Multi-area generation scheduling algorithm with regionally distributed optimal power flow using alternating direction method

Electrical Power and Energy Systems 33 (2011) 1527–1535

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Multi-area generation scheduling algorithm with regionally distributed optimal power flow using alternating direction method K.H. Chung a, B.H. Kim b,1, D. Hur c,⇑,2 a

Korea Electrotechnology Research Institute, Naeson-ro 138, Uiwang-city, Gyeonggi-province 437-808, Republic of Korea School of Electronic & Electrical Engineering, Hong-Ik University, Wowsan Rd. 85, Mapo-gu, Seoul 121-791, Republic of Korea c Dept. of Electrical Engineering, Kwangwoon University, Kwangwoon Rd. 26, Nowon-gu, Seoul 139-701, Republic of Korea b

a r t i c l e

i n f o

Article history: Received 26 July 2009 Received in revised form 3 September 2010 Accepted 23 November 2010 Available online 22 January 2011 Keywords: Alternating direction method Distributed optimal power flow Generalized Benders decomposition Generation scheduling Interconnected power system

a b s t r a c t This paper calls attention to the core issue as to the multi-area generation scheduling algorithm in interconnected electric power systems. This algorithm consists in deciding upon on/off states of generating units and their power outputs to meet the demands of customers under the consideration of operational technical constraints and transmission networks while keeping the generation cost to a minimum. In treating the mixed integer nonlinear programming (MINLP) problem, the generalized Benders decomposition (GBD) is applied to simply decouple a primal problem into a unit commitment (UC) master problem and inter-temporal optimal power flow (OPF) sub-problems. Most prominent in this work is that the alternating direction method (ADM) is introduced to accomplish the regional decomposition that allows efficient distributed solutions of OPF. Especially, the proposed distributed scheme whose effectiveness is clearly illustrated on a numerical example can find the most economic dispatch schedule incorporated with power transactions on a short-term basis where utilities are less inclined to pool knowledge about their systems or to telemeter measured system and cost data to the common system operator and nevertheless the gains from trade such as economy interchange are vital as well. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Since there is a strong possibility that utilities’ profits should be vastly changeable due to the complicated interactions between generation and transmission system variables in interconnected power systems, the generation scheduling procedure will have to provide more accurate information on these interactions [1]. For the generation scheduling in interconnected power systems, it seems reasonable to make use of an integrated model of unit commitment (UC) and optimal power flow (OPF). This integrated model can definitely yield the optimal hourly operating states by taking into account the transmission system configuration which is fully reproduced by the OPF problem. Therefore, the generation scheduling problem in interconnected power systems is mathematically formulated as a large-scale mixed integer nonlinear programming (MINLP) problem which contains the continuous variables representing the power outputs and various system states at a specified instant as well as the binary variables indicating the start-up/shut-down (on/off) status of each generating unit in the course of dispatch. Unfortunately, the batch processing of ⇑ Corresponding author. Tel.: +82 31 420 6134; fax: +82 31 420 6009. E-mail addresses: [email protected] (K.H. (B.H. Kim), [email protected] (D. Hur). 1 Tel.: +82 2 320 1462; fax: +82 2 320 1110. 2 Tel.: +82 2 940 5473; fax: +82 2 940 5141.

Chung),

[email protected]

0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2010.11.013

the MINLP problem is highly vulnerable to divergent solutions as well as computational inefficiency. The most typical strategy for solving this MINLP problem is to use the generalized Benders decomposition (GBD) technique [2,3], which divides an original MINLP problem into a master problem consisting of binary variables and sub-problems of continuous variables by eliminating coupling constraints concerned with both binary and continuous variables. The goal is straightforward: it can enhance the computational efficiency by reducing the dimension of a MINLP problem. This paper is intended to explore the generation scheduling algorithm based on GBD. The proposed scheme also includes the distributed OPF using regional decomposition technique which is really implemented from the alternating direction method (ADM) [4,5]. Since the alternating direction method suitable for the competitive multi-utility environment through its power transactions admits no modifications to the augmented Lagrangian function in the inter-temporal OPF separated by geographical boundaries, it is remarkably faster than any other distributed methodologies using the approximations to the augmented Lagrangian function in terms of computation speed [6]. 2. Application of generalized Benders decomposition As described in [7–10], the generation scheduling problem is mathematically defined as a large-scale MINLP problem which

