to real power, reactive power is not a factor in the objective

function; the flat relationship we see corresponds well to our

earlier findings regarding the highly elastic nature of reactive

power values. However, the interesting aspect of the reactive

power graph is that it appears quadratic in the immediate

neighborhood of the global optimum. This behavior lends itself

to a convex relationship.

Like reactive power, voltage magnitude is also flat, which

should be expected given its absence from the objective function. Unlike real power and reactive power, voltage magnitude

does not display an even, symmetric curvature around the

optimal value. Rather, most buses display a relatively flat

curvature with voltage levels that are below optimal, while

voltage magnitudes that are too high increase the cost almost

exponentially. Also, it is interesting to note that, like in Figure

3, most voltages have their optimal value at the higher end of

their feasible range.

2) Irregular Graphs: Some variables display irregular behavior with respect to cost, particularly in the form of infeasibile

points within the standard bounds. An examples of this can be

seen in Figure 4:

In the case of power at generator 1, our algorithms find a

single feasible point when power is fixed at 83% of its upper

bound. However, between the 79th and 82nd percentage level,

we fail to achieve convergence. In particular, our non-linear

solver failed to converge after 400+ iterations and 10,000 function evaluations, while the 83% level converged in 8 iterations.

This specific irregularity has no ready explanation.

For bus 12, the optimal solution is strictly less than the upper

Figure 4. Graphs of Reactive Power at Generator 1 Plotted Against Objective

Function Value.

8.9 8.92 8.94 8.96 8.98 9 9.02

x 10

-18

8081.5263

8081.5263

8081.5263

8081.5263

8081.5263

8081.5263

8081.5263

8081.5263

8081.5263

8081.5263

8081.5263

Q

F value

Objective Function Values for 14 Bus Case

Against Generator 1 Q Levels

-10 0 10 20

0.8

1

1.2

1.4

x 10

4

bound (1.06 in this case), yet is infeasible when pushed beyond

the optimal value. In fact, with a 0.1% perturbation in the

positive direction, our non-linear solver fails to find a solution

after 500 iterations. It is unclear what aspect of the network

topology contributes to this infeasibility.

V. CONCLUSIONS

At the conclusion of our investigation, we see no strong

evidence to suggest that the ACOPF problem is nearly convex.

However, many of our observations suggest that the problem

has some convex properties, but too many irregularities to be

considered reasonably convex.

Across every problem in the IEEE test set, we fail to find any

feasible convex combinations of feasible points. Additionally, if

the convex combination favors one point over the other, we find

that our infeasibility metric increases. This result suggests that

not only is the region non-convex, but very immediately so. If

we look at our elasticity results, we see that our optimization

variables can vary greatly without having a significant impact

on the objective function value.

Combining this characteristic with our feasible point exploration, we can conclude that the region is globally flat but

locally very dynamic for any constricted region. Our results

lend support to the frequent observation of local optima; because

the area around the true optimum is flat, solvers may quickly

converge to values close to the optimum. However, the local

topology would make it very easy for the solver to find a

feasible, sub-optimal point and declare optimality.

Finally, our visual examination of the pair-wise relationships

between variable values and cost suggest that many variables

display smooth quadratic behavior, at least in the neighborhood

of the optimal solution. However, the presence of some highly

irregular points and behavior indicative of local optima contradicts the claim that the variables are convex in nature.

REFERENCES

[1] “IEEE Power Systems Test Archive,” College of Engineering University of Washington [Online]. Available:

http://www.ee.washington.edu/research/pstca.

[2] R. D. Zimmerman, C. E. Murillo-Sánchez, and R. J. Thomas, "MATPOWER Steady-State Operations, Planning and Analysis Tools for Power

Systems Research and Education," Power Systems, IEEE Transactions on,

vol. 26, no. 1, pp. 12-19, Feb. 2011.
7

[3] J. Carpentier, “Contribution á l’étude du dispatching économique,” Bulletin

de la Société Française des Électriciens, ser. 8, vol. 3, pp. 431-447, 1962

[4] Javad Lavaei, Steven H. Low, “Convexification of Optimal Power Flow

Problem,” Forty Eighth Annual Allerton Conference on Communication,

Control, and Computing, Monticello, IL, Oct. 2010

[5] A. Klos, J. Wojcicka, “Physical Aspects of the Nonuniqueness of Load

Flow Solutions,” Electrical Power and Energy Systems, 1991