An efficient decomposition approach for multi-area production costing

ELSEVIER

Electrical Power & Energy Systems, Vol. 18, No. 4, pp. 259-270, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0142-0615(95)00071-2 0142-0615/96/$15.00 + 0.00

An efficient decomposition approach for multi-area production costing C Singh Electrical Engineering Department, Texas A&M University, College Station, Texas 77843, USA N Gubbala ABB Systems Control, 2550 Walsh Avenue, Santa Clara, California !)5051, USA

are several approaches in the literature for single area production costing. The most commonly used production costing approach has been the single area approach introduced by Baleriaux et aL 1 and Booth2. In this method, the equivalent load is defined as the summation of system load and capacity on outage for each generating unit. System load can be represented by a load duration curve (LDC) and then an equivalent load duration curve (ELDC) is obtained every time the probability distribution of a unit capacity outage is convolved with the LDC in the commitment order of generating units, so that the expected energy supplied by the generating unit can be found from the ELDC. To improve the computational efficiency, an alternative approach using the concept of cumulants and Gram-Charlier expansion has been used to approximate the ELDC by a continuous distribution'34 . Later, several other methods based on continuous distributions5-9 and z transforms l° were proposed. Several authors have extended the continuous distribution approach to two areas 11'12. Multi-area production costing for more than two areas, however, has received only limited attention 13. A new and efficient approach for multi-area configurations is presented in this paper. This approach is based on the SIDES (simultaneous decomposition-simulation) approach which has been developed for reliability analysis of interconnected power systems. One of the approaches for multi-area reliability analysis is the DES (decomposition-simulation approach) 14. This approach, however, uses only a snapshot of area loads. An improvement over this method is the extended decompositionsimulation approach 15 which uses a multi-area load model obtained from hourly load data by clustering algorithms 16. The clustering procedure captures the temporal correlation between area loads. In this method, the decomposition-simulation is performed for each of the load states and the reliability indices are aggregated. The most recent developments are the SIDES approach and its more efficient version, PRESIDES 17-19

This paper presents a new approach for multi-area production costing called SEG-GSIDES (segmented global simultaneous decomposition-simulation). It is based on the SIDES (simultaneous decomposition-simulation) approach which has been developed for computing reliability indices in a multi-area configuration. The global state space is obtained from the maximum number of units committed in each area and thus contains the state space corresponding to any set of committed units. In the GSIDES approach, the global state space is decomposed and the resulting sets are used to compute the required indicesfor any subset of units committed. The energy supplied by each unit is calculated by the difference between the expected unserved energy before and after unit commitment. In some cases it may be possible to obtain indices by decomposition alone but generally a decomposition phase isfollowed by a simulation phase. When the units committed are close to base loaded units, their contribution to unserved energy from the decomposition phase is ,tow. The major contribution in this case comes from the simulation phase and this introduces an approximation due to convergence. To reduce this approximation the decomposition is performed sequentially on different segments of the global state space. In this way, the major part of the indices is obtained from the decomposition phase and this leads to a more accurate determination of energy supplied by the units. The concepts are illustrated using a small example system. Case studies are presented for the three area and four area IEEE-RTS systems. Copyright © 1996 Elsevier Science Ltd. Keywords." production costing, multi-area, indices, generation model, cluster load model

reliability

I. I n t r o d u c t i o n The principal analytical tools in resource planning applications are production costing simulation models. There Received23 March 1995; accepted 10 July 1995

259

260

Decomposition approach for multi-area production costing. C. Singh and N. Gubbala

which perform the decomposition process for all the planned maintenance intervals and load states simultaneously, using the concept of reference load state. The reliability indices are then extracted from the resulting sets by decomposing the sets back into each planned maintenance interval and each load state. An interesting feature of this approach is that the number of load states in the load model can be increased without a significant increase in CPU time. The SIDES and PRESIDES are highly efficient approaches suitable for many areas interconnected with each other. At the conceptual level, multi-area production cost can be calculated by performing SIDES before and after each unit addition in the system. However, the number of decomposition-simulations becomes too large depending on the number of units. This paper proposes an efficient approach which performs decomposition only once globally and then extracts the indices before and after each unit commitment. This method is ideal if the decomposition is performed completely. But in practice, for large systems the decomposition may be stopped at a predetermined threshold probability and then simulation follows. This can lead to computational problems which are discussed in the paper and an improved approach SEGGSIDES (segmented global simultaneous decompositionsimulation) is offered to solve these problems.

II.

studies is important. The load model should be capable of including hourly load variation and also the temporal correlation between area loads. This paper uses a discrete joint distribution of area loads to incorporate the correlation between area loads. Let the load for the hour h, in interval k, be represented by, Lh

h Lh = ( L hk,l,Lk,2,''', k,na)

where h Lk,j = load for hour h in area j during interval k

na = number of areas.

These hourly loads can be represented, using clustering algorithms 15,2°, by a discrete joint probability distribution vector, dik = (dk,I, i d ki, 2 , . . . ,

d k,na) i

i = 1 , 2 , . . . , nl

where dik,j = load in area j in interval k in load state i nl -- number of load states in the load model.

The distribution is then represented by, Pr(D = d~) = probability of load state d~. Methods to determine these discrete joint distributions of area loads are explained in References 15 and 20.