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Nomenclature Parameters X off number of hours the generating unit m was off-line unm ðtÞ til time period t. X on number of hours the generating unit m was on-line until m ðtÞ time period t Di(t) load demand at bus i for time period t RS(t) spinning reserve requirement for time period t Bij susceptance of transmission line connecting bus i and bus j Lij maximum capacity of transmission line joining bus i and bus j MDTm minimum shut-down time of generating unit m MUTm minimum start-up time of generating unit m Pmax upper bound on power output of generating unit m m lower bound on power output of generating unit m Pmin m SDRm shut-down ramp rate of generating unit m SURm start-up ramp rate of generating unit m constant value of variable sm(t) computed from kth iterskm ðtÞ ation of unit commitment master problem

nkm ðtÞ

index for determining on/off states of generating unit m for time period t derived from inter-temporal OPF subproblems at iteration k qki ðtÞ lagrange multiplier (or bus incremental cost) of bus i for time period t pm(t) power produced by generating unit m for time period t voltage angle at bus i for time period t di(t) TCk total generation cost at iteration k Zk continuous variable which approximates fuel costs at kth iteration of unit commitment master problem FCm(pm(t)) fuel cost of generating unit m at power output pm(t) for time period t SCm(t) start-up cost of generating unit m for time period t Sets M N T K Xi

set set set set set

of of of of of

all generating units all buses all dispatch time periods all iterations generating units located at bus i

Variables and functions sm(t) binary variable: sm(t) = 1 if generating unit m is on; sm(t) = 0 if not

gracefully deals with not merely binary variables dictating the start-up/shut-down (on/off) status of each generating unit but also continuous variables related to optimal operating states in power systems. Once the MINLP problem concurrently determines both binary and continuous variables, it may produce a solution but waste valuable computer time, which would be unacceptable to operations engineers. Worse still, the slow oscillation of successive solutions eventually deteriorates into divergence. All these unwelcome effects have to be overcome in any successful techniques, and in this context the GBD is highly reliable. When the GBD is applied to the generation scheduling problem, an original problem is totally split into a UC master problem to determine the short-term operations plans for generation resources while taking account of startup costs and time delays and inter-temporal OPF sub-problems subject to the fixed UC schedule. The optimal dispatch solutions are accordingly created through the iterative procedures between the UC master problem and inter-temporal OPF sub-problems. The inter-temporal OPF sub-problems are computed with respect to the UC schedule derived from the master problem at the current iteration step. The Benders cuts are generated according to the results of sub-problems and then appended to the master problem at the next iteration. Here the UC master problem coordinates the turn-on and turn-off schedules of a set of electrical power generating units, depending on the Benders cuts and then gives the intertemporal OPF sub-problems back its results. This iterative process will continue as far as the objective value of a master problem becomes at least greater than the minimum operating costs of sub-problems so far enumerated [11–14]. Obviously, this iterative procedure requires a starting UC schedule to solve its inter-temporal OPF sub-problems. As mentioned earlier, to find a combination of globally optimized generation schedule, the results deduced from inter-temporal OPF subproblems are delivered to the UC master problem. As an initial UC schedule, we select the schedule that forces as many generators as possible to be committed among any different schedules satisfying operating constraints of sub-problems, thereby assuring the feasibility of the subproblem at the first iteration. The simplified flowchart focusing on the application of GBD to the generation scheduling problem is vividly shown in Fig. 1.

2.1. Formulation of master problem In a compact form, the UC master problem is described below [10]:

min

Zk þ

T X M X

SC m ðtÞ

ð1Þ

t¼1 m¼1

s:t: Z k P

T X M X 

  skm ðtÞ  FC m pkm ðtÞ

t¼1 m¼1

þ

T X M h X

 i nkm ðtÞ  sm ðtÞ  skm ðtÞ for 8 k 2 K

ð2Þ

t¼1 m¼1 M X 

N X  sm ðtÞ  Pmax Di ðtÞ þ RSðtÞ for 8 t 2 T P m

m¼1

ð3Þ

i¼1

M  N  X X sm ðtÞ  Pmin Di ðtÞ þ RSðtÞ for 8 t 2 T 6 m m¼1

ð4Þ

i¼1

ðX on m ðt  1Þ  MUTm Þ  ðsm ðt  1Þ  sm ðtÞÞ P 0

for 8 m 2 M; 8 t 2 T ð5Þ

ðX off m ðt  1Þ  MDTm Þ  ðsm ðtÞ  sm ðt  1ÞÞ P 0 for 8 m 2 M;

