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Economic Dispatch for a Distribution Network with Intermittent Renewables and Tap Changers

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Economic Dispatch for a Distribution Network with
Intermittent Renewables and Tap Changers
Mikhail A. Bragin, Bing Yan, Yan Li, Peter B. Luh, Peng Zhang
Department of Electrical and Computer Engineering
University of Connecticut
Storrs, USA
[email protected]
Abstract—Urban power distribution networks (UDNs) play an
important role but they have not been designed to sustain the everincreasing growth of distributed generation such as solar. Because
of the intermittency of such generation, UDNs are suffering from
voltage and frequency fluctuations. Moreover, to maintain power
quality and reliability, grid devices such as transformer taps are
forced to be adjusted frequently, and rapidly reach their end of
lives. In this paper, an urban network with one solar farm and
several distributed generators is considered. It is assumed that the
network is well-balanced, and economic dispatch (ED) is
performed. With significant levels of solar penetration, such ED
is challenging since (1) the intermittent nature of solar generation
makes the problem stochastic and complicated; (2) AC power flow
and tap changer equations make the problem highly nonlinear;
and (3) discrete decisions (tap positions) makes the problem
combinatorial. These difficulties will be resolved by (1) handling
uncertainties through the use of Markov chains; (2) novel dynamic
linearization through the use of absolute-value functions; (3) a
decomposition and coordination approach with accelerated
convergence. Testing results on a simple 3-bus and a modified 34bus system demonstrate that the method converges fast, and has
the potential to solve practical distribution ED problems.
Keywords— Distribution Network Economic Dispatch; AC power
flow; Solar Generation; Absolute-Value Linearization; Surrogate
Lagrangian Relaxation
I. INTRODUCTION
The proliferation of intermittent renewables resources, such as
solar farms, in urban power distribution networks (UDNs)
frequently leads to voltage and frequency fluctuations. To
maintain power quality and reliability, grid devices such as
transformer taps [1] are forced to be adjusted frequently, and
rapidly reach their end of lives. As an example, Hawaii utilities
reported that their on-load tap changer (OLTC) transformers,
traditionally maintenance-free during a 40-year lifespan, would
be maintained every three months and retire within two years
because the PV-induced voltage fluctuations made them
adjusted over 300 times per day [2]. The consideration of tap
changers to control voltage is therefore crucial. To achieve this,
Economic Dispatch (ED) has been recognized in recent years as
an important mechanism for distribution networks [3-7]. Such
an ED problem, however, is challenging since (1) the
intermittent nature of solar generation makes the problem
stochastic and complicated; (2) the consideration of AC power
flow and tap changer equations to control voltage fluctuations
make the problem highly nonlinear; and (3) the consideration of
discrete decisions (tap positions) makes the new ED problem
mixed-integer and combinatorial.
The purpose of this paper is to formulate the economic
dispatch problem while considering AC power flow and tap
changers, and to develop the novel solution methodology. In
Section II, the literature review will be provided. Existing
literature of ED problems, AC power flow and tap changers will
first be reviewed. Then, relevant optimization methods such as
Lagrangian relaxation and branch-and-cut that have been used
in power industry will be reviewed. While Lagrangian
relaxation has been traditionally used to exploit separability, the
presence of cross products of decision variables within AC
power flow, the problem is not separable in the traditional sense.
Moreover, the traditional LR requires solving the entire relaxed
problem thereby entailing high computational effort and
zigzagging of multipliers and resulting in slow convergence.
Our recent breakthrough – surrogate Lagrangian relaxation
(SLR) which overcomes all major difficulties of standard LR
will also be reviewed.
In Section III, the ED problem formulation whereby the
expected cost of tap changes is co-optimized together with the
expected generation cost while satisfying generator capacity,
ramp-rate, AC power flow, line capacity and tap changer
constraints.
