A transmission-constrained unit commitment method in ... - CiteSeerX

Ž .
Decision Support Systems 24 1999 297–310
A transmission-constrained unit commitment method in power
system scheduling
Chung-Li Tseng a,) , Shmuel S. Oren a , Carol S. Cheng b , Chao-an Li b ,
Alva J. Svoboda b , Raymond B. Johnson b
a Department of Industrial Engineering Operations Research, UniÕersity of California, Berkeley, CA 94720, USA
b Pacific Gas Electric Company, San Francisco, CA 94177, USA
Abstract
This paper presents a transmission-constrained unit commitment method using a Lagrangian relaxation approach. Based
on a DC power flow model, the transmission constraints are formulated as linear constraints. The transmission constraints, as
well as the demand and spinning reserve constraints, are relaxed by attaching Lagrange multipliers. A three-phase
algorithmic scheme is devised including dual optimization, a feasibility phase and unit decommitment. A large-scale test
problem with more than 2200 buses and 2500 transmission lines is tested along with other test problems. q 1999 Elsevier
Science B.V. All rights reserved.
Keywords: Power system scheduling; Unit commitment; Transmission-constrained unit commitment; Unit decommitment
1. Introduction
The unit commitment is an optimization problem that economically schedules generating units over a
short-term planning horizon subject to the satisfaction of demand and other system operating constraints. The
unit commitment problem is considered to be in the class of NP-hard problems w14 x. Many optimization methods
have been proposed to solve the unit commitment problem, e.g., see Ref. wx 4 for a survey. These methods
include priority list methods wx 3 dynamic programming methods w10,11,18x sequential method wx 8 and Law2,4–6
x, etc. Lagrangian relaxation methods are now among the most widely used
grangian relaxation methods
approaches to solving unit commitment.
Because generating units of a utility company are normally located in different areas interconnected via
transmission lines, power flows are subject to thermal limit of transmission lines. However, the transmission
constraints were usually left out in the unit commitment problems. If the transmission constraints are not
considered, the schedule obtained might cause some transmission lines to be overloaded. This may result in
) Corresponding author. Department of Civil Engineering, University of Maryland, College Park, MD 20742, USA. Tel.: q1-301-405-
1341; fax: q1-301-405-2585; e-mail: chungli@eng.umd.edu
0167-9236r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.
Ž .
PII: S0167-9236 98 00072-4

298
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C.-L. Tseng et al.rDecision Support Systems 24 1999 297–310
rescheduling of some generating units and may incur significant costs. This paper presents a method for solving
the unit commitment problem which takes transmission constraints into account. A first attempt to incorporate
AC load flow constraints in unit commitment optimization was detailed in Ref. wx 9 with promising although
limited computational testing. At present, the computational requirements of that approach would be prohibitive
for practical size problems but that might change with the rapid development in computation technology.
In this paper, the transmission constraints will be formulated as linear constraints based on a DC power flow
model. Pang et al. have considered the transmission constrained unit commitment problem in Ref. w11x
using a
dynamic programming method. In Ref. w12 x, Shaw has proposed a practical method for solving the security-constrained
unit commitment problem using the Lagrangian relaxation approach. This approach relaxes not only the
demand constraints and the spinning reserve constraints, but also the transmission constraints using multipliers.
Shaw describes two methods in his paper w12 x, a direct method and an indirect method. The former takes full
account of the transmission constraints in the optimization phase, while the latter does so only in locating a
feasible solution. The conclusion of Ref. w12x
favors the direct method.
In this paper, we employ a three-phase algorithmic scheme w14–16x
to solve the unit commitment problem.
The three phases include dual optimization, a feasibility phase and unit decommitment. Algorithmically, the
dual optimization phases of most Lagrangian relaxation-based approaches are alike. These approaches differentiate
in obtaining a feasible solution. In this paper we will present a robust feasibility method using linear
programming. The obtained feasible schedule will then be improved by a unit decommitment method. It has
been shown in Ref. w15x
that unit decommitment not only improves solution quality generally, but also mitigates
unpredictable effects due to heuristics in the first two phases. In this paper, we apply our method to a practical
test problem, which involves more than 2200 buses and 2500 transmission lines, and other small-scale test
problems as well.
This paper is organized as follows: Section 2 gives the formulation of the unit commitment problem. A
three-phase Lagrangian relaxation algorithm is presented in Section 3. Section 4 gives some numerical results.
We conclude the paper in Section 5.
2. Problem formulation
We first define the following nomenclature.
t: Index for time Ž ts0, PPP , T .
b: Index for the number of buses Ž bs1, PPP , B.
L b:
Index set of units at bus b
i: Index for the number of units Ž igL , bs1, PPP , B.
b
l : Index for transmission lines Ž ls1, PPP , L.