8t2T

ð6Þ

The Benders cuts of (2) serve to constrain the feasible region to a relatively small solution space of a master problem so that total generation cost in the master problem can be less than that inferred from the inter-temporal OPF sub-problems. In (2), the index nkm ðtÞ is concisely written as follows:

( nkm ðtÞ ¼

k

FC m ðPkm ðtÞÞ  qki ðtÞ  Pkm ðtÞ ifsm ðtÞ ¼ 1 FC m ðPmin m Þ

 qki ðtÞ 

Pmin m

k

ifsm ðtÞ ¼ 0

for 8 m 2 Xi ð7Þ

Evidently, the transmission network is included in deciding on/ off states of generating units by adopting the index of (7) where bus incremental costs incurred by transmission capacity limits of

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Initialize all binary variables

Set iteration number k = 1

Solve distributed inter-temporal optimal power flow

Create Bender’s cuts

k = k+1

Solve unit commitment problem

Yes Objective value improved ? No List optimal dispatch schedule Fig. 1. Flowchart for generation scheduling using generalized Benders decomposition (GBD).

lines are fairly connected with fuel costs. If a generating unit m committed at time period t of the previous iteration is shut down at time period t of the current iteration, its generation will be shifted to other generating units in the system. The term qki ðtÞ  Pkm ðtÞ represents the payment for purchasing insufficient power due to the shut down of this generating unit. When the unit m was off at time period t of the previous iteration and is required to be synchronized to the system at time period t of the current iteration, the expected generation is assumed as its lower generation limit to calculate the index of (7) since the accurate generation cannot be measured. Undoubtedly, the positive value of this index means that the shut down of unit m makes a lot of sense from an economic perspective since the fuel cost of the unit m to produce the power output is by far higher than the purchase payment of the same power. On the contrary, the negative value of this index is axiomatic that the commitment of unit m is profitable since the fuel cost is eventually lower rather than the purchase payment.

min max k1 sk1 m ðtÞ  P m 6 pm ðtÞ 6 sm ðtÞ  P m

for 8 m 2 M;

8t2T ð10Þ

Lij 6 Bij  ðdi ðtÞ  dj ðtÞÞ 6 Lij

for 8 i; j 2 N; i–j;

8t2T

ð11Þ

k1 sk1 m ðtÞ  sm ðt  1Þ  ðpm ðtÞ  pm ðt  1ÞÞ 6 SUR m

for 8 m 2 M;

8t2T

ð12Þ

k1 sk1 m ðt  1Þ  sm ðtÞ  ðpm ðt  1Þ  pm ðtÞÞ 6 SDR m

for 8 m 2 M;

8t2T

ð13Þ

3. Regional decomposition for inter-temporal OPF sub-problems 3.1. Implementation of algorithm-ADM

2.2. Formulation of sub-problems The objective of the inter-temporal OPF sub-problems is to minimize total generation costs of committed units that supply the load demand subject to a variety of operational constraints along with the transmission network. The inter-temporal OPF subproblem at the kth iteration is represented as follows:

min

TC k ¼

T X M h X

i k1 sk1 m ðtÞ  FC m ðpm ðtÞÞ þ SC m ðtÞ

ð8Þ

t¼1 m¼1

s:t:

X

k1 ðsm ðtÞ  pm ðtÞÞ  Di ðtÞ 

m2Xi

for 8 i 2 N;

N X

Bij  ðdi ðtÞ  dj ðtÞÞ ¼ 0

j¼1 j–i

8t2T

ð9Þ

To implement the distributed algorithm, consider Fig. 2 where a power system is divided by two contiguous regions connected by a single tie-line. The inter-temporal OPF variables within each region, called core variables, are denoted by xa and xb, respectively. The variables assigned to the border bus between a pair of interconnected regions, namely border variables, are denoted by y. If we define the copies of border variables to be ya and yb pertaining to the region a and region b, region a has state vector (xa, ya), while region b has state vector (xb, yb). In our scheme, we expect a solution matching the constraint ya = yb. In this scheme, hypothetical generating units, called dummy generators, are put on the border buses in one region, while dummy loads, set equal to the magnitude of the power output from the corresponding dummy generators, are linked with the border buses of the other region so that the power outputs from the dummy generators and dummy loads are canceled out at the optimum. Besides, core and border vari-