In Section IV, the solution methodology is developed
building upon our recent SLR. After formulating the relaxed
problem, the convergence is accelerated through the use of
quadratic penalties following the concept of Augmented
Lagrangian Relaxation. Then the resulting nonlinear relaxed
problem is linearized through the novel use of dynamic
linearization and absolute-value functions. After decomposing
the resulting linearized relaxed problem into subproblems,
subproblem solutions are coordinated by updating Lagrangian
multipliers following the SLR framework.
In Section V, through the use of a 3-bus system with meshed
topology and a modified 34-bus system with radial topology, it
is be demonstrated that the new method is computationally
efficient and can generate high-quality feasible solutions. For
these problems, good feasible solutions are obtained within the
matter of seconds.
II. LITERATURE REVIEW
In this Section, the modeling aspects of the problem such as
modeling of AC power flow and tap changers will be reviewed
first. Then, existing and relevant methods that have been used
in power systems will be reviewed. Since the problem under
This work was supported by the National Science Foundation under Award No. CNS-1647209. Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the author(s) and do not necessarily reflect the views of the NSF.
978-1-5386-7703-2/18/$31.00 ©2018 IEEE
consideration is non-linear, existing linearization methods
together with MILP methods will also be reviewed.
A. Modeling of Distriburtion System with AC power flow and
Tap Chnagers
The importance of an Economic Dispatch problem within
the distribution network has been recognized in recent years to
handle renewable energy uncertainties as well as to meet the
stability requirement of the distribution network [3-7].
Different aspects of ED within a distribution network have been
considered such as consideration of wind turbines and their
probabilistic modeling [3], consideration of power losses [4],
evaluation of dynamic pricing and peak power limiting based
on demand response [5], and unbalanced three-phase AC power
flow [6] by using a “semi-definite relaxation” of AC power
flow in considered, which is accurate for radial topology. For
the general topology, AC power flow in polar and rectangular
coordinates has been considered. Within polar coordinates the
power flow involves three way products of voltage decision
variables at two adjacent buses and a trigonometric function of
a phase angle between these two voltages. As compared to a
formulation of AC power flow in polar coordinates, a
formulation of the AC power flow model in rectangular
coordinates [8] and [9] is modeled using complex voltages and,
therefore, involves cross-products of two only real or imaginary
components of voltages at adjacent buses. Phase angles can
then be computed based on complex voltage values as needed.
To reduce voltage deviations, tap changers have been used
[1]. Generally, the goal is to keep the voltage amplitude within
the pre-defined limits. With high levels of renewable
penetration, however, to maintain power quality and reliability,
transformer taps are forced to be adjusted frequently. As a
result, they can rapidly reach their end of lives or suffer from
premature failures. As an example, Hawaii utilities reported
that their on-load tap changer (OLTC) transformers,
traditionally maintenance-free during a 40-year lifespan, would
be maintained every three months and retire within two years
because the PV-induced voltage fluctuations made the OLTC
be adjusted over 300 times per day [2].
As a result, with significant levels of solar penetration, an
ED problem is challenging because the intermittent nature of
solar generation, the consideration of nonlinear AC power flow
and tap changer equations to control voltage fluctuations and
the consideration of discrete decisions to model tap positions.
In the following, optimization methods that have been used in
power systems will be reviewed.
B. Relevant Optimization Methods
Lagrangian relaxation. Lagrangian relaxation (LR) in
combination with subgradient methods [10-11] was
traditionally used to solve mixed-integer linear programming
(MILP) problems by exploiting separability. After relaxing
system-wide coupling constraints such as demand or
transmission capacity constraints, and decomposing the relaxed
problem into subproblems associated with individual
subsystems, subproblem solutions are coordinated by iterative