R e v i e w of the SIDES approach

In this section the SIDES approach, which forms the basis for the new method, is briefly reviewed. The year is assumed to be divided into intervals. During each interval, the generation mix, the unit commitment order and the load models are assumed to be fixed. The division into intervals could be either due to maintenance scheduling or change in unit commitment order. First, the generation models, the load models and tie line models are explained. Then the formation of the composite state space is explained, which enables decomposition to be performed for all the intervals and load states in one step. Finally, the decomposition and simulation phases are briefly explained.

11.3 Tie line model An equivalent probabilistic multi-state tie line model is formed for all the transmission line flows possible between any two given areas. This multi-state tie line model can also incorporate common mode failure of transmission lines between any two given areas. The tie line model is represented by the tie line capacity vector tr, i = (fr, i, br, i), and Pr(t,,i) = probability of tie line state, tr, i

where r = subscript denoting the tie line where,

I1.1 Generation model The generation models are built using a unit addition algorithm for each area and each interval. Let Cki = capacity vector for area i for interval k Pki = probability vector associated with cki

such that Pr(Cki ) = Pki" The generation models are built or rounded off to fixed increments and thus the capacity vectors need not be stored, but only the minimum capacity states are required. cmina = minimum capacity state in area a ngi, a = number of generation states in area a and

interval i The above information along with probability vectors defines the generation models. The capacity of any state x can then be given as cmina + (x - 1) × s, where s is the increment step size in MW. 11.2 L o a d m o d e l Proper load modelling in multi-area production costing

r= l,2,...,ntl ntl = number of tie lines in the system i = subscript denoting tie line state where, i= l,2,...,nts, rigs r = number of states in the tie line r fr, i

tie line capacity in forward direction

br, i = tie line capacity in backward direction.

11.4 C o m p o s i t e state s p a c e Formation of a composite state space is the key to the SIDES approach. If there are ni intervals and nl load states in each interval, then using DES 14, decomposition would need to be performed nl x ni times. Thus for 20 load states and 12 intervals, DES would be repeated 240 times. Using the concept of composite state space, this problem can be transformed into a single decomposition 17'18. The whole procedure is based on two concepts. The first is that, if at a given node all generation states and the given load state are shifted by the same amount, the flow calculations are not changed. Hence the reliability and

Decomposition approach for multi-area production costing: C. Singh and N. Gubbala production cost indices remain unchanged. The second concept is that decomposition depends on the state capacities and not on the state probabilities. Using these two concepts, the problem of nl × ni load states (n! is the number of load states and ni is the number of intervals) and ni generation systems can be transformed into one load state (called the reference load state) and nl x ni generation models. These models are then combined into a composite set. Basically a capacity state in a composite set represents capacity states corresponding to n! × ni generation models. Then the decomposition is performed on the composite generation model using the reference load state. Thus the number of decompositions can be reduced from nl x ni to one. There is some effort involved in recovering indices from the composite state space but this is negligible compared with nl × ni decompositions. 11.4.1 Selection of reference load state The first step in the dewdopment of a composite state space is the selection of the reference load state. Any load state could be selected as the reference load state. For the sake of computational convenience, the reference load state is taken as the maximal load in each area over all the planned maintenance intervals. aref = (dr, d~ , . . . , dn*a)

(1)

where

*=

dJ

max

l<~i<.nl, l<~k<~ni

ti

{ak,j}

The decomposition is performed by using this reference load state. 11.4.2 Modification of generation model The generation capacities in each area and in each interval are shifted by the differences between the original load state and the reference load state. Consider an area a, interval i and load state l. The load value in area a is given by d~,a. The corresponding generation capacity states range from 1 to ngi, a where ngi,a is the number of generation states in area a, in interval i. The area load in load l to change the load state state I is increased by (d~ - d i,a) to the reference load state. To keep the equivalence, capacities of all the generation states in that area are also increased by the same amount. This amounts to shifting the generation model in area a by the number of states, £l~a = (d; - dil,a ) / S

(2)

Therefore for each interval, there are nl modified generation models resulting in nl × ni number of modified generation capacity models for the whole study period. Although the state capacities are modified, their associated probabilities remain the same as in the original generation models. 11.4.3 Composite generation model The nl x ni number of shifted generation capacity models can be merged into a single composite generation model. As there will be at least one load value identical to the reference value in each area, there will be no shift in the corresponding generation capacities. Consequently the composite generation model of an area starts from state number 1. The number of capacity states in the area a in the composite model is obtained by the following

261

equation, nc a =

max

l <~i <~ni, l <~l <~nl

{ngi, a -~- ( d ; - d l a ) / S }

(3)

The composite state space is now formed by the composite generation state space and tie line state space which is given below.

nil ... ...

nCna

1

nCna + l

i ncni+n]

(4)

1

where

1 , . . . , na ----generation arcs (na + 1),..., (na + ntl) = tie line arcs.

The number of tie line states ntsr is renamed as ncn~+r for notational convenience. 11.5 D e c o m p o s i t i o n - s i m u l a t i o n The composite state space is recursively decomposed 17'18 using the reference load state into acceptable (A) sets, loss of load (L) sets and unclassified (U) sets until all the U-sets fall below a predetermined threshold probability. The L-sets are then recursively decomposed into identical area loss of load (B) sets and unclassified loss of load (W) sets until all the W-sets fall below a predetermined threshold probability. After the decomposition phase, the sets of interest can be decomposed back into each interval and each load state easily, for evaluating the reliability indices. The loss of load probability of the system is obtained by the sum of probabilities of L-sets. Area indices like loss of load probability and expected unserved energy are calculated from the B-sets. At the end of the decomposition phase there are undecomposed sets whose probabilities are below the threshold. The indices from these sets can be efficiently approximated using simulation.