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Marginal Costs

Original system Region a

Region b y

xa

C1

λ b0 xb

πa

λ b1

πb La

β

Lb

1

λ a1 λ a0

Decomposed system Region a

0

Region b

xa

y b1

yb

y b*

(a) 1st iteration

xb Marginal Costs

La

ya

yb

Lb

λ b1

Fig. 2. Decomposition with dummy generator and dummy load (DGDL) scheme.

ables are associated with real power flows through the buses and phase angles at the buses over a set of time period since the inter-temporal OPF is formulated by the DC model. In principle, dummy generator and dummy load (DGDL) scheme is best modeled as a sequential computation algorithm, albeit, under specific situations, it can be modeled as a parallel computation scheme. In our implementation, first, the regional OPFs for region a and region b are executed with no dummy generators or dummy loads. Then a dummy generator is put in the region experiencing higher Lagrange multipliers on the borders (e.g., region b in this paper) to produce power. The iteration begins with the regional OPF for the region with the dummy generator, followed by the regional OPF for the region with the dummy load. The production cost function of the dummy generator is assumed to be:

Fðyb Þ ¼ b  yb þ c  y2b

ð14Þ

where yb is the power produced by the dummy generator and c is on the order of 1/N times the average of 2nd order coefficient in the quadratic fuel cost [15], ci, over all the generators in the region a, given by

c¼

N 1 X

N2

ci

ð15Þ

i¼1

where N is the number of generators in region a. In each iteration, the coefficient b is updated by the following formula [15]:

bkþ1 ¼

kka þ kkb  2  g  ykb 2

C1

λ b0

ð16Þ

where kka and kkb are the Lagrange multipliers at the border buses in region a and region b, respectively and g is a problem-dependent parameter governing the rate of convergence. The mechanism of the DGDL scheme with g = 1 is illustrated in Fig. 3, where the upper picture presents the 1st iteration, while the lower picture presents the 2nd iteration. In the figure, curves pa and pb represent respectively the values of marginal costs at the border buses in the region a and region b. The slope of the straight lines C1 and C2 equals the first order derivative of the production cost function of the dummy generator (in region b here). For the 1st iteration (in the upper picture), the value of b1 takes the

β

1

λb

2

C2

πa πb

λ a2

β2

λ a1

λ a0 0

y b1

y b2

y b*

yb

(b) 2nd iteration Fig. 3. Determination of power output by dummy generator in DGDL Scheme.

average of k0a and kkb , where k0a and k0b are the Lagrange multipliers at the border buses obtained from the regional OPFs without either the dummy generator (on the Bus 2b) nor the corresponding dummy load (on the Bus 2a). After the 1st iteration, y1b is determined at the intersection of the curve C1 and the curve pb. With the load L1a ð¼ y1b Þ applied to region a, the new Lagrange multiplier k1a is obtained after performing the regional OPFa. Then update b from k1a ; k1b , and y1b to get b2 using (16). Again perform the regional OPFb to determine the dummy generation y2b with the Lagrange multipliers k2a and k2b . We repeat this procedure until the solution converges to the optimal solution kb . It is noted that the line Ck moves in parallel toward the optimal point yb as iteration proceeds, always passing through the center of the line segment between kka and kkb . Notice that the DGDL scheme presented here is a variant of algorithm-ADM. To illustrate the implementation of DGDL scheme using ADM, we present an example below. As depicted in Fig. 2, there are a local demand in region a of La = 0.5 and a local demand in region b of Lb = 0.5. In addition, all transmission lines are assumed to be lossless. The cost function for each generator is given by:

1 2 x for region a 2 a fb ¼ x2b for region b

fa ¼

ð17Þ ð18Þ

We minimize the total production cost to meet 1.0 (p.u.) of system demand. Then we duplicate the border bus and the vector of the border variables. The dummy generator in region b and dummy load in region a are put on the duplicated border buses in each re-

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gion since it has been already known that the Lagrange multiplier at the border bus in region b is somewhat higher than that at the border bus in region a. Each individual region solves an OPF that includes its own region and the borders it shares with the other neighboring region.