updating of Lagrangian multipliers (“shadow prices” or simply
“prices.”). The major difficulties of standard LR with
subgradient methods are that (1) they require solving all
subproblems to update multipliers, leading to high
computational effort and zigzagging of multipliers; and (2) the
1
The smallest convex set containing feasible solutions.
convergence proof requires the knowledge of the optimal dual
value, which is generally unknown. While convergence can be
achieved in practice by adaptively estimating the optimal dual
value, such process is typically inefficient. As a result of the
above difficulties, overall convergence may be very slow.
When it comes to solving problems with AC power flow,
relaxed problems within the method are not separable in a
traditional sense because of cross-products of voltage of
adjacent buses.
Surrogate Lagrangian Relaxation (SLR) [12]. As presented
above, traditional LR suffers from several major difficulties.
All these difficulties have now been overcome by SLR whereby
after decomposing a relaxed problem, it is sufficient to solve
one or multiple subproblems with much reduced effort subject
to the simple “surrogate optimality condition” to update prices.
The resulting “surrogate” subgradient directions are smoother
as compared to subgradient directions, and multiplier
zigzagging is much alleviated. Moreover, convergence was
proved without requiring the knowledge of the optimal dual
value. This was achieved with a constructive process based on
the contraction mapping concept whereby distances between
Lagrange multipliers at consecutive iterations strictly decrease,
and as a result, multipliers converge to a unique limit. At the
same time, stepsizes are kept sufficiently large to avoid
premature algorithm termination. Additionally, a constructive
stepsizing formula satisfying these criteria has been developed.
The new method also generates a valid lower bound to the
optimal cost at convergence, thereby providing a measure of
solution quality.
While the problem under consideration is a mixed-integer
nonlinear programming problem, there are linearization
methods through Taylor series expansion or Benders
decompositions methods that typically decompose the mixed
integer nonlinear problem into mixed-integer linear master
problem and nonlinear subproblems. To solve resulting mixedinteger linear problems, branch-and-cut is frequently used.
Branch-and-cut. When solving a linear programming (LP)
problem, its convex hull1 is piece-wise linear and is exactly
equivalent to the feasible set. An optimal solution is at one of
its vertices. When solving an MILP problem, the feasible set of
its LP relaxation is typically not equivalent to the convex hull
because of integrality restrictions on some of the decision
variables, and the resulting LP solution is typically not feasible
with respect to the original problem. Within branch-and-cut,
after relaxing integrality restrictions the method attempts to
obtain the convex hull by cutting off LP regions outside of it by
using linear “valid inequalities” (or cuts) without cutting off
feasible solutions. However, the problem of obtaining the
convex hull is known to be difficult, the method frequently
relies mostly on time-consuming branching operations with
questionable performance.
III. PROBLEM DESCRIPTION AND FORMULATION
Consider a distribution network with I (i = 1,…,I) buses, L
(l = 1,…,L) lines, one solar farm and G (g = 1,...,G) generators.
Each line l is defined by using two nodes: s(l) whereby line l
starts and e(l) whereby line l ends. Solar states are modeled by
using N discrete states [13] with associated probabilities ϕn(t) at
each time period t (= 1,…,T), transition probabilities 𝜋𝑛,𝑚 from
𝑆𝑜𝑙𝑎𝑟
state n to state m. The solar generation levels are 𝑃𝑖,𝑛
(𝑡).
Each generator g (=1,…,G) has its cost function Cg, which
generally depends on generation levels pg,n(t), which in turn,
may depend on the state n (=1,…,N). Assume that there is one
tap changer associated with one solar farm and let 𝑑𝑛 (𝑡) be an
integer decision variable denoting the tap position at time t and
𝑢𝑝
state n, and 𝑑𝑛 (𝑡) and 𝑑𝑛𝑑𝑜𝑤𝑛 (𝑡) denote the change of tap
position up and down, respectively. Each node i (=1,…,I) is
𝑅𝑒
associated with complex voltage 𝑉𝑖,𝑛 (𝑡) = 𝑉𝑖,𝑛
(𝑡) + 𝑗 ·
𝐼𝑚
𝑉𝑖,𝑛 (𝑡), and AC power flow in each line is complex Fl,n(t) =
Pl,n(t) + j·Ql,n(t).
A. Objective function
The objective is to co-optimization the expected generation
cost and the expected tap-changer cost:
 