G l b: Line flow distribution factor for transmission line l due to the net injection at bus
b
F l :
The transmission capacity on the transmission line l
u it:
Zero-one decision variable indicating whether unit i is up or down in time period
t.
x it:
State variable indicating the length of time that unit i has been up or down in time
period t
ti
on :
The minimum number of periods unit i must remain on after it has been turned on
ti
off :
The minimum number of periods unit i must remain off after it has been turned
off
p it:
Variable indicating the amount of power unit i is generating in time period t
pi
min :
Minimum rated capacity of unit i
p max :
Maximum rated capacity of unit i
i

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C.-L. Tseng et al.rDecision Support Systems 24 1999 297–310 299
C Ž p .
i it : Fuel cost for operating unit i at output level pit
in time period t
S Ž x , u , u .
i i,ty1 it i,ty1 : Start-up cost associated with turning on unit i at the beginning of time period t
D bt:
Forecast demand requirement at bus b in time period t
R bt:
Spinning capacity requirement at bus b in time period t
The transmission-constrained unit commitment problem is formulated as the following mixed-integer
programming problem: Žnote that the underlined variables are vectors in this paper, e.g. the components of u are
all legitimate u .
it .
Minimize the total generating cost:
T
B
Ý Ý Ý i it it i i,ty1 it i,ty1
min C Ž p . u qS Ž x , u , u . Ž 1.
u,x, p ts1 bs1 igLb
subject to the following constraints:
Demand constraints
B
Ý Ý Ý
bs1 igLb
B
p u sD ' D , ;t, Ž 2.
it it t bt
bs1
Spinning capacity constraints
B
B
max
Ý Ý pi uitGR t' Ý R bt, ;t, 3
bs1 igLb
bs1
Transmission constraints
B
Ý
ž Ý /
yF F G p u yD FF , ; l , t, Ž 4.
l l b it it bt l
bs1 igL b
Local reserve constraints
Ý it it bt bt bt
igL b
p u Gr D , r g w0,1 x,;b, Ž 5.
max
Ý p i u it Gs bt R bt , s bt g w0,1 x,;b, Ž 6.
igL b
where sbt
and r bt are scalars used to define the local minimum generation level and local spinning capacity
level within bus b. Although constraints Ž. 5 and Ž. 6 are normally considered within an area instead of a bus, we
int to present a method suitable for a generalized multi-area model.
Local unit constraints Unit capacity constraints, for all igL ;b and ts1,...,T:
pi min FpitFpi
max , Ž 7.
the state transition equations
maxŽ x i ,ty1,0.
q1, if uits1,
x its Ž 8
½
.
minŽ x i ,ty1 ,0.
y1, if u it s0,
the minimum uptime and downtime constraints,
° 1, if 1Fx i ,ty1 -ti
on ,
u s~ off
it 0, ify1Gx )yt ,
Ž 9.
i ,ty1 i
¢ 0 or 1, otherwise,
and the initial conditions on x
it
at ts0 for ;i.
b,
Ž .

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C.-L. Tseng et al.rDecision Support Systems 24 1999 297–310
For simplicity, we do not consider ramp constraints in this paper. Regarding incorporating different kinds of
ramp constraints in the unit commitment, the interested reader is directed to Refs. w13,14 x. If considered, the
ramp constraints should be treated as are other local unit constraints such as Eq. Ž. 7 within the corresponding
bus subproblem to be detailed later.
3. Three-phase Lagrangian relaxation algorithm
The Lagrangian relaxation Ž LR.
approach relaxes the demand constraints, the spinning reserve constraints
and the transmission constraints using Lagrange multipliers. Algorithmically, the dual optimization of most
Lagrangian relaxation based approaches are alike. The dual approach presented in this closely corresponds to the
it direct method in Ref. w12 x.
3.1. Phase 1: the dual optimization
Letting l , m be the corresponding nonnegative Lagrange multiplier of Eqs. Ž. 2 and Ž.
t t
3 , respectively. And
a , b are corresponding to Eq. Ž. 4 , n and r to Eqs. Ž. 5 and Ž.
tl tl bt bt
6 , respectively. We now have the following
dual problem:
Ž D. max d l,m,a,b,n,r Ž 10.
where
Ž l,m,a,b,n,r.
G0
dŽ l,m,a,b,n,r.
T
B
Ž .