OPFa

 1 2 xa fxa ¼La þyb g 2 min

ð19Þ

OPF b

min

fxb þyb ¼Lb g



 2

x2b þ b  yb þ c  yb

ð20Þ

As is well known, algorithm-ADM requires the regional OPFs for region a and region b to be performed sequentially. In solving OPF in region b; xkþ1 and ykþ1 are given by b b

xkþ1 ¼ b

bkþ1 þ 2c  Lb 2 þ 2c

ð21Þ

ykþ1 ¼ Lb  xbkþ1 b From (19), the solution xkþ1 is directly calculated. a

Table 1 DGDL scheme with g = 1 as a variant of Algorithm-ADM.

xkþ1 ¼ La þ ybkþ1 a k+1

k

xa

xb

yb

kka

kkb

b

0 1 2 3 4 5 .. . 11

.5000 .5714 .6122 .6355 .6489 .6565 .. . .6666

.5000 .4285 .3877 .3644 .3511 .3434 .. . .3333

.0000 .0714 .1122 .1355 .1489 .1565 .. . .1666

.5000 .5714 .6122 .6357 .6489 .6565 .. . .6666

1.0000 .8571 .7755 .7286 .7022 .6869 .. . .6666

.7500 .6071 .5255 .4786 .4522 .4369 .. . .4166

jkka  kkb j .5000 .2857 .1632 .0932 .0533 .0304 .. . .0000

ð22Þ

Furthermore, the coefficient b is updated by (16). This process is repeated until convergence criteria are met. The results for the first few iterations and the final solutions with g = 1 are summarized in Table 1. 3.2. Inter-temporal OPF sub-problems using algorithm-ADM In fact, the inclusion of ramp rate limits makes it difficult to decompose the inter-temporal OPF sub-problems since these

Region b

Region a

Fig. 4. A test system divided by two regions.

Hourly Load Demand 7000

6040

Load [MW]

6000

4690

5000 4000

5000

4870

4060 3280

3000

T2

2860

3070

3140

T3

T4

T5

4480

3980

3000 2000 1000 0

T1

T6

T7

Hour [h] Fig. 5. Hourly load demand curve.

T8

T9

T10

T11

T12

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constraints continuously influence the power system operations over the dispatch time horizon. Once these sub-problems are solved by a single indecomposable nonlinear programming technique, it may require a considerable amount of running computation time and have an adverse impact on the convergence. For this reason, the distributed OPF using a regional decomposition is

Objective value vs. Iteration 1.06

Master problem

1.04

Subproblems

1.02 1 0.98 0.96

min

1

2

3

4

5

ðxa ;ya Þ2a;ðyb ;xb Þ2b

6

ffa ðxa Þ þ fb ðxb Þ : ya ¼ yb g

Fig. 6. Convergence between master problem and sub-problems.

01GEN040 01GEN150 02GEN040 02GEN150 07GEN300 13GEN600 15GEN060 15GEN155 16GEN155 18GEN400 21GEN400 22GEN300 23GEN310 23GEN350 25GEN040 25GEN150 26GEN040 26GEN150 31GEN300 37GEN600 39GEN060 39GEN155 40GEN155 42GEN400 45GEN400 46GEN300 47GEN310 47GEN350 T1

T2

T3

ð23Þ

where the cost functions fa and fb are convex approximations to the actual cost functions in region a and region b, respectively. Then the augmented Lagrangian function for problem (23) is aptly defined as

Iteration

Gen ID

Million $

1.08

introduced [15]. The proposed distributed inter-temporal OPF is based on the alternating direction method (ADM) which converges at least at linear rate [16,17] and, under special condition, superlinear rate can be achieved [4]. Though the minimization steps in ADM cannot be carried out independently and this restricts its potential advantage in parallel implementations, it seems quite competitive to the other decomposition schemes which employ the approximated augmented Lagrangian function. Above all, this regional decomposition is most applicable to the generation scheduling in the multi-utility system in that it can ensure the optimal operations of the overall power system without any need for collecting information on each utility’s generation cost at one place. To sum up the procedure of solving inter-temporal OPF by aid of algorithm-ADM, consider a convex program with separable structure of the form:

T4

T5

T6

T7

T8

T9

Hour [h] Fig. 7. Optimal on/off schedules of generators for half a day.