G

 T N   n t   Cg pg , n t   
min     g 1
.
t 1n 1  t  С  c t   b t  
n
n
 n


(1)
The problem is to determine generation levels of generators
while nodal constraints, AC power flow constraints and tap
changer constraints, which will be explained next.
B. Nodal constraints
Generation Capacity Constraints. When a node contains a
generator g, the generator is subject to generation capacity
constraints for all possible levels n:
p gmin  p g ,n t   p gmax , g = 1,…,G, n = 1,…,N,
(2)
t = 1,…,T.
Ramp-rate constraints. For an online generator g, the change
of generation levels from time period t to t+1 cannot exceed its
ramp rate Δg. Because of the presence of renewable generation,
there are stochastic states and all transitions with nonzero
probabilities should be feasible and satisfy the following ramprate constraints:
 g  pg ,n t   pg ,m t  1   g , g = 1,…,G,
(3)
n = 1,…,N, n = 1,…,N, t = 1,…,T:  n,m  0 .
Nodal Power Balance. For every node i, the net power
generated and transmitted to the node should be equal to the net
power consumed and transmitted from the node:
t  
pi , n t   Pi ,Solar
n
L

Pl , n t   Li t  
L

Pl , n t  ,
(4)
i = 1,…,I, n = 1,…,N, t = 1,…,T.
If there are no generators or loads at bus i, then power generated
𝑆𝑜𝑙𝑎𝑟
pi,n(t), 𝑃𝑖,𝑛
(𝑡) and the load Li(t) will be trivially zero.
l 1:e ( l )  i
l 1:b ( l )  i
Voltage Restrictions. The complex voltage within each node
is subject to the following restrictions:

 

Im
Vg , min  Vg , n t   VgRe
, n t   Vg , n t   Vg , max ,
2
2
2
(5)
g = 1,…,G, n = 1,…,N, t = 1,…,T.
C. AC power flow constraints in rectangular coordinates
Following [8] and [9], AC power flow is modeled in
rectangular coordinates by using complex voltages 𝑉𝑖,𝑛 (𝑡) =
𝑅𝑒
𝐼𝑚
𝑉𝑖,𝑛
(𝑡) + 𝑗 · 𝑉𝑖,𝑛
(𝑡) at bus i and state n. Then active and
reactive power in a line l that connects buses s(l) and e(l) are:
Re
Re
Im
Pl ,n (t )  g s (l ),e(l )VsRe
(l ),n (t )Ve (l ),n (t )  bs (l ),e (l )Vs (l ),n (t )Ve(l ),n (t ) 
Re
Im
Im
bs (l ),e(l )VsIm
(l ),n (t )Ve (l ),n (t )  g s (l ),e(l )Vs (l ),n (t )Ve (l ),n (t ),
(6)
Re
Re
Im
Ql ,n (t )  bs (l ),e(l )VsRe
(l ),n (t )Ve (l ),n (t )  g s (l ),e (l )Vs (l ),n (t )Ve (l ),n (t ) 
Re
Im
Im
g s (l ),e(l )VsIm
(l ),n (t )Ve (l ),n (t )  bs (l ),e (l )Vs (l ),n (t )Ve(l ),n (t ),
(7)
l = 1,…, L, n = 1,…, N, t = 1,…, T.
Here bi,j is susceptance of branch i,j, ViRe and ViIm are real and
imaginary parts of the voltage at bus i respectively, and gi,j is
conductance of branch i,j.
The power flow in each line l should satisfy the following
line capacity constraints:
Pl , n t 2  Ql , n t 2  Fl , max , l = 1,…,L,
(8)
n = 1,…,N, t = 1,…,T.
D. Tap Changer Constraints
Following reference [11], tap changer constraints can be
written as:
Vn t  
1
an t 
V in  an t 
ZT ( a )  S
V 
in *
, n = 1,…,N,
(9)
t = 1,…,T,
where Vin is a voltage input and Vn voltage is a decision
variable, a is the transformer ratio defined under no load
conditions as Vin/Vn, ZT(a) is transformer leakage impedance,
which generally is a function of a, but in this paper it is assumed
that ZT(a) is a complex number, and S is the transformer load.
Equation (9) can be equivalently written as:
 
an t  V in Vn t   V1  an t   ZT ( a )  Sn ,
*
2
2
n = 1,…,N, t = 1,…,T.
The transformer ratio is controlled as
an t   a0  d n t  a , n = 1,…,N, t = 1,…,T,
(10)
(11)
where a0 is the rated turn ratio (usually 1), Δa is the single tap
position change, and d is integer decision variable denoting the
current tap position as
d n t   d n t  1  d nup t   d ndown t  , n = 1,…,N,
(12)
t = 1,…,T.
With significant levels of solar penetration, the ED
problem formulated above is challenging since (1) the
intermittent nature of solar generation makes the problem
stochastic and complicated; (2) the consideration of AC power
flow and tap changer equations to control voltage fluctuations
make the problem highly nonlinear; and (3) the consideration
of discrete decisions (tap positions) to traditional ED makes the
new problem combinatorial and difficult to be optimized.
IV. SOLUTION METHODOLOGY
This section is on development of the novel methodology
building upon our recent Surrogate Lagrangian Relaxation
motivated by accelerated convergence of Augmented
Lagrangian relaxation in subsection A.
To overcome
nonlinearity issues, the linearization scheme is introduced in
subsection B.
A. Surrogate Lagrangian Relaxation
The general idea of our recent Surrogate Lagrangian
relaxation [12] is decomposition and coordination. After
relaxing constraints an decomposing the resulting relaxed
problem into subproblems, subproblem solutions are
coordinated by updating Lagrangian multipliers after solving
one a few subproblems based on constraint violations.
Convergence of the method is improved motivated by
Augmented Lagrangian relaxation [14] and [15] by penalizing
violations of relaxed constraints.
In terms of the ED problem under consideration, after
introduction of slack variables, relaxation and penalization of
nodal power balance (4), line capacity (8) and tap-changer (10)
constraints, the Augmented relaxed problem becomes:


min Laug
c  ,  , ; p, P, Q, c, b  




 С  d nup t   d ndown t   

T N 


G


t 1n 1    t   C p t  

g
g ,n
 g 1 n







I N T
I N T
     t   g NFB t      c g NFB t 2  
i,n
i,n
min i 1n 1 t 1 i , n
,
i 1n 1 t 1 2
L N T

L N T c
     t   g LC t      g LC t 2  
l,n
l,n
l,n
l 1n 1 t 1

l 1n 1 t 1 2
N T

N
T
   n t   g nTCH t     c g nTCH t 2

n 1 t 1

n 1 t 1 2
 

a line begins, and fixing decision variables of other nodes
obtained at the previous iteration k-1 as:
Pˆl (t ) 
Re,k 1
Im,k 1
g b(l ),e(l )VbRe
(t )  bb(l ),e(l )VbRe
(t ) 
( l ) (t )Ve ( l )
( l ) (t )Ve (l )
Re,k 1
bb(l ),e(l )VbIm
(t ) 
( l ) (t )Ve ( l )
If node b(l) is also an end node (e(l’)) with respect to another
line l’, then the linearization is performed in the same way by
fixing decision variables associated with nodes b(l’).
To resolve nonlinearity of (15), each quadratic term will be
approximated as:
Pˆl ,n t 2 ~ Pˆl k,n1 t  Pˆl ,n t  and Qˆ l ,n t 2 ~ Qˆ lk,n1 t  Qˆ l ,n t  .
(13)
Therefore, following [16], a piece-wise linearization of (25) is:
Pˆl k, n1 t   Pˆl , n t   Qˆ lk, n1 t   Qˆ l , n t   zl , n t   Fl , max .
 
 
  ank 1 t   V in Vn t     an t   V in Vnk 1 t  
*
2
giNFB
,n t  

L
l 1:e ( l ) i
 
Pl ,n t   Li t  
L

l 1:b ( l ) i

glLC
, n t   Pl , n t   Ql , n t   zl , n t   Fl , max ,
2
2
Pl ,n t ,
(14)
(15)
g nTCH t  
 
an t   V in Vn t   V in  an t 2  ZT ( a )  Sn ,
*
2
(16)
are constraint violations corresponding to nodal flow balance
(NFB), line capacity (LC), and tap changer (TCH) constraints.
The problem (13) is highly nonlnear because quadratic penalty
terms, quadratic functions within linear capacity constraints,
and because of cross products within AC power flow
constraints and within tap changer constraints.
Linearization of quadratic terms can be performed, e.g., by
using Taylor Series expansion while keep only linear terms, and
resulting linearized relaxed problem can be decomposed into
subprobelms, which can then be coordinated. However, the
fundamental characteristic of quadratic penalties is altered
through such linearization, and the efficiency is significantly
reduced. One our recent way to overcome this difficulty is to
use absolute-value linearization of quadratic penalties [16] that
has been used within the SLR framework as will be discussed
in the following subsection B.
B. Absolute-Value Lagrangian Relaxation with Dynamic
Linearization of AC Power Flow
One way to accurately linearize quadratic penalty terms is
through the use of absolute-value penalties [16]. For example,
quadratic penalties corresponding to (14) can be approximated
using absolute value function as:
,k 1
t   giNFB
giNFB
,n
,n t  .
(20)
*
V in  a0  d n t   a  a0  d nk 1 t   a  ZT ( a )  S n ,

