Engineering Optimization
Vol. 00, No. 00, Month 20XX, 1–17
Experiments with hybrid Bernstein global optimization algorithm for the OPF problem in power systems
Bhagyesh V. Patila ∗, L.P.M.I. Sampathb, Ashok Krishanc, J. M. Maciejowskid, K. V. Lingc, H. B. Gooic
aCambridge Centre for Advanced Research and Education in Singapore (CARES) bInterdisciplinary Graduate School, Nanyang Technological University, Singapore
cSchool of Electrical & Electronic Engineering, Nanyang Technological University, Singapore dDepartment of Engineering, University of Cambridge, United Kingdom
(Received 00 Month 20XX; final version received 00 Month 20XX)
This paper presents an algorithm based on the Bernstein form of polynomials for solving the optimal power flow (OPF) problem in electrical power networks. The proposed algorithm combines local and global optimization methods and is therefore referred to as a ‘hybrid’ Bernstein algorithm in the context of this work. The proposed algorithm is a branch-and-bound (B& B) procedure wherein a local search method is used to obtain a good upper bound on the global minimum at each branching node. Subsequently, the Bernstein form of polynomials is used to obtain a lower bound on the global minimum. The performance of the proposed algorithm is compared with the previously reported Bernstein algorithm to demonstrate its efficacy in terms of the chosen performance metrics. Furthermore, the proposed algorithm is tested by solving the OPF problem for several benchmark IEEE power system examples and its performance is compared with generic global optimization solvers such as BARON and COUENNE. The test results demonstrate that the algorithm HBBB delivers satisfactory performance in terms of solution optimality.
Keywords: Bernstein polynomials; Global optimization; Power systems; Optimal power flow; Network optimization; Nonconvex problems.
Nomenclature
1
(A) Sets
2
N Set of all buses.
3
G Set of generator buses.
4
L Set of all lines.
5
N Set of natural numbers.
6
R Set of real numbers.
7
IR Set of compact intervals.
8
S Set of all vertices of an array (bI(x)).
9
S0 Subset of S comprising only index vertices of an array (bI(x)).
10
(B) Parameters
11
n Total number of system buses.
12
PDk, QDk Active and reactive load demands at the kth bus.
13
Yik Line admittance of the transmission line in between buses i and k.
14
ck0, ck1, ck2 Coefficients for the generator cost curve in $/h, $/MWh, and $/MW2h,
15
16 respectively. 17
Gik, Bik Conductance and susceptance of the line in between ith and kth bus.
18
PGkmin, PGkmax Limits on the active power generation capacity at the kth bus.
19
Qmin
Gk, QmaxGk Limits on the reactive power generation capacity at the kth bus.
20
Vkmin, Vkmax Limits on the absolute value of the voltage at the kth bus.
21
Sikmax Limit on the absolute value of the apparent power flow through
22
the line connecting any two buses i and k such that (i, k) ∈ L.
23 x An interval or box. 24 w(x) Width of an interval x. 25 m(x) Midpoint of an interval x. 26
aI Coefficients of polynomial in the power form.
27
BIN Ith Bernstein basis polynomial of degree N .
28
bI Bernstein coefficients.
29
(bI(x)) Array of the Bernstein coefficients.
30
B(x) Bernstein range enclosure.
31
Bgi(x), Bhj(x) Bernstein range enclosures for an inequality and equality constraints.
32
conv Convex hull.
33
t Termination tolerance.
34
zero Tolerance on the equality constraint satisfaction.
35
M ax Subdiv Maximum number of subdivisions for B& B scheme.
36
Iter Count B& B iterations.
37
LBD, U BD Lower and upper bounds.
38
LIter Count List at the Iter Count iteration.
39
t Computational time in seconds.
40
(C) Variables
41
Vdk, Vqk Real and imaginary values of the voltage phasor at the kth bus.
42
PGk, QGk Active and reactive power generation at the kth bus.
43
fk(PGk) A quadratic fuel cost function.
44
Sik Apparent power flow on the line (i, k) ∈ L.
45
l Total number of decision variables.
46
f∗ Global minimum.
47
x∗ Global minimizers.
48
fIter Countlocal Upper bound at the Iter Count iteration.
49
xlocalIter Count Upper bound solution at the Iter Count iteration.
50
fIter Countglobal Lower bound at the Iter Count iteration.
51
1. Introduction
52
Numerical optimization algorithms play a vital role in ensuring the stable and reliable
op-53
eration of modern electric power systems (Kundur (1994); Capitanescu (2016)). Among
54
other applications, optimization algorithms are used in network expansion planning
prob-55
lems and generator scheduling problems. The OPF problem is one such well studied
56
problem in the power systems community. The OPF problem aims at optimizing
net-57
work operations by finding optimal operating points for the electric generators in the
58
system. It achieves this by minimizing the total power generation cost subject to
cer-59
tain network constraints. Some of these constraints include generator active and reactive
power generation limits, bus voltage magnitudes, and network constraints. An excellent
61
recent survey about the OPF problem can be found in Capitanescu (2016).
