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

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