(19)
Lastly, penalty the tap-changer constraints will be piece-wise
linearized as follows:
where
t  
pi ,n t   Pi ,Solar
n
(18)
Im,k 1
g b(l ),e(l )VbIm
(t ).
( l ) (t )Ve ( l )
(17)
The linearization of the absolute-value function in will be
performed in a standard way based upon [17, p. 28].
The resulting function will still be nonlinear because of
nonlinear AC power flows within (14). This difficulty will be
resolved by considering one node at a time, say, node b(l) where
(21)
where the linearization of cross products is performed by fixing
one variable as a time and then taking a weighted average of
resulting terms using coefficients α and β so that α + β = 1.
Quadratic terms are linearized using absolute value functions as
explained above. Penalization of (20) and (21) is then performs
as in (17), and the corresponding penalty terms are linearized
following the standard practice [17, p. 28].
After using these linearization principles, the relaxed
problem is decomposed into linear nodal subproblems by fixing
decision variables obtained at the previous iteration k-1
associated with other nodes. Linear nodal subproblems are then
solved one by using branch-and-cut at a time and Lagrangian
multipliers are updated using surrogate subgradient directions:
,k
,k
~ TCH , k t  .
t , g~lLC
g~ k  g~iNFB
(22)
,n
, n t , g p , n
The components of (22) are constraint violations (14)-(16)
evaluated at subproblem solutions. To guarantee convergence,
stepsizes are used following [12] based on the notion of
contraction mapping as:
s k 1 g~ k 1
sk  k
, 0  k  1 .
(23)
g~ k
A specific formula for setting k to guarantee convergence
without requiring the optimal dual value is:
k 1
1
1
,   1  r , M  1, 0  r  1 .