T10

T11

T12

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c Lc ðxa ; xb ; kÞ ¼ fa ðxa Þ þ fb ðxb Þ þ kT ðya  yb Þ þ jjya  yb jj2 2

b is organized by 24 buses, starting from bus 25 and ending by bus 48. In this case study, the generation costs of region b is assumed to be exactly 1.4 times higher than those of region a. The case has been implemented by the GAMS (General Algebraic Modeling System) optimization tool [18]. Fig. 5 is displayed to represent a 12-h load demand pattern of a particular day. The most significant feature of Fig. 6 is that the optimal solution of the generation scheduling problem for this system converges within six iterations between the UC master problem and intertemporal OPF sub-problems. For better understanding, the optimal on/off schedules of generator units in the whole system for half a day are portrayed in Fig. 7. It is pointed out that the first two-digit number of a generator index on the left-hand side in Fig. 7 indicates an index of a bus to which the generator is attached and that the last three-digit number stands for the maximum capacity of the corresponding generator. Plus, a black square denotes on state of the generating unit during that hour while a white square indicates off state of the generating unit. From this result, three types of generators, i.e., base, medium, and peak generators, are observed. Base plants, which are relatively economic and largely located in region a, are committed for all time periods. On the other hand, peak plants, which are scattered over region b, are operated only at peak hours to balance supply and demand. All in all, 39GEN012 with the highest generation cost is decommitted all half a day long because of tremendous economic inefficiency. To prove the viability of the distributed implementation for the generation scheduling problem, the comparison of generation costs between the centralized and distributed OPFs is presented in Table 2. The centralized implementation is solved as a single system on one processor with no decomposition whereas the distributed implementation is solved on multiple processors using the regional decomposition technique. Each region is assigned to a separate process, or a parallel computation. First of all, the start-up cost provoked by the ADM based distributed implementation is in accordance with that of the centralized

ð24Þ

where kT denotes the transpose of Lagrange multipliers and c is taken as the average of 2nd order coefficient over all the core generators in the two adjacent regions. The algorithm-ADM produces the following sub-problems in each region:

n o c k T kþ1 k 2 ðxkþ1 a ; ya Þ ¼ arg min fa ðxa Þ þ jjya  yb jj þ ðk Þ ya 2 ðxa ;ya Þ2a

ð25Þ

n o c k T kþ1 kþ1 2 ðxkþ1 b ; yb Þ ¼ arg min 2 b fb ðxb Þ þ jjyb  ya jj  ðk Þ yb 2 ðyb ;xb Þ

ð26Þ

Meanwhile, the Lagrange multipliers k is updated by

kkþ1 ¼ kk þ s  ðykþ1  ykþ1 a b Þ

ð27Þ

where s is the updating parameter with the following relationship:

s¼

c 2

ð28Þ

4. Illustration on sample system A test system, sketched in Fig. 4, is used to illustrate the proposed generation resource schedule and dispatch model. Overall, this sample system is composed of 28 generators, 80 transmission lines, and 50 buses, and divided into two regions which are connected by two tie-lines (see Appendix A). Individual regions refer to the modified IEEE 24-bus reliability test system. Region a is made up of 24 buses ranging from bus 1 to bus 24 while region

Table 2 Comparison of generation costs between centralized and ADM based distributed OPF.

Total generation cost ($) Start-up cost ($)

Centralized OPF

Distributed OPF

Error (%)

1017,645 293.3

1017,653 293.3

0.00079 0

Table 3 Generating unit data. Gen ID

Pmax (MW)

Pmin (MW)

a ($)

b ($/MW)

c ($/MW2)

Ramp rate (MW/h)

Start-up cost ($)

01GEN040 01GEN150 02GEN040 02GEN150 07GEN300 13GEN600 15GEN060 15GEN155 16GEN155 18GEN400 21GEN400 22GEN300 23GEN310 23GEN350 25GEN040 25GEN150 26GEN040 26GEN150 31GEN300 37GEN600 39GEN060 39GEN155 40GEN155 42GEN400 45GEN400 46GEN300 47GEN310 47GEN350