III.

Production costing

II1.1 P r o b l e m f o r m u l a t i o n Multi-area production costing problem ion can be solved by the decomposition-simulation (DES) approach. In this method, the expected unserved energy (EUE) needs to be calculated before and after each unit commitment and therefore DES needs to be performed for each unit. Further, for every EUE calculation, decompositionsimulation needs to be performed for every interval and load state• This leads to substantial computational effort and makes the procedure impractical. A simultaneous decomposition-simulation approach can be used to reduce the computational effort in the decomposition phase. In this method decomposition and simulation need to be performed only once for all the intervals and load states. However, SIDES has to be performed for every unit addition. To overcome the need to perform SIDES for each unit, a global simultaneous decomposition-simulation (GSIDES) approach is proposed. In this approach only one decomposition needs to be performed and the EUE can be computed for any combination of unit commitment. The GSIDES approach is explained in the next sub-section.

Decomposition approach for multi-area production costing: C. Singh and N. Gubbala

262

Table 1. Approximate time comparison between various approaches Time for Decomposition Indices phase evaluation

Approach DES SIDES GSIDES SEG-GSIDES n = ni = nl = nd = td = tc = to =

Misc. time

n x n i x n l x ta n x n i x nl x t C n x ta nxnixnlxt c to ta

na x ta

n x ni x n l x tc to n x n i x nl × t c t o

n u m b e r of non-base loaded generating units number of planned maintenance intervals in the study period number of discrete load levels in the load model number of decomposition phases <
There are, however, some accuracy problems with this approach if exhaustive decomposition is not performed. To eliminate these problems, several decompositions are performed on different segments of the global state space and this approach is called segmented global simultaneous decomposition-simulation (SEG-GSIDES) approach. The computational times for various approaches are compared in Table 1. As GSIDES and SEG-GSIDES are based on the SIDES method, there are certain advantages they inherit. Preferential simultaneous decomposition-simulation (PRESIDES) 19, is an improvement in the basic decomposition process which reduces the number of sets generated tremendously and consequently reduces the CPU time. The computer program for GSIDES and SEGGSIDES is based on PRESIDES. PRESIDES and SIDES methods are theoretically feasible for any number of areas and interconnection. Case studies involving 3, 5, 8 and 10 are reported in Reference 19. The energy supplied by base loaded units can be evaluated directly as follows. li

E S / = NHR × ~ { C A P q x Pr(j)}

(5)

j=l

where ESi = energy supplied by unit i during the study period NHR = total number of hours in the study period ti =

number of capacity states of unit i

CAPij = capacity of unit i in state j Pr(j') = probability of state j. 111.2 G S I D E S a p p r o a c h The key to the GSIDES approach is that the decomposition depends only on state capacities and is independent of state probabilities. Therefore the production costing problem is solved as follows. First, the decomposition is performed on the global state space which contains the state space corresponding to any set of committed units and is obtained by committing the maximum number of units in each area. The sets obtained by this state space are valid for any set of committed units if the capacity probability tables corresponding to the committed unit

are used. Thus the indices can be calculated for any combination of unit commitment from the decomposed sets by considering corresponding probability distributions. This approach is ideal and gives accurate results for cases where the decomposition can be performed exhaustively. However, exhaustive decomposition may not be practical for all the cases and a stopping criterion based on threshold probabilities is generally used. This leads to some accuracy problems which are discussed below. Consider an example where each area has a capacity of 3400 MW corresponding to full commitment of all units. First, global decomposition is performed on the state space with all units committed. Now say we want to find the EUE with a certain number of units committed in each area such that the capacities in the three areas are 1100, 1200, 1250 MW. To make this calculation, we need the loss of load {B} sets and their probabilities. As decomposition depends only on capacities, the sets obtained from global decomposition are valid. However, the generation models used to calculate their probabilities should be the ones corresponding to the units actually committed. If global decomposition is performed completely, then the results obtained by this method are exact. For large systems global decomposition will be stopped when the probability of undecomposed sets is below a threshold value. For the purpose of this probability calculation, generation systems with all units committed may be used. When only subsets of units are considered, the probability of a given undecomposed set becomes higher than with all units committed. This results in a larger contribution from the simulation phase. This approximation leads to problems in production costing calculations for the following reasons. As mentioned earlier, the energy supplied by any unit is calculated by the difference between the expected unserved energy before and after the unit commitment. The energy supplied by the first unit is Ul - u2, where ul and uz are the EUEs before and after the first unit commitment. If both results converge with a variance of x% in the simulation phase there is a variance of x% in both ul and u2 which leads to 2x% variance in the energy supplied with respect to ul or u2. However, this variance is very large with respect to the energy supplied, as u~ or u2 >> (ul - u2). This problem is severe only for the units committed closer to the base loaded region and disappears towards the peak units because the contribution from the simulation phase decreases as the committed capacity increases. Owing to these problems, the GSIDES approach cannot accurately compute the production cost when exhaustive decomposition is not performed. In order to solve the above problems, the decomposition is performed more than once on different segments of the global state space. This is done to reduce the contribution of indices from the simulation phase near base loaded units. 111.3 S E G - G S I D E S a p p r o a c h The global state space can be broken into a number of segments based on either fixed capacity or fixed number of units in each area. The generation model is first built with base loaded units and stored as g s m o. The composite state space for g s m o , using equation (3) is shown as CSS0 in Figure 1. The generation units in the first segment are added to g s m o and stored as GSM1. The composite state space (CSS1 in Figure 1) is computed for GSMI, using equation (3) and decomposition is performed on it. To

Decomposition approach for multi-area production costing." C. Singh and N. Gubbala

.