Mk
k
(24)
V. NUMERICAL TESTING
The method is implemented using CPLEX 12.7.1 on a Dell
Precision 7720 laptop with processor Inter® Xeon® CPU E31535M v6 @ 3.10GHz with installed memory (RAM) 32.0 GB
and 64-bit operating system Windows 10. To demonstrate the
coordination aspect and the near-optimal performance of the
method, a simple 3-bus system with meshed topology and a
modified 34-bus system with radial topology are considered.
Example 1: A 3-bus system
Consider a 3 bus triangular system with all busses interconnected. Assume that at bus 1 there is a generator, a solar
farm of five stochastic states, and a tap changer, while the other
two buses have one generator each.
Susceptance and
conductance for each branch are chosen to be -0.4 and 0.4,
respectively. The time interval is 5 minutes and the planning
horizon is twelve. Results are shown in Fig. 1 below. The
feasible cost of $954.96 is obtained under 10 seconds, and the
corresponding duality gap is 1.5%.
Feasible Cost ($)
965
960
955
950
945
CONCLUSION
With increasing levels of intermittent renewable generation
such as solar, the importance of an ED problem within
distribution networks has been recognized. To control voltage
fluctuations, AC power flow together with tap changers are
considered thereby making the ED problem difficult because of
nonlinearities, stochasticity and combinatorial nature. These
difficulties have been successfully overcome by modeling solar
generation using Markov processes, by novel absolute value
linearization to handle nonlinearity and by decomposition and
coordination. The method is generic and has a broad
applicability to solve difficult mixed-integer nonlinear
programming problems.
REFERENCES
[1]
940
0
2
4
6
8
10
CPU time (sec)
Fig. 1. Results for Example 1. Feasible solutions are shown by x’s and a
lower bound by a triangle
Example 2: A modified IEEE 34-bus system
Consider the 34-bus system of [18] with slight simplification
that there is only one solar farm with three stochastic states and
one tap changer at bus 800. Three generators are located at buses
816, 856 and 834. Other transformers are removed for
simplicity. The time interval is 5 minutes and the planning
horizon is six.
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
Fig. 2. Topology of a modified IEEE 34-bus system.
[11]
The results shown in Fig. 3 below indicate that a feasible
solution with cost 121.1 is obtained under 6 seconds, and the
duality gap of 4.8% is obtained after 20 seconds.
[12]
350
[13]
Feasible Cost ($)
300
250
200
[14]
150
[15]
100
[16]
50
0
0
5
10
15
20
25
CPU time (sec)
Fig. 3. Results for Example 2. Feasible solutions are shown by x’s
[17]
[18]
B. Kasztenny, E. Rosolowski, J. Izykowski, M M. Saha, and B. Hillstrom,
“Fuzzy logic controller for on-load transformer tap changer,” IEEE
Transactions on Power Delivery, Vol 13, no. 1, pp. 164-170, 1998.
T. Sokugawa and M. Shawver, “Big data for renewable integration at
Hawaiian electric utilities,” in Proceedings of DistribuTech Conference
& Exhibition, Orlando, FL, pp. 9-11, 2016.
M. Farsadi, et al., “Economic dispatch problem for distribution network
including wind turbines and diesel electric generators,” IEEE 2015 9th
International Conference on Electrical and Electronics Engineering
(ELECO).
W. Zheng, W. Wu, B. Zhang, Z. Li, and Y. Liu, “Fully distributed multiarea economic dispatch method for active distribution networks,” IET
Generation, Transmission & Distribution, vol. 9, no. 12, pp. 1341-1351,
2015.
M. P. Anand, et al., “Impact of economic dispatch in a smart distribution
network considering demand response and power market,” in Energy
Economics and Environment (ICEEE), 2015 IEEE International
Conference, pp. 1-6, 2015.
Dall'Anese, E., Giannakis, G. B., & Wollenberg, B. F. “Economic
dispatch in unbalanced distribution networks via semidefinite
relaxation,” arXiv preprint arXiv:1207.0048, 2012
Q. Li, G. Zhang, and C. Jiang, “Economic dispatch of active distribution
network considering renewable energy uncertainties,” in CIRED
Workshop, pp. 1-4, 2016.
X. Bai, H.Wei, K. Fujisawa, and Y.Wang, “Semidefinite Programming
for Optimal Power Flow problems,” Int. J. Elect. Power Energy Syst., vol.
30, no. 6-7, pp. 383-392, 2008.
Q. Li , L. Yang and S. Lin, “Coordination Strategy for Decentralized
Reactive Power Optimization Based on a Probing Mechanism,” IEEE
Trans. Power Syst., vol. 30, no. 2, pp. 555-562, Mar. 2015.
X. Guan, P. B. Luh, H. Yan, and J. A. Amalfi, “An optimization-based
method for unit commitment,” Int. J. Elec. Power Energy Syst., Vol. 14,
no. 1, pp. 9–17, 1992.
X. Guan, P. B. Luh, H. Yan, and P. M. Rogan, “Optimization-based
scheduling of hydrothermal power systems with pumped-storage units,”
IEEE Trans. Power Syst., Vol. 9, no. 2, pp. 1023-1031, 1994.
M. A. Bragin, P. B. Luh, J. H. Yan, N. Yu and G. A. Stern, “Convergence
of the Surrogate Lagrangian Relaxation Method,” Journal of Optimization
Theory and Applications, vol. 164, no. 1, 2015, pp. 173-201, DOI
10.1007/s10957-014-0561-3.
P. B. Luh, Y. Yu, B. Zhang, E. Litvinov, T. Zheng, F. Zhao, J. Zhao and
C. Wang, “Grid Integration of intermittent wind generation: A Markovian
Approach,” IEEE Transactions on Smart Grid, vol. 5, no. 2, pp. 732-740,
2014
M. R. Hestenes, “Multiplier and gradient methods,” J. Optim. Theory
Appl., Vol. 4, no. 5, pp. 303-320, 1969.
M. J. D. Powell, “A method for nonlinear constraints in minimization
problems,” in: Optimization, (R. Fletcher, ed.), Academic Press, 1969.
X. Sun, P. B. Luh, M. A. Bragin, Y. Chen, J. Wan, and F. Wang, “A
decomposition and coordination approach for large-scale security
constrained unit commitment problems with combined cycle units,” in
Proc. 2017 IEEE Power Energy Soc. Gen. Meeting, pp. 1-5.
D. G. Luenberger, “Linear and Nonlinear Programming,” Addison
Wesley, Reading, MA, 1984.
J. O. Owuor, J. L. Munda and A. A. Jimoh, “The IEEE 34 Node Radial
Test Feeder as a simulation testbench for Distributed Generation,” in
IEEE Africon, Livingstone, 2011.
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