ÝÝ Ý i it it i i,ty1 it i,ty1
' min C Ž p . u qS Ž x , u , u .
u it, x it, pit ts1 bs1 igL b
ž / ž /
ž / ž /
max
ž / ž /
T B
max
y ÝÝ lt Ý pituityDbt qmt Ý pi uityR
bt
ts1 bs1 igLb
igLb
ž / ž /
T L B B
ÝÝ Ý Ý Ý Ý
y a F q G p u yD qb F y G p u yD
t l l l b it it bt t l l l b it it bt
ts1 ls1 bs1 igLb bs1 igLb
T
B
ÝÝ Ý Ý
y n p u yr D qr p u ys R Ž 11.
bt it it bt bt bt i it bt bt
ts1 bs1 igLb
igLb
subject to initial conditions, and the unit constraints. After rearrangement of the terms in Eq. Ž 11 ., the
separability of dŽ l, m, a, b, n, r.
appears.
where
B
Ž . Ý Ý i Ž . Ž .
d l,m,a,b,n,r s d b;l,m,a,b,n,r qs l,m,a,b,n,r , Ž 12.
bs1 igL b
ž / ž /
T T L B B
Ž . Ý Ý Ý Ý Ý
s l,m,a,b,n,r s Ž l D qm R . y a F y G D qb F q G D
t t t t t l l b l bt t l l b l bt
ts1 ts1 ls1
bs1 bs1
T B
ÝÝ bt bt bt bt bt bt
ts1 bs1
q Ž n r D qr s R . Ž 13.

is a constant term given the multipliers, and
Ž . Ý
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C.-L. Tseng et al.rDecision Support Systems 24 1999 297–310 301
T
d b;l,m,a,b,n,r s min c Ž p . qS Ž x , u , u . yl p ym p max
i i it i i,ty1 it i,ty1 t it t i
u it, x it, pit ts1
L
max
Ý t l t l l b it bt it bt i it
ls1
y Ž a yb . G p yn p yr p u . Ž 14.
Once again, the minimization in Eq. Ž 14.
is subject to initial conditions and the unit constraints.
Note that di
is a unit subproblem corresponding to unit i. We can further define a bus subproblem db
such
that
d l,m,a,b,n,r ' d b;l,m,a,b,n,r Ž 15.
b
Ž . Ý iŽ .
igL b
Ž .
and the dual objective 12 is equivalent to
B
Ž . Ý b Ž . Ž .
d l,m,a,b,n,r s d l,m,a,b,n,r qs l,m,a,b,n,r Ž 16.
bs1
Each unit subproblem d ŽEq. Ž 14.. can be solved using dynamic programming. The dual function dŽ
i
l, m, a,
b, n, r. is concave, but not necessarily differentiable at all points Že.g. Ref. wx. 1 . A subgradient method is
employed to solve the dual maximization problem Ž D ..
A subgradient of the dual function d is an ascent direction of d. Let g be a point in the dual space, i.e. a
vector of all multipliers, an iterative relation can be implemented to maximize the dual function d:
g kq1 §g k qs k D g k , Ž 17.
where the superscript denotes iteration k, s is the step size and Dg is a subgradient. The subgradient Dg k used
here is a vector of the mismatches of the relaxed constraints evaluated at g k
Žwx. 1 . The step size used in our
implementation has the following form,
k
s sm r5 g k
5, m )0, Ž 18.
k
k
where mk
is an adaptive constant. Our stopping criteria used in the algorithm combine the maximum number of
iteration, the change of norm of subgradients at two consecutive iterations and the number of iterations without
improvement in the dual objective value. For details of the dual optimization algorithm, the reader is referred to
Ref. w16 x.
3.2. Phase 2: the feasibility phase
Due to separability, the dual problem presented in the previous can be efficiently solved. Unfortunately,
solving the dual problem is not necessarily equivalent to solving the primal original problem. This can be seen
in a fact that the solution obtained through the dual optimization of the unit commitment is generally not
feasible. A feasibility phase is therefore required to obtain a feasible solution.
Definition 1 With respect to a given spinning capacity requirement R 4
t ,areserÕe-feasible commitment is a
unit commitment u
4 that satisfies the spinning reserve capacity constraints Ž.Ž. 2 , 6 , unit constraints Ž.Ž.
it
8 , 9 and
the initial conditions x i0.
Definition 2 With respect to a given load requirement D 4, a unit commitment u
4
t
it is said to be dispatchable if
there exists a set of dispatches p 4 such that Eqs. Ž.Ž.Ž. 2 , 4 , 5 and Ž.
it
7 can be satisfied.B
In Ref. w19 x, a method for obtaining feasible solutions to the single-area unit commitment problem is
proposed. The basic idea is to obtain a reserve-feasible commitment in the single-area case, ignore Eq. Ž. 6 in the
definition. In Ref. w19x
a reserve-feasible commitment is found by projecting the subgradient onto the hour

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C.-L. Tseng et al.rDecision Support Systems 24 1999 297–310
corresponding to the most unsatisfied capacity constraint and increasing only the multiplier mt
of this hour. This
approach can be modified to simultaneously updating the multipliers m t’s of it all hours that the spinning
capacity requirement is violated. Such a modification can speed up the feasibility phase at the cost of
overcommitment w5,14 x. Under the three-phase scheme, this overcommitment effect can nevertheless be
corrected in the final unit decommitment phase to be addressed in a later section.