62
The complexity involved in the OPF problem is mainly two-fold: (i) the size of
real-63
world OPF problems for which a direct solution approach is prohibitive due to memory
64
and computational time limitations and (ii) nonconvex problem structure resulting from
65
highly nonlinear power balance equations, which demand good global optimization
pro-66
cedures to determine the optimal operating points for the generators. In this work, we
67
primarily focus on addressing (ii) with specific application to benchmark IEEE power
68
system examples.
69
Several deterministic solution approaches have been proposed for solving the OPF
70
problem. Prominent among these are sequential linear and quadratic programming,
La-71
grangian relaxation, and interior-point methods (see, for instance Phan and Kalagnanam
72
(2014); Momoh, El-Hawary, and Adapa (1999a); Momoh, El-Hawary, and Adapa (1999b);
73
Gopalakrishnan et al. (2012)). However, as noted above, the OPF problem is nonconvex
74
in nature with multiple equilibrium points (cf. Bukhsh et al. (2013)). Consequently, the
75
aforementioned solution approaches, which typically rely on a ‘convexity’ assumption
76
of the optimization problem, may fail to find the good optimal solution in practice. In
77
addition to the aforementioned solution approaches, semidefinite programming (SDP)
78
relaxation is another popular method which is widely used for solving the OPF problem
79
(Bai et al. (2008)). However, the exactness of the SDP relaxation can only be
guaran-80
teed for radial networks (see, for instance, Kocuk, Dey, and Xu. A. Sun (2016)). Other
81
research directions in the context of the OPF problem are based on the development
82
of convex envelopes (Zhijun, Hou, and Chen (2015)) and decomposition based global
83
optimization methods (Li and Li (2016)).
84
Similarly, in the past decade, a number of non-deterministic solution approaches have
85
also been investigated for solving OPF problems. A few examples of such approaches are
86
ant colony optimization (Soares et al. (2011)), genetic algorithm (Todorovski and Rajicic
87
(2006)), differential evolution (A. A. Abou El Ela, Abido, and Spea (2010); Shaheen,
88
El-Sehiemy, and Farrag (2016)), particle swarm optimization (Abido (2002); Vaisakh
89
and Srinivas (2011); Mohamed et al. (2017)), simulated annealing (Roa-Sepulveda and
90
Pavez-Lazo (2003)), bacterial foraging algorithm (Edward et al. (2013)), and imperialist
91
competitive algorithm (Ghasemi et al. (2014a); Ghasemi et al. (2014b); Ghasemi et al.
92
(2015)). A detailed survey of deterministic and non-deterministic solution approaches for
93
solving the OPF problem can be found in Frank, Steponavice, and Rebennack (2012a)
94
and Frank, Steponavice, and Rebennack (2012b).
95
We note that the last two decades have witnessed the emergence of interval form based
96
B& B has emerged as a promising framework to solve nonconvex optimization problems
97
(Vaidyanathan and M. El-Halwagi (1996); Hansen and Walster (2005)). This is evident
98
from the seminal work on αBB relaxation by Adjiman, Androulakis, and Floudas (1998)
99
which had yielded B& B implementations, such as BARON (Tawarmalani and Sahinidis
100
2005) and COUENNE (Belotti et al. 2009). The impressive performances of BARON
101
and COUENNE on a wide variety of optimization problems has been well documented.
102
In recent times, various modifications of the aforementioned B& B implementations have
103
also been reported in the literature (see, work reported by Grimstad and Sandnes (2016),
104
Gerard, Kppe, and Louveaux (2017), Castro (2017), and references therein). This has
mo-105
tivated us to investigate an alternative interval form based Bernstein global optimization
106
algorithm to solve the polynomial OPF problem.
107
This work explores the well-known Bernstein form of polynomials (Ratschek and Rokne
108
(1988)), and uses several attractive ‘geometrical’ properties associated with the Bernstein
109
form (refer to Section 3.1). Optimization procedures based on the Bernstein form, also
110
called Bernstein global optimization algorithms, have shown good promise in solving hard
111
(nonconvex) nonlinear programming (NLP) and mixed-integer nonlinear programming
(MINLP) problems (see, for instance, Nataraj and Arounassalame (2011); Patil, Nataraj,
113
and Bhartiya (2012)). Recently, a Bernstein global optimization algorithm was also
pro-114
posed to solve the OPF problem for small power networks (see Patil et al. (2016)). As
115
such, we believe that further investigations in the context of the OPF problem using the
116
Bernstein global optimization approach seems to be a promising research direction.