M3 . M1 m3 "A l

.

• Set 3 does A' not exist.

i

~ m 1

•

:::::::::::::::::::::::::::::::::::::

i I

i::::::::::::::::::::::::::::::::::::: :~:~:~:~:~:~:~:~:i:~:~:!:~:i:~:!:!:i

::::::::::::::::::::::::::::::::::::: ................ w ............................... ,.... ...................

CSS 1

.... . . . . . . . . . . . . . . . . . . . .

================================

............................... .............. •:,:.:.:+:.:+:.:.:.:.:+:.:+: . . . . . . . . . . . . . . . . . . . ::::::::::::::::::::::::::::::::::::: .:.:.:.:+:.:.:.:.:+:.:+:..:.: ::::::::::::::::::::::::::::::::

•

:

::

:

:i

:

:

:::i. ::

.

.

ii:iil :

:

M~

. . . .

ii

::i:

:: : i

,qlate. 1

:: : :::::: : ::iiT::i:: .... : : : : : :

::iii:

i.i

:

iili

:

:i:iiiii i

:.i:ii

:

:

:

m

.i

:::::

i

iiiii:

cl i,a

CSS 0 CSS l CSU

Ni-1

Area

i

3

= Composite state space for base loaded unit = Composite state space for units in segment 1 = Composite state space used for N iq units

Figure 1. Segmented composite state space for SEGGSl DES approach

evaluate the energy supplied by the ith unit, the generation model, gsrn i_ l using Ni-i units is built and a composite state space, CSUN~_, is defined as shown in Figure 1. The EUESi_ 1 :is computed using the generation model g s m i_ 1 and sets generated from the decomposition of CSSI. One way to compute the EUES is to re-adjust the boundaries of all the decomposed sets to exclude the states between CSUwi_, and CSS1. After this the evaluation of the indices is similar to the SIDES method. Another efficient way to calculate the EUES is discussed in the next sub-section. The energy supplied by the unit i is given by ESi = EUESi_ 1 - EUESi

(b)

(c)

(a) Original generationstates for inverval 'i' and load state T (b) Composite state space{A,B,C} (c) Original generationstates {B'}

:::

2

¢'J

l.I

:

: : : ::::::::: : . . . . . . . . . . . . . . . . . . . . . . . .

1

_

:

(a) i ii:il

]

...........l

CSUN i-1

•

' / 7 ! ~

:

I

iiiiiiiiiiiiiil

:'::::+:':'::::':::'::: :~:i:~:~:i:~:~:i:~:~:i:~:~:~:~:i:i:i .......................................................................... :::::::::::::::::::::::::::::::: ::: ::::::::: ::::::::::::::::::::::::::::::::::: : : .... .:: ::::::::::::::::::::::::::::::::::::.

.~

.!

=================================== ::::::::::::::::::::::::::::::::::::

.:.:.:.:+;.:,:.:.:+:.:+:+:.:

CSS 0

M'Im' 1

I

:::::::::::::::::::::::::::::::::::::

263

(6)

The energy supplied by all the units in the first segment can be computed in a similar manner. Then units in the second segment are added to GSM1 and decomposition is performed on the second segment. This decomposition phase can be used to calculate energy supplied by the units in the second segment. This process is continued for the rest of the segments and the total production cost is computed. It should be noted that decomposition is performed on each segment only once. Then using the generation models corresponding to various sets ofcommired units, EUE can be extracted.

111.4 Extraction of indices from the decomposition phase The process of re-adjusting the maximum and minimum states of the sets after every unit addition can be efficiently incorporated in the SIDES method without any overhead. As mentioned before, for evaluating the indices, the sets need to be re-decomposed into original sets corresponding to each planned maintenance interval and each

Figure 2. Conversion from composite states to original states load state. The process of decomposing the sets into original sets in the SIDES method is shown in Figure 2. For simplicity only the states in one area are shown. The original generation states for interval i are shown in Figure 2a. During the development of the composite state space, these states are shifted to region B in Figure 2b for load state l. It should be noted that state space B is the original set of states shifted up by cl~a as given by equation (2). The other modified generation models may partly overlap with B and the nonoverlapping portions will generate regions A and C. The region {A, B, C) comprises the entire composite state space. The maximum and minimum states of any decomposed set can lie anywhere in {A, B, C}. These maximum and minimum states should be converted to include only the original states in planned maintenance interval i and load state l. As explained in Section II.4.2, to create composite state space the original states are shifted up by cli,1a in Figure 2b and hence to recover them the composite state space needs to be shifted down by the same amount. Figure 2c shows the composite state space {A, B, C} shifted down by cli l a and represented as {A,t B,t l l I ! C }. It should be noted that A , B, C are the same as A, B, C expect that they are shifted down by Cl~a. Accordingly a composite set should also be shifted down by the same amount. The shifted set will now lie anywhere in {A', W, C t} but the set should be corrected to exclude region {A/, C'). In other words, the set has to be in the original generation states, i.e, B'. The maximum and minimum state of the set should be corrected after shifting down as follows. Maximum state: If it falls in Region A': It should be forced to the maximum state of the region B' i.e., nga, i. Region B': No correction required. Region C~: The set does not exist since the maximum state of the set falls in the infeasible region C t'