The feasibility phase presented in this paper contains two parts. Part 1 seeks a reserve-feasible commitment
by increasing the multipliers mt
of all hours with insufficient spinning capacities. Starting from this reserve-
feasible commitment obtained, Part 2 solves a linear program Ž LP.
for each hour to verify if this commitment is
also dispatchable at this hour. If not, information can be extracted from the solution of the LP to determine how
much additional spinning capacity at this hour is required so that a dispatchable commitment exists. The
spinning reserve requirement of this hour is thus uplift. Part 1 is then rerun with respect to the new Ž uplift.
spinning reserve requirements. Suppose Part 1 of the feasibility phase terminates at iteration k with an obtained
commitment u k
4
it . Let qbt
be the variable representing the total generation to be dispatched to bus b in time t.
k
4
min min k max max k
Given a commitment uit let qbt sÝig L pi uit and qbt sÝig L pi u it, which form an immediate
b
b
lower bound and upper bound for q bt.
ŽLPŽ t..
Ž .
min Ý B y yq max Ž 19.
q bt, ybt
bs1 bt bt
s.t. qbt min FqbtFy bt, ;b Ž 20a.
q max Fy FÝ p max , ;b Ž 20b.
bt bt ig L b i
Ýbs1 B qbtsDt
Ž 20c.
yFlFÝbs1 B G l bŽ qbtyDbt. FF l , ; l Ž 20d.
q Gr D , ;b Ž 20e.
bt bt bt
Ž Ž ..
max
In LP t , y is a variable upper bound of both q and q . Let y ) be the solution of ŽLPŽ t ...Ifu k
4
bt bt bt bt it is
max
dispatchable in time t, y )sq solves ŽLPŽ t.. and the optimal objective value of ŽLPŽ t.. is zero. If u k
4
bt bt it is
not dispatchable in time t, ŽLPŽ t..
yields a positive objective value, which indicates that to be dispatchable,
more units should be committed in time t, and the target value of the new bus capacity requirement is now
increased to y bt) for each bus b in time t. That is, an additional bus spinning capacity requirement is imposed
to each bus subproblem d Ž P .:
Ý i it bt
b
p max u Gy ). Ž 21.
igL b
Ž .
Part 1 of the feasibility phase is therefore rerun with respect to the updated capacity requirement 21 . The phase
2 algorithm is summarized below.
3.2.1. Phase 2 algorithm
Step 0: k§0; m 0 and r 0 and other multipliers are from Phase 1.
k k
Ž . Ž k k .
Step 1: Given m and r and other multipliers, solve the dual subproblems 14 to obtain u , p .
Ž. Ž.
Step 2: If 3 and 6 are satisfied, go to Step 5 if Step 5 has not been visited, otherwise stop.
kq1 k k
Ž
B max k
.
kq1 k k
Step 3: For ;b,t, m §m qs Pmin 0, R yÝ Ý p u ; r §r qs PminŽ
t t t bs1 i g L i it bt bt 0,sbtRbty
b
max
Ý p u k
..
i g L b i it

Step 4: k§kq1, go to Step 1.
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C.-L. Tseng et al.rDecision Support Systems 24 1999 297–310 303
k
Step 5: Solve ŽLPŽ t.. for all t. If the optimal value of ŽLPŽ t.. is 0 for all t, stop and u is reserve-feasible and
dispatchable. Otherwise, commit more units in any bus b and time t such that y bt))qbt
max to satisfy Eq.
Ž 21 ., where y ) solves ŽLPŽ t ..
bt
.
In the algorithm, the loop between Step 1 and Step 4 corresponds to the Part 1 of the feasibility phase, and
Step 5 corresponds to Part 2.
The feasibility phase is nevertheless heuristic. An important issue is how robust the feasibility phase of a
transmission-constrained unit commitment method is. To test the robustness of the feasibility phase, one can
apply it to an instance in which each single bus has enough capacity to handle its own load over the planning
horizon. Then the transmission line capacities are gradually reduced to zero. In such a problem, a feasible
solution obviously exists, and it is equivalent to solving many single-area unit commitment problems. It can be
easily verified that our feasibility phase reduces to solving many single-area problems in such an instance.
3.3. Economic dispatch
With the DC power flow model, the transmission constraints are formulated as linear constraints. If the fuel
cost is represented by a quadratic function of the power generation, the transmission-constrained economic
dispatch is a quadratic programming problem. Many methods can be used to solve this type of problem, e.g., the
reduced gradient method. In our implementation, the transmission-constrained economic dispatch is solved by
the commercial software MINOS, which does general nonlinear programming.