117
In this work, we propose a hybrid1 branch-and-bound (B& B) algorithmic scheme.
118
Specifically, we use the Bernstein polynomial form in conjunction with a local NLP
solv-119
ing technique to form a new hybrid Bernstein global optimization algorithm(hereinafter
120
referred to as algorithm HBBB). The algorithm HBBB uses an iterative subdivision
pro-121
cedure in a B& B scheme, wherein a series of upper and lower bounding subproblems
122
are solved at each node of the B& B tree. We obtain the upper bound using MATLAB’s
123
‘fmincon’ as a local NLP solver and the lower bound using the minimum Bernstein
coef-124
ficient value (see Theorem 3.1). Furthermore, we follow the principle of interval analysis,
125
wherein iterative subdivisions are performed at each step of a B& B scheme. This
en-126
ables the B& B scheme to converge the upper and lower bounds within a user-specified
127
accuracy. The overall schematic of the proposed approach is depicted in Figure 1.
128
We first show with a simple nonlinear optimization problem the effectiveness of the
129
algorithm HBBB over the previously reported Bernstein algorithm in (Nataraj and
130
Arounassalame (2011)), and the state-of-the-art BARON solver. The performance
com-131
parison is made on the basis of the number of boxes processed, and the computational
132
time required to locate the correct global solution. Subsequently, we assess the scalability
133
and performance of the algorithm HBBB over the OPF problem for the several
bench-134
mark IEEE power system network examples. The performance of the proposed algorithm
135
HBBB is compared with the generic global optimization solvers BARON (Tawarmalani
136
and Sahinidis (2005)) and COUENNE (Belotti et al. (2009)).
137
Figure 1. The hybrid Bernstein B& B scheme illustrating the lower (L) and upper (U) bounding processes followed by subdivision. P represents the original (nonconvex) problem, whose global minimum is to be sought and R is the convex relaxation obtained (in our case using the Bernstein polynomial form).
1The word hybrid in this context means that our algorithm is a combination of local and global optimization
methods. To the best of the authors’ knowledge, this is the first work which explores the use of local solving techniques for the early pruning of nodes in a B& B tree in the context of Bernstein global optimization algorithms.
The remainder of this paper is organized as follow. The classical OPF formulation for
138
the power network first is first introduced in Section 2. Next, the Bernstein polynomial
139
form is briefly introduced in Section 3. This is followed by a description of our proposed
140
algorithm HBBB in Section 4. The results from numerical studies performed with our
141
algorithm HBBB on some benchmark IEEE power system network examples are reported
142
in Section 5. The results of the numerical studies are also compared with those obtained
143
using well established global optimization solvers in Section 5. Finally, some concluding
144
remarks and directions for future research are given in Section 6.
145
2. Optimal power flow problem
146
In this section, we briefly present the classical OPF formulation along the lines of Molzahn
147
et al. (2013) which is in terms of the rectangular power and voltage co-ordinates. The
148
objective of the OPF problem is to minimize the cost of real power generation. The
149
problem is subject to constraints such as the power balance, satisfaction of bus voltage
150
limits, active and reactive power generation limits, and line-flow limits.
151
Consider an n-bus power system, where N = {1, 2, . . . , n} represents the set of all
152
buses; G represents the set of generator buses and L represents the set of all lines. Let
153
PDk and QDk represent the active and reactive power demands respectively at each bus
154
k ∈ N . Let Vk = Vdk+ jVqk represent the voltage phasor in rectangular coordinates at
155
each bus k ∈ N . Let PGk and QGk represent the active and reactive power generations
156
respectively at each generator bus k ∈ G. Let Sik represent the apparent power flow and
157
Yik = Gik+ jBik denote the line admittance of the line (i, k) ∈ L respectively.
158
The quadratic fuel cost function associated with each generator k ∈ G representing a
159
$/h operating cost is given below.
160
fk(PGk) = ck2PGk2 + ck1PGk+ ck0 ∀ k ∈ G (1) Then, the classical OPF optimization problem can be stated as follows:
min PGk,QGk,Vdk,Vqk f =X k∈G fk(PGk) (2) subject to PGk− PDk = Vdk n X i=1 (GikVdi− BikVqi) + Vqk n X i=1 (BikVdi+ GikVqi) ∀ k ∈ N (3) QGk− QDk = Vdk n X i=1 (−BikVdi− GikVqi) + Vqk n X i=1 (GikVdi− BikVqi) ∀ k ∈ N (4) PGkmin 6 PGk6 PGkmax ∀ k ∈ G (5) QminGk 6 QGk6 QmaxGk ∀ k ∈ G (6) Vkmin26 Vdk2 + Vqk2 6 (Vkmax) 2 ∀ k ∈ N (7) Pki = Gik Vdk2 + Vqk2 − Gik(VdkVdi+ VqkVqi) + Bik(VdiVqk− VdkVqi) ∀ k ∈ N (8) Qki = Bik Vdk2 + Vqk2 − Gik(VdiVqk− VdkVqi) − Bik(VdkVdi+ VqkVqi) ∀ k ∈ N (9) q Pki2 + Q2ki6 Skimax ∀ (i, k) ∈ L (10)
The objective function (2) is the minimization of the total operating cost of the system.