40.0 150.0 40.0 150.0 300.0 600.0 60.0 155.0 155.0 400.0 400.0 300.0 310.0 350.0 40.0 150.0 40.0 150.0 300.0 600.0 60.0 155.0 155.0 400.0 400.0 300.0 310.0 350.0

32.0 30.0 32.0 30.0 75.0 200.0 12.0 54.3 54.3 100.0 100.0 0.0 100.0 140.0 32.0 30.0 32.0 30.0 75.0 200.0 12.0 54.3 54.3 100.0 100.0 0.0 100.0 140.0

132.162 167.400 132.162 167.400 347.340 389.630 132.162 325.672 325.672 721.922 721.922 0.000 325.672 703.500 132.162 167.400 132.162 167.400 347.340 389.630 132.162 325.672 325.672 721.922 721.922 0.000 325.672 703.500

18.79 20.24 18.98 20.44 22.67 23.32 19.17 15.88 16.04 17.82 18.00 0.00 16.20 14.25 18.79 20.24 18.98 20.44 22.67 23.32 19.17 15.88 16.04 17.82 18.00 0.00 16.20 14.25

0.05971 0.01866 0.06031 0.01885 0.01516 0.00456 0.06090 0.00922 0.00931 0.00921 0.00930 0.00000 0.00940 0.00786 0.05971 0.01866 0.06031 0.01885 0.01516 0.00456 0.06090 0.00922 0.00931 0.00921 0.00930 0.00000 0.00940 0.00786

8.0 120.0 8.0 120.0 225.0 400.0 48.0 100.7 100.7 300.0 300.0 300.0 210.0 210.0 8.0 120.0 8.0 120.0 225.0 400.0 48.0 100.7 100.7 300.0 300.0 300.0 210.0 210.0

62 230 62 230 780 13,000 62 1500 1500 4300 4300 0 2100 12,500 62 230 62 230 780 13,000 62 1500 1500 4300 4300 0 2100 12,500

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Table 4 Transmission line data. ID

From bus

To bus

Capacity (MW)

Susceptance (S)

Conductance (S)

Shunt (S)

LINE1 LINE2 LINE3 LINE4 LINE5 LINE6 LINE7 LINE8 LINE9 LINE10 LINE11 LINE12 LINE13 LINE14 LINE15 LINE16 LINE17 LINE18 LINE19 LINE20 LINE21 LINE22 LINE23 LINE24 LINE25 LINE26 LINE27 LINE28 LINE29 LINE30 LINE31 LINE32 LINE33 LINE34 LINE35 LINE36 LINE37 LINE38 LINE39 LINE40 LINE41 LINE42 LINE43 LINE44 LINE45 LINE46 LINE47 LINE48 LINE49 LINE50 LINE51 LINE52 LINE53 LINE54 LINE55 LINE56 LINE57 LINE58 LINE59 LINE60 LINE61 LINE62 LINE63 LINE64 LINE65 LINE66 LINE67 LINE68 LINE69 LINE70 LINE71 LINE72 LINE73 LINE74

1 1 1 2 2 3 3 4 5 6 7 8 8 9 9 10 10 11 11 12 12 13 14 15 15 15 15 16 16 17 17 18 18 19 19 20 20 21 25 25 25 26 26 27 27 28 29 30 31 32 32 33 33 34 34 35 35 36 36 37 38 39 39 39 39 40 40 41 41 42 42 43 43 44

2 3 5 4 6 9 24 9 10 10 8 9 10 11 12 11 12 13 14 13 23 23 16 16 21 21 24 17 19 18 22 21 21 20 20 23 23 22 26 27 29 28 30 33 48 33 34 34 32 33 34 35 36 37 38 37 38 37 47 47 40 40 45 45 48 41 43 42 46 45 45 44 44 47

500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500

69.51 4.4382 11.096 7.3969 4.8813 7.8758 11.91 9.0394 10.617 15.7 15.263 5.6772 5.6772 11.91 11.91 11.91 11.91 20.669 23.531 20.669 10.184 11.373 25.289 56.884 20.076 20.076 18.952 37.993 42.572 68.376 9.3431 37.993 37.993 24.841 24.841 45.531 45.531 14.51 69.51 4.4382 11.096 7.3969 4.8813 7.8758 11.91 9.0394 10.617 15.7 15.263 5.6772 5.6772 11.91 11.91 11.91 11.91 20.669 23.531 20.669 10.184 11.373 25.289 56.884 20.076 20.076 18.952 37.993 42.572 68.376 9.3431 37.993 37.993 24.841 24.841 45.531