3.4. Phase 3: unit decommitment
A Ž single area. unit decommitment Ž UD. method was developed as a postprocessing method for the Ž LR.
approach for solving the unit commitment problem in Ref. w15 x. Given the demand requirement D t, reserve
requirement R and a feasible solution Ž u, p. Ž satisfying D, R and all physical constraints .
t ˜ ˜
t t
, the unit
decommitment method improves Ž u, ˜ p˜.
while maintaining feasibility. Thus, the unit decommitment can be
regarded as a primal approach. A transmission-constrained unit decommitment algorithm can be devised by
repeatedly solving the Ž single area.
unit decommitment to each bus. Suppose at iteration k, we have a feasible
schedule Žu, ˜
k p ˜
k .. The schedule of bus b can be improved by the unit decommitment with respect to the it
current load assigned to be the bus demand at this iteration, i.e., Ýi g L p˜it k u˜it
k . The unit decommitment method
b
can obtain a more efficient schedule, if possible, which generates the same load Ýi g L p˜it k u˜it
k . After the
b
commitments of all buses are refined, a transmission-constrained economic dispatch is performed to redispatch
the units in all buses. The decommitment procedure can thus proceed until no improvement in the total cost is
made. Since at each iteration the decommitment procedure improves the commitment without affecting the
aggregated bus generation, the transmission feasibility is retained at each iteration. The algorithm is as follows.
3.4.1. Transmission-constrained UD algorithm
Step 0: A feasible schedule Žu ˜
0 , p˜
k . is given, k§0.
Step 1: Given Žu,p
˜
k ˜
k . each bus performs the unit decommitment presented in Ref. w15x
independently with
the current load assigned to be the demand requirement Ýi g L p˜it k u˜it
k also subject to the satisfaction of Eq.
b
k
Ž. 6 . The resultant commitment is denoted by u ˆ .
Step 2: Apply the transmission-constrained economic dispatch with respect to uˆ
k to obtain a dispatch pˆ
k . Let
Žu ˜ kq1 , p ˜ kq1 .§ Žu,p ˜ k ˜
k ..

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C.-L. Tseng et al.rDecision Support Systems 24 1999 297–310
Step 3: If the total cost of Žu ˜ kq1 , p˜ kq1 . is no better than that of Žu,p ˜ k ˜
k ., stop; otherwise k§kq1, and go to
Step 1.
The above method works best for the cases with many generators in each bus. For other transmission-conw14
x.
strained unit decommitment approaches, the interested reader is referred to Ref. 4. Numerical results
The transmission-constrained unit commitment algorithm presented in this paper has been implemented in
FORTRAN on an HP 700 workstation. This presents numerical results of some test problems.
Table 1
Generating unit parameters
a min max up down cold
b_i Type pi pi xi0 ti ti ti ai0 ai1 ai2 Si
1_1 C 4 20 1 1 1 2 63.999 20.000 1.000=10 40
y6
1_2 C 4 20 2 1 1 2 63.999 20.000 1.000=10 40
y2
1_3 S2 10 76 3 6 6 12 133.919 16.193 1.508=10 45
y2
1_4 S2 10 76 y2 6 6 12 133.919 16.193 1.508=10 45
y6
2_1 C 4 20 1 1 1 2 63.999 20.000 1.000=10 40
y6
2_2 C 4 20 2 1 1 2 63.999 20.000 1.000=10 40
y2
2_3 S2 10 76 10 6 6 12 133.919 16.193 1.508=10 45
y2
2_4 S2 10 76 y2 6 6 12 133.919 16.193 1.508=10 45
y2
7_1 S1 15 100 10 10 10 20 199.124 12.468 1.532=10 45
y2
7_2 S1 15 100 10 10 10 20 199.124 12.468 1.532=10 110
y2
7_3 S1 15 100 10 10 10 20 199.124 12.468 1.532=10 110
y3
13_1 S1 20 197 12 12 12 24 209.546 13.928 2.085=10 100
y3
13_2 S1 20 197 12 12 12 24 209.546 13.928 2.085=10 100
13_3 S1 20 197 12 12 12 24 209.546 13.928
y3
2.085=10 100
15_1 S1 3 12 4 2 2 4 21.145 16.193 9.553=10 y2 30
y2
15_2 S2 3 12 y9 2 2 4 21.145 16.193 9.553=10 30
y2
15_3 S1 3 12 4 2 2 4 21.145 16.193 9.553=10 30
y2
15_4 S1 3 12 3 2 2 4 21.145 16.193 9.553=10 30
y2
15_5 S1 3 12 4 2 2 4 21.145 16.193 9.553=10 30
y3
15_6 S2 20 155 y20 12 12 24 275.606 12.360 8.898=10 100
y3
16_1 S2 20 155 5 12 12 24 275.606 12.360 8.898=10 100
y4
18_1 S3 40 400 100 48 48 60 577.537 14.253 7.365=10 440
y4
21_1 S3 40 400 100 48 48 60 577.537 14.253 7.365=10 440
y2
22_1 S2 10 76 y2 6 6 12 133.919 16.193 1.508=10 45
y2
22_2 S2 10 76 y2 6 6 12 133.919 16.193 1.508=10 45
y2
22_3 S2 10 76 y2 6 6 12 133.919 16.193 1.508=10 45
y2
22_4 S2 10 76 y2 6 6 12 133.919 16.193 1.508=10 45
y2
22_5 S2 10 76 y2 6 6 12 133.919 16.193 1.508=10 45
22_6 S2 10 76 y2 6 6 12 133.919 16.193
y2
1.508=10 45
23_1 S2 20 155 5 12 12 24 275.606 12.360 8.898=10 y3 100
23_2 S2 20 155 3 12 12 24 275.606 12.360
y3
8.898=10 100
23_3 S2 35 350 11 24 24 36 517.669 11.892
y3
5.220=10 250
a S1: Steam: fossil-oil; S2: Steam: fossil-coal; S3: Steam: nuclear; C: Combustion turbine.