161
Equations (3) and (4) are the real and reactive power balance constraints at each bus
162
k. Equations (3) and (4) are formulated considering the Kirchoff’s laws of power flow
163
through branches attached to buses. Active and reactive power generation capability
164
margins are considered in (5) and (6) respectively. Equations (7) and (10) represent the
165
voltage security margins and the line apparent power flow capacities respectively.
166
Remark 1 We note that the constraints (3)-(4) possess multilinear terms in the real and
167
imaginary voltage components. Hence, the OPF problem turns out to be a nonconvex
168
nonlinear programming (NLP) problem, albeit polynomial in nature (i.e., (2)-(4) are
169
always polynomials in the power form shown in (11)).
170
3. The Bernstein polynomial approach
171
In this section, we introduce some notions related to interval analysis and the theory
172
pertaining to the Bernstein form of polynomials presented in Patil, Nataraj, and Bhartiya
173
(2012). Interested readers may also refer to Ratschek and Rokne (1988) and Moore,
174
Kearfott, and Cloud (2009) for more details about this topic.
175
3.1 Bernstein form
176
Let l ∈ N be the number of variables and x = (x1, x2, ..., xl) ∈ Rl. A multi-index I is
de-177
fined as I = (i1, i2, ..., il) ∈ Nl and the multi-power xI is defined as xI = (xi11, xi22, ..., xill).
178
Another multi-index N is defined as N = (n1, n2, ..., nl). Inequalities I ≤ N for
multi-179
indices are meant component-wise. With I = (i1, ..., ir−1, ir, ir+1, ..., il), we associate the
180
index Ir,k given by Ir,k = (i1, ..., ir−1, ir+k, ir+1, ..., il), where 0 ≤ ir+k ≤ nr. Also we
181 write NI for n1 i1 · · · nl il
and (N/I) for (n1/i1, n2/i2, ..., nl/il) provided that 0 < ik,
182
k = 1, 2, . . . , l.
183
A real, bounded and closed interval x is defined as follows: x = [x, x] := [inf x, sup x] ∈ IR,
where IR denotes the set of compact intervals. Let w(x) denote the width of x, that
184
is w(x) := x − x, and m(x) denote the midpoint of x, that is m(x) := (x + x)/2. For
185
an l-dimensional interval vector or box x = (x1, x2, . . . , xl) ∈ IRl, the width of x is
186
w(x) := max(w(x1), w(x2), . . . , w(xl)).
187
We can write an l-variate polynomial p in the power form as shown below.
188
p(x) = X I≤N
aIxI, x ∈ Rl, (11)
with N being the degree of p. We expand a given multivariate polynomial p into Bernstein
189
polynomials to obtain the bounds for its range over an l-dimensional box x. The Ith
190
Bernstein basis polynomial of degree N is defined as follows:
191 BNI (x) = Bn1 i1 (x1) · · · B nl il (xl), x ∈ R l, (12)
where for ij = 0, 1, ..., nj, j = 1, 2, . . . , l 192 Bnj ij (xj) = nj ij (x j− xj)ij(xj− xj)nj−ij (xj− xj)nj . (13)
The Bernstein coefficients bI(x) of p over the box x are given by the following equation:
193 bI(x) = X J ≤I I J N J w(x) J X K≤J K J (inf x)K−JaK, I ≤ N . (14)
The Bernstein form of a multivariate polynomial p is defined by
194
p (x) = X I≤N
bI(x) BIN(x) . (15)
The Bernstein coefficients are collected in an array (bI(x))I∈S, where S = {I : I ≤ N }.
195
We denote S0 as a special subset of the index set S comprising indices of the vertices of
196
this array, i.e.
197
S0 := {0, n1} × {0, n2} × · · · × {0, nl}.
Theorem 3.1 (Range enclosure property) Let p be a polynomial of degree N , and let
198
p(x) denote the range of p on a given box x ∈ IRl. Then,
199
p(x) ⊆ B(x) :=min (bI(x))I∈S, max (bI(x))I∈S . (16) Proof : See Garloff (1993).
200
Remark 2 The above theorem states that the minimum and maximum coefficients of the
201
array (bI(x))I∈S provide lower and upper bounds for the range. This forms the Bernstein
202
range enclosure defined by B(x) in (16).