Decomposition approach for multi-area production costing." C. Singh and N. Gubbala

264

(4)

Minimum state: If it falls in

Region At: The set does not exist as the minimum state of the set falls in the infeasible region A t. Region Bt: No correction required. Region C': It should be forced to the minimum state of the region Bt i.e., 1. For example assume that, three composite sets shown in Figure 2b are to be decomposed to the original sets corresponding to a planned maintenance interval i and a load state l. The first composite set has maximum and minimum states as Ml and ml. The maximum state MI lies in region A' and the minimum state ml in region Bt. Thus no correction is needed to the minimum state and the maximum state is forced to the maximum state of B'. The corresponding set in original state space is M'I and mtl. The second composite set has its maximum state in region B' and minimum state in region C t. Thus no correction is needed to the maximum state and the minimum state is forced to the minimum state of Bt, resulting in set M~ - m[. For set three, the minimum state lies in region A' making this set non-existent. Using this process B-sets are decomposed into original subsets and the mean curtailment of load in the system is computed for every planned maintenance interval and load state. The expected unserved energy of the system is obtained by aggregation of mean load curtailment for each B-set. 111.5 Algorithm for GSIDES a n d SEG- GSIDES The overall algorithm for multi-area production costing is given below. If the number of segments given is one, then the approach is GSIDES, otherwise it is the SEGGSIDES approach. Exhaustive decomposition can be performed by specifying a threshold probability of 0.0. Let TPC represent the total production cost and PCUg represents the production of unit energy from unit i. (1)

(2) (3)

Build the base loaded generation model for the system in each planned maintenance interval using all base loaded units and store the generation model in gsm i. Set the total production cost, TPC = 0.0 and the segment numberp = 1. Calculate the production cost of all the base loaded units by equation (5) and TPC = TPC + ESi x PCUi.

(5) (6) (7) (8) (9)

Build the generation model GSMp by adding units from the pth segment to the generation model gsmi and compute the composite state space using equation (3). Perform the decomposition on the composite state space of the pth segment. If this is the first segment phase, then compute the EUESi_ 1 by using the base loaded generation model gsm i. Add the next unit on priority to the generation model gsm i. Using the updated generation model gsmi evaluate the indices from decomposition and simulation phases. Compute the energy supplied by the unit, ESi = EUESi_ 1 EUESi. Update the production cost: TPC = TPC + ES; x PCUi and assign EUESi to EUESi_ 1. If there are more units in thepth segment then go to step 7. Increment segment number, p = p + 1 and go to step 4. Print the overall and individual production cost of the units and stop. -

(10) (11) (12) (13)

-

The algorithm is implemented in a computer program and is based on the PRESIDES approach which is similar to SIDES. The PRESIDES approach requires less memory and CPU time.

IV.

Example system

The main steps in the production costing algorithm are illustrated by means of a small system consisting of three areas with each area having three generators. The three areas are connected to each other by tie lines. IV.1 Area generation m o d e l s Each of the three generators in an area is assumed to have a capacity of 50 MW and a failure probability of 0.12005. The generation model for the whole system is shown in Table 2. IV.2 Load m o d e l A two state load model shown in Table 3, is assumed for the system.

Table 2. Generation models for the system

Area 1

Area 2

Area 3

State

Cap (MW)

Prob.

Cap (MW)

Prob.

Cap (MW)

Prob.

4 3 2 1

150 100 50 0

0.681356 0.278868 0.038046 0.001730

150 100 50 0

0.681356 0.278868 0.038046 0.001730

150 100 50 0

0.681356 0.278868 0.038046 0.001730

Table 3. Load model for the system

Load state 1

2

Table 4. The line model for the system

Area 1 (MW)

Area 2 (MW)

Area 3 (MW)

Prob.

50 100

50 100

50 100

0.8 0.2

Tie line

From

To

Capacity

Prob.

1 2 3

1 1 2

2

50 50 50

1.0 1.0 1.0

3 3

Decomposition approach for multi-area production costing: C. Singh and N. Gubbala

265

IV.3 Tie line mode/ For simplicity, inter-are, a tie lines are assumed to be perfectly reliable with the same capacity in both directions and are shown in Table 4.

IV.6 Composite state space The composite state space is obtained by taking composite generation state space and tie line state space and is shown in Table 8.

IV.4 Selection of reference load state Any load state can be a,;sumed to be the reference load state, but as indicated by equation (1), the maximum load state i.e., (100, 100, 100) is taken as the reference load state.

IV.6.1 Decomposition of the global state space The global state space is decomposed recursively into acceptable (A) sets, loss of load (L) sets and undecomposed (U) sets. The L-sets are decomposed recursively into identical area loss of load (B) sets and unclassified loss of load (W) sets. Expected unserved energy (EUE) and loss of load probability (LOLP) are computed from the B-sets. For this small system the state space was decomposed exhaustively resulting in A and B sets. All the B-sets generated are tabulated in Table 9. The arcs M1 to M6 indicate the maximum state whereas ml to m6 indicate the minimum state of a set. The first three arcs are the generation arcs and the next three are tie line arcs. These sets which are generated from the global state space can be used for the generation model from any subset of units.