y6

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C.-L. Tseng et al.rDecision Support Systems 24 1999 297–310 305
4.1. IEEE test system
The first test problem is based on an IEEE test problem wx 7 . This test system contains 24 buses, 34
transmission lines and 32 generating units. The parameters of the generating units are summarized in Table 1, in
which the unit fuel costs are assume to be a quadratic function of power generation as:
CiŽ pit. sai0 qai1 pitqai2 pit
2 , Ž 22.
and the unit startup costs S are assumed to be constants. This test system contains steam units and combustion
i
turbines as indicated in Table 1.
The transmission line parameters are given in Table 2. In Table 2 two sets of line capacities are presented.
The capacities of the first set of transmission lines are generally larger than those of the second set. Therefore,
the second set of transmission lines would yield more serious congestion than the first set. A 24-h planning
horizon is adopted in the test problem, the system load information can be found in Table 3. The load taken here
corresponds to the day with peak load in the year in the IEEE test problem. In this test, the spinning capacity
Table 2
Transmission line parameters
l From bus To bus X Ž p.u. . Ž i. Fl
Ž MW. Ž ii. Fl
Ž MW.
1 1 2 0.0139 175 175
2 1 3 0.2112 175 175
3 1 5 0.0845 175 175
4 2 4 0.1267 175 175
5 2 6 0.1920 175 175
6 3 9 0.1190 175 175
7 3 24 0.0839 400 175
8 4 9 0.1037 175 175
9 5 10 0.0883 175 175
10 6 10 0.0605 175 175
11 7 8 0.0614 175 175
12 8 9 0.1651 175 175
13 8 10 0.1651 175 175
14 9 11 0.0839 400 175
15 9 12 0.0839 400 136
16 10 11 0.0839 400 200
17 10 12 0.0839 400 250
18 11 13 0.0476 400 198
19 11 14 0.0418 400 250
20 12 13 0.0476 400 150
21 12 23 0.0966 400 166
22 13 23 0.0865 400 250
23 14 16 0.0389 400 220
24 15 16 0.0173 400 250
25 15 21 0.0245 400 290
26 15 24 0.0519 400 270
27 16 17 0.0259 400 270
28 16 19 0.0231 400 270
29 17 18 0.0144 400 270
30 17 22 0.1053 400 270
31 18 21 0.0129 400 270
32 19 20 0.0198 400 198
33 20 23 0.0108 300 270
34 21 22 0.0678 400 270

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Table 3
Load information
System load
Bus load
Hour Load Ž MW .
% of peak Bus % of peak
1 2223.0 78 1 3.8
2 2052.0 72 2 3.4
3 1938.0 68 3 6.3
4 1881.0 66 4 2.6
5 1824.0 64 5 2.5
6 1852.5 65 6 4.8
7 1881.0 66 7 4.4
8 1995.0 70 8 6.0
9 2280.0 80 9 6.1
10 2508.0 88 10 6.8
11 2565.0 90 11 0
12 2593.5 91 12 0
13 2565.0 90 13 9.3
14 2508.0 88 14 6.8
15 2479.5 87 15 11.1
16 2479.5 87 16 3.5
17 2593.5 91 17 0
18 2850.0 100 18 0
19 2821.5 99 19 6.4
20 2764.5 97 20 4.5
21 2679.0 94 21 0
22 2622.0 92 22 0
23 2479.5 87 23 0
24 2308.5 81 24 0
requirements are set to be R ts1.07 Dt
for all t. The proposed algorithm is applied to solve this test problem.
The test result is summarized in Table 4.