203
Lemma 3.2 (Convex hull property) Let (bI(x)) be an array of Bernstein coefficients for
204
a polynomial p(x) on a given box x ∈ IRl. Then, the following property holds:
205
conv (x, p(x)) ⊆ (I/N, bI(x) : I ∈ S) , where conv (x, p(x)) denotes the convex hull of p.
206
Remark 3 The above lemma states that the range of p(x) is contained in the convex hull
207
of the control points (I/N, (bI)). Figure 2 illustrates this fact, wherein the polynomial
208
function is represented as p(x) and Bernstein coefficients are denoted as b0, b1, b2, b3, b4,
209
and b5. The dotted lines in Figure 2 define the convex hull. Furthermore, this Bernstein
210
range enclosure can be successively sharpened by the continuous domain subdivision
211
procedure. This is illustrated in Figure 3.
Figure 2. A polynomial function p, its Bernstein coefficients and the convex hull over a box x = [0, 1].
Figure 3. Improvement in the range enclosure of p with a subdivision of the original box x = [0, 1]. (b10, b11, b12, b13, b14, b15), and (b20, b21, b22, b23, b24, b25) are the Bernstein coefficients over x1 = [0, 0.5] and
x2= [0.5, 1], respectively.
The following properties follow immediately from Theorem 3.1.
213
Lemma 3.3 Let B(x) be the Bernstein range enclosure for a polynomial p(x) on a
214
given box x ∈ IRl. Then, the following properties hold
215 (1) B(x) ≤ 0 ⇒ p(x) ≤ 0 for all x ∈ x. 216 (2) B(x) > 0 ⇒ p(x) > 0 for all x ∈ x. 217 (3) 0 /∈ B(x) ⇒ p(x) 6= 0 for all x ∈ x. 218
(4) B(x) ⊆ [−, ] ⇒ p(x) ∈ [−, ] for all x ∈ x, where > 0.
219
4. Hybrid Bernstein global optimization algorithm
220
In this section, we outline the proposed algorithm HBBB to solve polynomial NLP
prob-221
lems. We first briefly describe the algorithm. Subsequently, we demonstrate the strength
222
of the algorithm HBBB over the previously reported Bernstein algorithm (Nataraj and
223
Arounassalame (2011)) on a nonlinear optimization problem. Furthermore, with the
op-224
timization problem, we also demonstrate the merits of the algorithm HBBB over the
225
BARON solver. Finally, in Section 5, the algorithm HBBB is used to determine the
op-226
timal solution (global minimum and minimizers) of the OPF problem (2)-(4) described
227
in Section 2
228
Briefly, the algorithm works as follows. At the outset, for the original problem, a
feasible solution (from the search box xIter Count) is computed using a local search
230
method. The obtained minimum is called a feasible upper bound (U BD). Next, a valid
231
lower bound (LBD) on the optimal objective function value is obtained using the
232
minimum Bernstein coefficient value. After establishing the upper and lower bounds on
233
the global minimum, we refine them. This is accomplished by successively subdividing
234
the initial box xIter Count at the midpoint along the longest side, resulting in two smaller
235
boxes (xIter Count,1, xIter Count,2). This procedure generates a nonincreasing sequence
236
for the upper bound and a nondecreasing sequence for the lower bound. Within a finite
237
number of subdivisions, the gap between U BD and LBD shrinks to the termination
238
accuracy t. Finally, the algorithm terminates with the current upper bounding solution
239
as the global solution.
240
241
Algorithm hybrid Bernstein: [f∗, x∗] = HBBB(f, g, h, x, t, zero, M ax Subdiv)
242
243
Inputs: The objective function (f ) and constraints (g, h) in the power form, the
244
initial search box x, parameter tas the termination accuracy, tolerance parameter zero
245
to which the equality constraints are to be satisfied, and M ax Subdiv as the maximum
246
number of subdivisions to be performed to locate the global solution.
247
248 Outputs: A global minimum f∗ and global minimizers x∗ over a box x. 249 250 BEGIN Algorithm 251 (1) {Initialization} 252
Set Iter Count ←− 0, Subdiv N o ←− 0, LBD ← −∞, U BD ← ∞, xIter Count ← x,
253
xcIter Count← m(xIter Count), LIter Count ← {}.
254
(2) {Upper bound computation}
255
Solve OPF (2)-(4) over xIter Countusing a local search method. We use xcIter Count as
256
an initial point to start the optimization for a local NLP solver. Denote the obtained
257
minimum as fIter Countlocal and minimzers as xlocalIter Count. If fIter Countlocal is feasible, and
258
fIter Countlocal < U BD, then update U BD as U BD ← fIter Countlocal .
259
(3) {Subdivision}
260
Subdivide the current box xIter Count into two smaller subboxes
261
(a) Subdiv N o ← Subdiv N o + 1.