IV.5 Composite generation model The composite generation model is obtained by modifying the capacities of the generation models for each load state. The generation capacities for load state 2 do not change because the difference between the reference load state and load state 2 is (0, 0, 0). These generation capacities are given in Table 5. The generation capacities for load state 1 change by the difference between reference load state 1 i.e., by a vector (50, 50, 50). As explained earlier this is done to keep the expected unserved energy and loss of load probability unchanged. The modified generation capacities are given in Table 6. The composite generation model is obtained by taking a union of the two modified generation models and is shown in Table 7. This composite model is also the global state space for generatzion states, as all the units are committed in each area.

IV.7 Production costing Let the generators in the system be numbered as shown in Table 10. The generators 1-3 are assumed to be base loaded units. Energy supplied by the base loaded units can be computed by equation (5). The energy supplied by non-base loaded units is computed by the difference between the expected unserved energy (EUE) of the system, before and after the unit is added. As there are six non-base loaded units, EUE needs to be computed for seven stages as shown in Table 11. The sets used in the EUEi calculation are from the global decomposition as shown in Table 9 but the probabilities of generation models are obtained only from the committed units in that state. The EUE is calculated for each B set and then these values are aggregated. Therefore EUE calculation from the global sets is illustrated for one set and the same procedure can be used for the other sets. An illustration of EUE computation for 34th B-set for stages 6 and 7 follows. In stage 6, three units are committed in areas 1 and 2 and two units are committed in area 3. The generation model for this is shown in Table 12. In stage 7, all units are committed and the generation model for this state is shown in Table 2. As the B-set is from the composite state space, it needs to be decomposed into one set for each load state. The B-set corresponding to load state 1 is given below. [221111 iJ 1 1 1 1

Table 5. Modified generation model for load state 2 State

Area 1

Area 2

Area 3

4 3 2 1

150 100 50 0

150 100 50 0

150 100 50 0

Table 6. Modified generation model for load state 1 State

Area 1

Area 2

Area 3

5 4 3 2

200 150 100 50

200 150 100 50

200 150 100 50

Table 7. Composite generation model for the system State

Area 1

Area 2

Area 3

5 4 3 2 1

200 150 100 50 0

200 150 100 50 0

200 150 100 50 0

This is obtained by subtracting vector (1, 1, 1) from the maximum and minimum states which is the difference between the reference load state and load state 1. An additional condition is that the minimum state cannot be less than 1. The probability of the set can be computed

Table 8. Composite state space for the system State

Area 1

Area 2

Area 3

Tie 1

Tie 2

Tie 3

Maximum Minimum

5 1

5 1

5 1

1 1

1 1

1 1

Decomposition approach for multi-area production costing: C. Singh and N. Gubbala

266

Table 9. The B sets obtained by decomposing the global state space Set

M1

M2

M3

M4

M5

M6

ml

m2

m3

m4

m5

m6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

2 1 2 1 2 ! 2 1 2 1 2 1 2 1 2 1 5 4 3 4 1 I 1 1 1 1 1 2 2 2 2 2 3 3 5 4 4 3 5 4 4 1 1 2 3 4 3 1

1 1 1 1 1 1 1 1 5 5 4 4 3 3 2 2 2 2 2 1 2 2 2 2 5 4 3 3 2 3 2 4 2 2 I 1 2 3 5 5 3 4 3 2 1 1 4 3

5 5 4 4 3 3 2 2 1 1 1 1 1 1 1 1 1 1 1 1 5 4 3 2 2 2 2 3 3 2 2 2 3 2 2 2 2 2 I 1 1 3 3 4 4 3 1 4

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 5 4 3 4 1 1 1 1 1 1 1 2 2 2 2 2 3 3 5 4 4 3 5 4 4 1 1 2 3 4 3 1

1 1 1 1 1 1 I 1 5 5 4 4 3 3 2 2 2 2 1 1 2 2 2 2 5 4 3 3 2 3 2 4 1 1 1 1 2 3 3 4 3 4 3 2 1 1 4 3

5 5 4 4 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5 4 3 2 2 2 2 3 3 2 2 2 3 2 2 2 2 1 1 1 1 3 3 4 4 3 1 4

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 t 1 1

1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Table 10. Unit numbering Area 1

Area 2

Area 3

Table 11. Unit commitment at each stage

1 4 7

2 5 8

3 6 9

Stage

Units committed

Generation model

EUE i

1 2 3 4 5 6 7

1-3 1-4 1-5 1-6 1-7 1-8 1-9

GENl GEN2 GEN3 GEN4 GEN5 GEN6 GEN 7

EUE1 EUEz EUE 3 EUE 4 EUE5 EUE6 EUE 7

f r o m the p r o b a b i l i t y o f the g e n e r a t i o n states f r o m GEN6, tie line states, a n d the load state. T h e state n u m b e r s in this set c o r r e s p o n d to those in T a b l e 12. T h e r e is only one state in this set a n d

Decompo.~rition approach for multi-area production costing: C. Singh and N. Gubbala

267

Table 12. Generation model G E N 6 for stage 6 Area 1

Area 2

Area 3

State

Cap (MW)

Prob.

Cap (MW)

Prob.

Cap (MW)

Prob.