In case Ž. i of the transmission line capacity F l , the test result shows that line 33 is always congested and line
11 becomes congested during peak hours. In this test case, it is noted that no combustion turbine is turned on
during the planning horizon. The duality gap is 0.47% in this case. Compared with the total cost in the
unconstrained case, the increased cost due to the transmission network is about 0.17% of the total cost of the
unconstrained case. In the second case Ž ii.
of F l , the congestion of the network involves seven lines. There are
12 h on the planning horizon in which there are 5 lines congested Ž during the peak hours ., 7 h with 4 lines, 4 h
with 2 lines and 1 h with 1 line congested. The increased cost due to the transmission system constraints
Table 4
Test result of IEEE test problem Ž direct approach.
Case Dual value Primal value Duality gap
Unconstrained US$898,619 US$898,683 0.00%
Constrained Fl
Ž. i US$895,980 US$900,206 0.47%
Constrained Fl
Ž ii.
US$899,662 US$912,650 1.43%
Case
CPU time Ž. s
Ph 1 Ph 2 Ph 3 Total
Unconstrained 3.72 0.37 2.32 6.41
Constrained Fl
Ž. i 7.81 5.77 6.28 19.86
Constrained Fl
Ž ii.
7.79 12.14 10.92 30.85

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C.-L. Tseng et al.rDecision Support Systems 24 1999 297–310 307
increases to 1.5% of the total cost of the unconstrained case. The congestion is much more serious than in the
case Ž. i . It can be observed that when the transmission network gets more congested, the algorithm takes longer
time to converge, specially in the dual optimization and in obtaining a feasible solution in the feasibility phase.
Congestion of the transmission network also yields poorer Ž larger.
duality gap. In this case, we also found that
the duality gap of 1.43% is primarily due to the poor dual objective value obtained. In another test run, tripling
the CPU time spent on the dual optimization improves the dual value to 902 758 with the primal value reduced
to 912 576 for the duality gap of 1.09%. Overall, our experience show that our feasibility phase is very robust in
obtaining feasible solutions.
4.2. Direct and indirect approaches
In Ref. w12 x, the direct and indirect methods are defined. Those methods which take transmission constraints
into account in the dual optimization are called direct methods and those which do not are called indirect.
Indirect methods ignore the transmission constraints in the dual optimization, but deal with them only in the
feasibility phase. In this section, we shall compare two algorithms. Using the same naming convention as in Ref.
w12 x, we will call them the direct approach and the indirect approach. The direct approach is the three-phase
algorithm proposed in this paper. The indirect approach is essentially the direct approach except that the
multiplier at lsbt ls0, for ;t, l through out the algorithm, and the transmission constraints are enforced
through Eq. Ž 20d.
during the feasibility phase.
The major motivations for applying the indirect approach are to speed up the dual optimization, and to save
memory space. Another reason for using the indirect approach may be to simulate a power market structure, like
that of the British power pool, where an initial unit commitment does not consider transmission constraints, and
transmission constraints are considered only in modifying this initial commitment. We also applied the indirect
approach to solve the IEEE test problem given in the previous. The result is summarized in Table 5. It can be
seen in Table 5 that the direct approach obtains a better solution and uses less CPU time. The indirect approach
only uses less CPU time in the dual optimization phase Ž Ph 1 ., but spends more time to locate a feasible
solution. Also the quality of the solution obtained from the feasibility phase of the indirect approach tends to be
worse than that from the direct approach, so that it might take longer for unit decommitment to improve the
solution. Our comparison between the direct and indirect approaches agrees with what is observed in Ref. w12 x.
4.3. A PG&E test problem
The proposed three-phase algorithm is also applied to a test problem based on PG&E’s system. The test
problem with more than 2500 lines, 2200 buses and 79 dispatchable generating units is derived with necessary
modifications to fit the scope discussed in this paper. The peak load is 3957 MW Žonly the percentage
Table 5
Test result of the indirect approach
Case Dual value Primal value Duality gap
Constrained Fl
Ž. i US$895,840 US$901,289 0.60%
Constrained Fl
Ž ii.
US$895,840 US$913,542 1.97%
Case
CPU time Ž. s
Ph 1 Ph 2 Ph 3 Total
Constrained Fl
Ž. i 5.69 21.74 10.45 37.88
Constrained Fl
Ž ii.
5.80 18.72 9.52 34.04

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Table 6
Test result of a large scale problem
Total cost Ž. $
CPU time Ž. s
Dual optimal: 199,216.56 Ph. 1:1292.47
Primal optimal: 200,214.32 Ph. 2:116.60
Duality gap: 1.469% Ph. 3:108.32
Total: 1512.39
contributed by the thermal units .. The test result is given in Table 6. We emphasize that these test results used a
program developed for research purpose with little effort spent in speeding up the algorithm performance. The
motivation is to see how the algorithm performs on a large scale problem. We observe: Ž. 1 the convergence of
the dual optimization is not directly affected by the increased number of multipliers corresponding to the
transmission constraints but by the increased number of nonzero multipliers. The more congested the network is,
the more iterations are required to reach a near-optimal dual solution. We observe that the dual objective value
tends to improve more slowly as the congestion of the network increases. Ž. 2 The CPU time required in this test
problem, although high, scales approximate linearly with problem size. Algorithm performance can be improved
by taking advantage of the sparsity of the matrix of the distribution factors G and the multipliers associated
with the transmission constraints. Along this direction, a large scale ‘bus-based’ model can be equivalent
reduced to a small sized ‘area-based’ one without violating problem optimality, and the algorithm performance
can be greatly accelerated w17 x.