262
(b) Choose a coordinate direction λ parallel to which xIter Count,1× · · · × xIter Count,l
263
has an edge of maximum length, that is λ ∈ {i : w(x) := w(xi), i = 1, . . . , l}.
264
(c) Bisect xIter Count normal to direction λ, getting boxes xIter Count,1 and
265
xIter Count,2, such that xIter Count = xIter Count,1∪ xIter Count,2.
266
(4) {Lower bound computation}
267
for k = 1, 2
268
(a) Find the Bernstein coefficients and the corresponding Bernstein range enclosure
269
of the objective function (f ) over xIter Count,k as
270
b0(xIter Count,k) and B0(xIter Count,k), respectively.
271
(b) Set fIter Countglobal := min Bo(xIter Count,k).
272
(c) If fIter Countglobal > U BD, then go to substep (g).
273
(d) for i = 1, 2, . . . , m
274
(i) Find the Bernstein coefficients and the corresponding Bernstein range enclosure
275
of the inequality constraint polynomial (gi) over xIter Count,kas bgi(xIter Count,k)
276
and Bgi(xIter Count,k), respectively.
277
(ii) If Bgi(xIter Count,k) > 0, then go to substep (g).
278
(iii) If Bgi(xIter Count,k) ≤ 0, then go to substep (e)
279
(e) for j = 1, 2, . . . , n
(i) Find the Bernstein coefficients and the corresponding Bernstein range enclosure
281
of the equality constraint polynomial (hj) over xIter Count,k as bhj(xIter Count,k)
282
and Bhj(xIter Count,k), respectively.
283
(ii) If 0 /∈ Bhj(xIter Count,k) then go to substep (g).
284
(iii) If Bhj(xIter Count,k) ⊆ [−zero, zero] then go to substep (f).
285
(f) Enter (xIter Count,k, fIter Countglobal ) into the list LIter Count such that the second
mem-286
bers of all the items of the list do not decrease.
287
(g) end (of k−loop).
288
(5) {Update iteration counter and lower bound}
289
(a) Set Iter Count ← Iter Count + 1.
290
(b) Update LBD to the minimum of the second entries over all the items in LIter Count.
291
Similarly, fetch the first entry corresponding to this minimum and denote it as
292
xIter Count. Also compute xcIter Count as xcIter Count← m (xIter Count).
293
(6) {Termination condition}
294
If Subdiv N o < M ax Subdiv or U BD − LBD > t, then go to step 2. Else go to
295
step 7.
296
(7) {Compute global solution}
297
Return the global minimum and global minimizers as f∗ ← U BD, and x∗ ←
298
xlocalIter Count, respectively.
299
END Algorithm
300
Remark 4 The algorithm HBBB follows a classical subdivision procedure for the
orig-301
inal box x. As such, the feasible region for x shrinks with each iteration. Furthermore,
302
the objective function value is a function of x. Hence, the sequence of upper and lower
303
bounds converge in the limit within a finite number of iterations (cf. Ratschek and Rokne
304
(1988)).
305
Remark 5 The subdivision of x aids in raising the lower bound (computed using the
306
Bernstein form) of the objective function value (cf. Patil, Nataraj, and Bhartiya (2012)),
307
thereby speeding up the convergence of the algorithm HBBB.
308
Theorem 4.1 The algorithm HBBB based on the upper and lower bounding schemes
309
converges to the global optimal solution.
310
Proof : The algorithm HBBB is both bound consistent (see Remark 4), and bound
311
improving (see Remark 5). Hence, it is also convergent (Tawarmalani and Sahinidis
312
(2002), Li and Sun (2010)).
313
314
Demonstrative Example: The strength of the algorithm HBBB is now
demon-315
strated with a nonlinear optimization problem adapted from (Lebbah, Michel, and
316
Rueher (2007)). We first demonstrate the strength of the algorithm HBBB with the
317
previously reported Bernstein algorithm from Nataraj and Arounassalame (2011) in
318
terms of the number of subdivisions and the computational time. Subsequently, we also
319
demonstrate the merit of the algorithm HBBB in terms of the solution optimality when
320
compared with the BARON solver.
321
Consider the following nonlinear optimization problem:
322 min x,y f = x subject to y − x2≥ 0 y − x2(x − 2) + 10−5 ≤ 0 x ∈ x = [−10, 10], y ∈ y = [−10, 10] (P)
Figure 4. Geometrical representation of an optimization problem (P) (Lebbah, Michel, and Rueher (2007)). Geometrically, this problem is shown in Figure 4. As pointed out by Lebbah, Michel,
323
and Rueher (2007), the solution of the problem (P) lies in the neighborhood of the point
324
x ≈ 3, y ≈ 9, with the global minimum as f∗ ≈ 3.