4 3 2 1

150 100 50 0

0.681356 0.278868 0.038046 0.001730

150 100 50 0

0.681356 0.278868 0.038046 0.001730

100 50 0

0.774312 0.211276 0.014412

Table 13. System and area EUE at each stage

Table 14. Energy supplied by each unit

Stage

System (MWh)

Area 1 (MWh)

Area 2 (MWh)

Area 3 (MWh)

1 2 3 4 5 6 7

420548.75 245211.56 145482.45 63935.44 21918.90 6294.62 1590.09

140183.11 26083.44 221542.49 211346.84 3.573.72 1805.62 530.71

140183.09 107607.13 22113.95 21287.76 7593.32 1264.77 529.62

140182.56 111520.98 100726.02 21300.84 10651.86 3224.23 529.76

Unit no.

Energy supplied (MWh)

1 2 3 4 5 6 7 8 9

385416.94 385416.94 385416.94 175337.19 99729.11 81547.02 42016.54 15624.28 4704.53

its probability Pr(2) × Prob(1) x Prob(1) x Prob (Load state 1) = 0.038046 × 0.001730 x 0.014412 x 0.8 = 7.58873 x 10 -7. The E U E in arc i for any B set can be computed from the mean unserved load for each area times the probability of the set. The EUEs in the three areas for this B set are given by the vector

1X2

(0, 3.794365 x 10 -5, 3.794365 x 10 -5) Because load state 2 i,; the same as the reference load state, the set for this state is the same as set 34. The probability of this set is given by

and the EUEs for the three areas are (0, 2.445455 x 10 -2, 2.343527 × 10 -2) The B-set corresponding to load state 2 and GEN6 is the same as the original set, as the load state is the same as reference load state. As previously described, the probability is computed to be 4.687054 x 10 -4 and the E U E equals (0.0, 2.4454:55 x 10 -2, 2.343527 × 10-2). The total probability is 4.69464 x 10-4 and the EUE contribution is

4

3

Three area system

Four area system

Figure 3. Multi-area systems used in case studies Table 15. Cluster load model for three area system

0.278868 x (0.038046 + 0.001730) x 0.211276 x 0.2 = 4.687054 x 10 -4.

2

State

Area 1 (MW)

Area 2 (MW)

Area 3 (MW)

Probability

1 2 3 4 5 6 7 8 9 10

1250.0 1550.0 1750.0 1950.0 2100.0 2250.0 2400.0 2500.0 2650.0 2750.0

1250.0 1550.0 1750.0 1950.0 2100.0 2250.0 2400.0 2500.0 2650.0 2750.0

1250.0 1550.0 1750.0 1950.0 2100.0 2250.0 2400.0 2500.0 2650.0 2750.0

0.2454212 0.1951694 0.1571658 0.1224817 0.0880266 0.0864240 0.0626145 0.0301053 0.0098443 0.0027473

(0.0, 0.0244925, 0.0234732) These values are obtained by the sum of individual probabilities and EUEs for each load state. For stage 7, the same sets as in stage 6 are used but probabilities are from GEN7 in Table 2. Proceeding in a similar manner with the GENT generation model will yield a !probability of 8.4501473 x 10 -5 and EUE of (0.0, 0.00439389, 0.00422507). Indices are computed for all the sets and aggregated to obtain the area loss of load probability and EUE. These are then multiplied by t]ae duration of the study period, which in this case is 8760 h. The EUE of the system and each area after each stage are given in Table 13. The energy supplied by each unit is computed as the difference between the EUE of the system in successive stages and shown in Table 14. Note that units 1-3 are

base loaded units and the energy supplied is computed by equation (5).

V.

Case studies

Case studies are performed with three area and four area systems as shown in Figure 3. All the areas in both the systems are assumed to be IEEE-RTS systems with 32 generating units in each area. The total capacity for each area is 3405 M W and it has a peak load of 2850 MW.

V.1 Threearea system Two studies are performed on this system. The first three units in each area are assumed to be base loaded units. The units are committed based on system wide pool

268

Decomposition approach for multi-area production costing: C. Singh and N. Gubbala

Table 16. Energy supplied by various units for three area system with GSIDES approach Cap.

Energy supplied (GWH)

(MW)

Area 1

Area 2

Area 3

400 400 350 197 197 197 155 155 155 155 100 100 100 76 76 76 76 50 50 50 50 50 50 20 20 20 20 12 12 12 12 12

3083.335 3083.335 2820.607 1638.640 1423.981 1167.766 739.818 615.590 480.530 346.852 145.116 100.054 61.086 26.934 15.652 9.063 4.611 1.671 0.984 0.617 0.368 0.212 0.128 0.029 0.026 0.019 0.017 0.009 0.008 0.007 0.006 0.005

3083.335 3083.355 2820.607 1572.505 1345.922 1080.340 702.867 558.535 427.211 301.749 129.819 84.819 51.589 23.198 13.314 7.583 ).888 1.485 0.887 0.552 0.330 0.193 0.118 0.029 0.025 0.019 0.017 0.009 0.008 0.007 0.006 0.005

3083.335 3083.335 2820.607 1500.789 1245.899 1025.281 666.545 521.825 391.590 266.857 115.858 72.845 42.431 20.140 11.372 6.310 3.275 1.320 0.803 0.499 0.297 0.177 0.110 0.028 0.024 0.018 0.016 0.009 0.008 0.007 0.006 0.005

dispatch in a cyclic order. The load model for the three area system consists of 10 load clusters and is shown in Table 15. These clusters are obtained by using the hourly load data of the IEEE-RTS system and using the clustering algorithm described in Reference 15. All inter-area tie lines are assumed to have two states, 0 and 200 MW. The probability of the 0 MW state is assumed to be 0.02.