13 1.147 2.862 1.915 1.263 2.038 0.326 2.336 2.74 3.607 3.952 1.468 1.468 0.326 0.326 0.326 0.326 2.648 3.039 2.648 1.307 1.459 3.25 7.233 2.581 2.581 2.446 4.84 5.528 8.547 1.197 4.84 4.84 3.199 3.199 5.902 5.902 1.86 13 1.147 2.862 1.915 1.263 2.038 0.326 2.336 2.74 3.607 3.952 1.468 1.468 0.326 0.326 0.326 0.326 2.648 3.039 2.648 1.307 1.459 3.25 7.233 2.581 2.581 2.446 4.84 5.528 8.547 1.197 4.84 4.84 3.199 3.199 5.902

0.4611 0.0572 0.0229 0.0343 0.052 0.0322 0 0.0281 0.0239 2.459 0.0166 0.0447 0.0447 0 0 0 0 0.0999 0.0879 0.0999 0.203 0.1818 0.0818 0.0364 0.103 0.103 0.1091 0.0545 0.0485 0.0303 0.2212 0.0545 0.0545 0.0833 0.0833 0.0455 0.0455 0.1424 0.4611 0.0572 0.0229 0.0343 0.052 0.0322 0 0.0281 0.0239 2.459 0.0166 0.0447 0.0447 0 0 0 0 0.0999 0.0879 0.0999 0.203 0.1818 0.0818 0.0364 0.103 0.103 0.1091 0.0545 0.0485 0.0303 0.2212 0.0545 0.0545 0.0833 0.0833 0.0455

1535

K.H. Chung et al. / Electrical Power and Energy Systems 33 (2011) 1527–1535 Table 4 (continued) ID

From bus

To bus

Capacity (MW)

Susceptance (S)

Conductance (S)

Shunt (S)

LINE75 LINE76 LINE77 LINE78 LINE79 LINE80

44 45 1 2 25 26

47 46 49 50 49 50

500 500 700 700 700 700

45.531 14.51 69.51 69.51 69.51 69.51

5.902 1.86 13 13 13 13

0.0455 0.1424 0 0 0 0

case since the parallelization of inter-temporal OPF sub-problems does not affect the determination of binary variables in the master UC problem. In contrast, the values of continuous variables from the ADM based distributed scheme are, to some extent, different from those of the centralized case, arising from the fact that some accuracy is damaged in the process of using approximations to continuous variables in other regions when the regionally partitioned OPF sub-problems are run by ADM. However, since the difference of total generation costs between the centralized case and the ADM based distributed implementation is approximately 0.00079%, the error is thought to be completely acceptable. To capitulate briefly, the distributed inter-temporal OPF by ADM does not distort the generation schedule even though a slight error is made in total generation costs.

5. Concluding remarks As we have seen, the main purpose of this paper has been to address an innovative short-term scheduling of power generation in interconnected power systems which is mathematically expressed by a large-scale, nonlinear, mixed-variable mathematical programming model. The foregoing analysis has demonstrated the validity of the decomposition structure of Bender whose basic idea is to decompose an original problem into the binary discrete variables and the continuous variables and alternate between solving a master problem and sub-problems at each iteration. As an outcome, the proposed approach includes the distributed optimal power flow (OPF) by a regional decomposition technique to cover transmission network constraints in the multi-utility system. It is indeed apparent that the alternating direction method (ADM) is seamless and well fitted for solving inter-temporal OPF sub-problems since it makes effective use of the primitive augmented Lagrangian function. At the same time, our regional decomposition technique is expected to find the most economic dispatch schedule reflecting power transactions of utilities without disclosing details of their operating costs to competitors. In this regard, it is worth emphasizing that our proposed generation scheduling algorithm should certainly facilitate the operations of interconnected power systems. Acknowledgements This work was funded by Ministry of Education and Human Resources Development as for BK21 (2nd phase) Project (Research on New Energy Resource & Power System Interface). This research was supported by Basic Science Research Program through the Na-

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