325
We observed that both the classical Bernstein algorithm and HBBB algorithm
con-326
verged to the correct global solution reported by the Lebbah, Michel, and Rueher (2007)
327
for the problem (P). However, the algorithm HBBB was found to be superior in term of
328
its performance during the global search process (cf. Table 1). The algorithm HBBB
re-329
quired approximately 66% fewer subdivisions, thereby reducing the computational time
330
required by approximately 30%.
331
Table 1. Performance comparison of the previously reported Bernstein algorithm in Nataraj and Arounassalame (2011) and the algorithm HBBB on the optimization problem (P).
Performance Bernstein algorithm HBBB metrics (Nataraj and Arounassalame (2011))
Number of 294 100
subdivisions
Computational 2.53 1.78
time (seconds)
Furthermore, the problem (P) was solved using BARON with an optimality tolerance
332
10−12 (i.e. in GAMS, set optca = 10E-12). BARON reported f∗= 0 as the global
min-333
imum, and x∗ = 0, y∗ = 0 as the global minimizers. This successfully demonstrates the
334
merit of the algorithm HBBB when compared with BARON for this particular
optimiza-335
tion problem.
336
5. Numerical results
337
In this section, we report results from solving the OPF problem for several benchmark
338
IEEE power system models with our hybrid Bernstein algorithm (HBBB). The
bench-339
mark IEEE power system models were adapted from Zimmerman, Murillo-Sanchez, and
340
Thomas (2011). We analyze the results from two perspectives. First, the performance
341
of algorithm HBBB for several test cases (3-, IEEE 9-, IEEE 14-, IEEE 30-, and IEEE
342
39-bus systems) is compared with the performance of general purpose global
optimiza-343
tion solvers like BARON (Tawarmalani and Sahinidis (2005)) and COUENNE (Belotti
344
et al. (2009)). Subsequently, we study the computational time growth of the algorithm
HBBB. This was achieved by increasing the number of subdivisions (M ax Subdiv) and
346
tightening the termination accuracy (t) in the algorithm HBBB.
347
The algorithm HBBB was implemented in MATLAB (R2014a). All experiments were
348
carried out on a desktop PC with an IntelrCore i7-5500U CPU processor running at 2.40
349
GHz with a 8 GB RAM. The termination accuracy t and equality constraint feasibility
350
tolerance zero were both specified as 10−3. For testing with BARON and COUENNE
351
solvers, all test cases were modeled in General Algebraic Modeling System (GAMS), and
352
solved using the NEOS server for optimization (NEOS server (2018)).
353
Case I (Performance comparison with BARON and COUENNE solvers)
354
Table 2 shows the OPF solutions obtained using the different solution approaches for
sev-355
eral benchmark test cases (3-, IEEE 9-, IEEE 14-, IEEE 30-, and IEEE 39-bus systems).
356
Table 2 shows the different test cases and their corresponding numbers of optimization
357
decision variables, apart from the following two performance metrics - computational time
358
in seconds and the optimal fuel cost in $/h. Specifically, we analyze the performance of the
359
algorithm HBBB by setting the number of subdivisions to 25 and termination accuracy t
360
to 10−3. Figure 5 illustrates the comparison between the algorithm HBBB, BARON and
361
COUENNE in terms of the computational time. It can be seen that algorithm HBBB was
362
computationally slower compared with BARON for most test cases. However, algorithm
363
HBBB performed exceptionally well for the IEEE 30-bus system test case wherein it was
364
94% faster than BARON. For some test cases (IEEE 14-, IEEE 30-, and IEEE 39-bus
365
systems), COUENNE was found to be the slowest. On an average, COUENNE was 96%
366
and 83% slower than algorithm HBBB and BARON, respectively. Furthermore, we also
367
found that the algorithm HBBB was competitive in terms of locating the correct optimal
368
solution for all the test cases when compared with with BARON.
369
Table 2. Comparison of the OPF cost (2) (f∗, in $/h) and computational time (t, in seconds) for benchmark IEEE test cases under different solution approaches.
Test Number of Performance Solver/Algorithm
case decision variables metric BARON COUENNE HBBB†
3-bus 12 f∗ 5703.52 5703.52 5703.52 t 0.5 0.2 0.45 IEEE-9 24 f∗ 5296.68 5296.68 5296.68 bus t 0.2 7.72 7.62 IEEE-14 38 f∗ 8081.53 8081.53 8081.53 bus t 0.3 371.72 13.64 IEEE-30 72 f∗ 576.89 576.89 576.89 bus t 396.93 1021.01 24.41 IEEE-39 98 f∗ 41864.18 41864.18 41864.18 bus t 4.25 1026.83 48.75
3 9 14 30 39 0 200 400 600 800 1000 1200 Test instances Time (sec) BARON COUENNE HBBB
Figure 5. Computational time comparison for five test cases (3-, 9-, 14-, 30-, and 39-bus systems) solved using BARON, COUENNE and algorithm HBBB.