5. Conclusions
In this paper we presented a transmission-constrained unit commitment algorithm using the Lagrangian
relaxation approach. The transmission constraints are formulated as linear constraints under the DC power flow
model. The Lagrangian relaxation approach relaxes the demand constraints, the spinning reserve constraints and
the transmission constraints using multipliers. In this paper, we employ a three-phase algorithmic scheme. Phase
1, the subgradient method is used to maximize the dual function to determine a near-optimal dual solution. A
feasibility phase follows to locate a feasible commitment. The feasibility phase is essentially an extension of that
in the single area unit commitment case. Initially a reserve-feasible commitment is located, then by solving
linear programs a dispatchable commitment is determined. Finally we devise a transmission-constrained unit
decommitment method, which serves as a post-processing method of the algorithm.
In limited numerical tests, the proposed algorithm is found to be efficient and robust. A large scale problem
based on PG&E system is also tested. The result suggests that the proposed algorithm can be used to deal with
practical-sized transmission-constrained unit commitment problems.
Acknowledgements
This work was partly supported by the National Science Foundation under Grant IRI-9120074 and PG&E’s
R&D Department. Their support is greatly appreciated.
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Chung-Li Tseng an Assistant Professor in Civil Engineering Department at the University of Maryland at College Park. He received a B.S.
in Electrical Engineering from National Taiwan University in 1988, and a M.S. in Electrical and Computer Engineering from U.C. Davis in
1992 and a Ph.D. in Industrial Engineering and Operations Research from U.C. Berkeley in 1996. He has worked as an operational research
consultant for PG&E from 1995 to 1996. Since 1997 he has been with Edison Enterprises as a senior risk management researcher. His
research interests include financial engineering, optimization applied to industrial applications and neural network computation.
Shmuel S. Oren is Professor of Industrial Engineering and Operations Research at the University of California at Berkeley. He holds a B.S.
and M.S. in Mechanical Engineering from the Technion in Israel and an M.S. and Ph.D. in Engineering Economic Systems from Stanford.
Prior to his current position he was on the faculty of the Engineering Economic Systems Department at Stanford University and he worked
for eight years as a research scientist at the Xerox Palo Alto Research Center. His research interests include Optimization theory, Modeling
and analysis of economic systems, Coordination and decentralization through market mechanisms and Electric utility planning and
operations.
Carol S. Cheng was born in Beijing China. She received B.S. in Electrical Engineering from Northern Jiaotong University in Beijing in
1982, M.S. in Mechanical Engineering from University of Cincinnati in 1986, and Ph.D. in Electrical Engineering from Georgia Institute of
Technology in 1991. From 1982 to 1985, she joined the faculty of Northern Jiaotong University as a Teaching Assistant. From 1992 to 1995
she worked with the Energy Systems Automation group of PG&E where she developed several advanced applications for distribution
systems. Currently she is a Systems Engineer in the Application Integration group responsible for the development of applications on
resource scheduling and generation heat rate modeling.
Chao-an Li graduated from Electric Power System Department of Moscow Energetic Institute, Moscow, USSR. He has broad interests in
power system optimization including hydrothermal coordination, economic dispatch, unit commitment, load forecasting, automatic
generation control, power flow and power system state estimation problems. and network analysis. He is currently working on projects
related to hydro-thermal optimization for PG&E.
Alva J. Svoboda received a B.A. in mathematics from U.C. Santa Barbara in 1980, and an M.S. and Ph.D. in Operations Research from
U.C. Berkeley in 1984 and 1992. He has worked on contract as an operations research analyst for Pacific Gas and Electric since 1986. His
current research interest is the extension of existing utility operations planning models to incorporate new operating constraints.

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Raymond B. Johnson received his B.A. in 1976 in Electrical Sciences from Trinity College, Cambridge University, and a Ph.D. in Electrical
Engineering from Imperial College, London University in 1985. His professional experience includes positions as a power system design
engineer with Hawker Siddeley Power Engineering from 1976 to 1980 and an EMS applications developer with Ferranti International
Controls from 1987 to 1989. Since 1989, has been with PG&E where he is currently a Systems Engineering Team Leader responsible for
resource scheduling and energy trading applications.