Case II (Computational time growth study)
370
In this case, we study the growth in computational time for algorithm HBBB with an
371
increasing number of subdivisions (50, 100, 150) and tightened termination accuracy t
372
(10−8). Table 3 reports the results of our experiments. From Figure 6, it is observed
373
that with an increase in the number of subdivisions, the computational time required
374
increases almost linearly. However, no improvement in terms of optimality was observed
375
when compared with the algorithm HBBB results reported in Case I. Furthermore, we
376
also analyzed the degree to which the equality constraints in (3)-(4) are satisfied for the
377
five OPF test cases considered in this work. This is particularly important as the power
378
supply and demand need to be balanced in real-time. The results are shown in Table 4.
379
We observed that at the optimal solution, the equality constraints are tightly satisfied
380
for all the test cases considered in this study.
381
Table 3. Comparison of the cost function (2) (f∗, in $/h) and computational time (t, in seconds) for
bench-mark IEEE test cases solved using the algorithm HBBB with increasing number of subdivisions and tightened termination accuracy t.
Test Performance Number of subdivisions (t= 10−8)
instance metric 50 100 150 3-bus f∗ 5703.52 5703.52 5703.52 t 3.44 7.95 17.90 IEEE-9 f∗ 5296.68 5296.68 5296.68 bus t 8.25 24.6 56.01 IEEE-14 f∗ 8081.53 8081.53 8081.53 bus t 32.43 52.47 79.15 IEEE-30 f∗ 576.89 576.89 576.89 bus t 52.42 125.27 322.82 IEEE-39 f∗ 41864.18 41864.18 41864.18 bus t 109.20 277.86 455.17
50 60 70 80 90 100 110 120 130 140 150 0 50 100 150 200 250 300 350 400 450 500 Number of subdivisions Time (sec) 3 bus IEEE 9 bus IEEE 14 bus IEEE 30 bus IEEE 39 bus
Figure 6. Computational time growth of the algorithm HBBB with increasing number of subdivisions (50, 100, 150) and tightened termination accuracy t (10−8).
Table 4. Equality constraint satisfaction at the optimal solution for five benchmark IEEE test cases solved using the algorithm HBBB under a tightened termination accuracy t= 10−8.
Test Equality constraint
case Mean Max
3-bus 7.04×10−11 4.95×10−10 IEEE-9 bus 3.57×10−12 1.74×10−11 IEEE-14 bus 8.97×10−12 1.10×10−10 IEEE-30 bus 2.05×10−14 1.39×10−13 IEEE-39 bus 1.19×10−14 3.89×10−13 6. Conclusions 382
In this work, we presented a new B& B scheme in the context of the OPF problem. Our
383
scheme was based on the concept of sequential improvement in the upper and lower
384
bounds of a B& B tree. The interesting feature of our approach was the use of the
Bern-385
stein polynomial form in conjunction with a local search method (a ‘hybrid’ algorithm
386
HBBB in our terminology). The efficacy of the algorithm HBBB was compared with
387
the previously reported Bernstein algorithm using a nonlinear optimization instance.
388
Furthermore, the same optimization instance was also used to demonstrate the merits
389
of the algorithm HBBB over the BARON solver. Further, to ascertain the
practicabil-390
ity of the algorithm HBBB, we tested it on several benchmark IEEE OPF instances
391
and compared its performance with well established global optimization solvers such as
392
BARON and COUENNE. In terms of computational time, the algorithm HBBB was
393
slower than BARON except for one test instance (IEEE 30-bus system), where it
per-394
formed exceptionally well. On the other hand, the algorithm HBBB was found to be
395
faster than COUENNE for most test cases. We note that the algorithm HBBB was able
396
to achieve the same optimality as BARON and COUENNE in terms of fuel cost for the
397
OPF problem.
398
The work reported in this paper can be extended in the following directions:
• The OPF problem in this work was restricted to small to medium-size power systems
400
(to be precise, 3- to IEEE 39-bus). It is well-known that the size of OPF problem
401
grows enormously with the size of the power system network. In such circumstances,
402
distributed optimization algorithms hold a lot of promise. As such, we plan to extend
403
the algorithm HBBB into a distributed framework.
404
• The problem formulation in this work considered a traditional fossil fuel based power
405
generation network. The inclusion of intermittent renewable energy sources makes the
406
OPF problem more challenging. In this scenario, solving the OPF problem requires
407
the adoption of robust optimization procedures with chance constraints. In future, we
408
plan to extend the algorithm HBBB to solve such problems.
409
Acknowledgement
410
This research is supported by the National Research Foundation, Prime Ministers Office,
411
Singapore under its CREATE programme.
412
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