V.1.1 GSIDES approach In this case, only one decomposition is performed exhaustively on the global state space instead of 87 (total units minus number of base loaded units) in the SIDES approach and 87 × 10 in the DES approach as shown in Table 1. The results are shown in Table 16. The CPU time for this case is 605 s on a Sun Sparc Server 1000. It should be pointed out that under exhaustive decomposition, SIDES and GSIDES are equivalent.

V. 1.2 SEG- GSlDES approach In this case, ten decompositions are performed on the global state space instead of 87 in the SIDES approach and 87 x 10 in the DES approach as shown in Table 1. The results as shown in Table 17 are close to the GSIDES approach and the small variation comes from the simulation phase. The threshold probability for stopping decomposition is assumed to be 10-8 .

Table 17. Energy supplied by various units for three area system with SEG-GSIDES approach using ten decomposition phases Cap.

Energy supplied (GWH)

(MW)

Area 1

Area 2

Area 3

400 400 350 197 197 197 155 155 155 155 100 100 100 76 76 76 76 50 50 50 50 50 50 20 20 20 20 12 12 12 12 12

3083.335 3083.335 2820.607 1638.596 1424.152 1167.697 739.821 615.586 480.530 346.853 145.116 100.054 61.086 26.934 15.652 9.063 4.611 1.671 0.985 0.618 0.368 0.212 0.128 0.029 0.025 0.019 0.017 0.009 0.008 0.007 0.006 0.006

3083.335 3083.355 2820.607 1572.494 1346.006 1080.342 702.866 558.532 427.211 301.749 129.819 84.819 51.589 23.198 13.314 7.583 3.887 1.486 0.887 0.552 0.331 0.193 0.118 0.029 0.025 0.019 0.017 0.009 0.008 0.007 0.006 0.005

3083.335 3083.335 2820.607 1500.821 1245.727 1025.280 666.549 521.830 391.592 266.857 115.858 72.845 42.432 20.141 11.372 6.310 3.276 1.319 0.802 0.499 0.297 0.177 0.110 0.028 0.024 0.018 0.016 0.009 0.008 0.007 0.006 0.005

The convergence criterion for simulation from U-sets is 5% and for L-sets and W-sets is 2.5%. The results are shown in Table 17 and the CPU time for this case is 575s on a Sun Sparc Server 1000 and is very close to the GSIDES approach. However, the exhaustive decomposition may be computationally intensive for many of the multi-area systems. If exhaustive decomposition is possible with reasonable computation, then it is preferred because there is no approximation involved due to elimination of the simulation phase.

V.2 Four area system The results for the four area system are shown in Table 18. The first three units in each area are assumed to be base loaded units. The units are committed based on a system wide pool in cyclic order. The load model is similar to the three area system with an additional fourth point in every load cluster. All inter-area tie lines are assumed to be perfectly reliable with 200 MW capacity. In this case, ten decompositions are performed on the global state space instead of 116 in the SIDES approach and 116 x 10 in the DES approach as shown in Table 1. The threshold probability for stopping the

Decomposition approach for multi-area production costing. C. Singh and N. Gubbala

269

Table 18. Energy supplied by various units for four area system with S E G - G S I D E S approach using ten decomposition phases

time. Therefore more states may be used if required without significantly affecting the computation time.

Cap.

Energy supplied ( G W H )

VII.

(MW)

Area 1

Area 2

Area 3

Area 4

The work reported in this paper was supported by National Science Foundation under grant ECS-9412712.

400 400 350 197 197 197 155 155 155 155 100 100 100 76 76 76 76 50 50 50 50 50 50 20 20 20 20 12 12 12 12 12

3083.335 3083.335 2820.607 1638.710 1402.144 1183.486 746.537 612.784 474.123 343.591 146.920 96.258 57.808 25.645 15.072 8.194 4.479 1.638 1.035 0.662 0.420 0.267 0.167 0.043 0.037 0.030 0.028 0.016 0.013 0.012 0.011 0.010

3083.335 3083.335 28120.607 1638.722 13,60.225 10:~3.027 724.684 5:30.634 441.681 315.025 1.35.159 37.450 52.497 23.394 13.742 7.552 4.111 1.556 0.994 0.638 0.407 0.266 0.163 0.042 0.037 0.030 0.028 0.016 0.013 0.012 0.011 0.010

3083.335 3083.335 2820.607 1547.804 1316.640 1050.106 689.875 553.423 414.851 290.660 125.664 79.964 46.848 21.250 12.556 6.912 3.792 1.481 0.945 0.613 0.394 0.257 0.160 0.042 0.037 0.030 0.028 0.015 0.013 0.012 0.011 0.010

3083.335 3083.335 2820.607 1529.898 1240.672 1018.578 659.076 530.572 391.686 267.160 115.057 72.445 41.627 19.515 11.450 6.345 3.520 1.415 0.914 0.595 0.385 0.251 0.155 0.042 0.037 0.030 0.027 0.015 0.013 0.012 0.011 0.010

decomposition phase is assumed to be 10 -9 . The convergence criterion fi)r simulation from U-sets is 5% and for L-sets and W-sets is 2.5%. CPU time for this case is 1930 s on a Sun Sparc Server 1000.

Vl.

Conclusions

A new methodology for multi-area production costing is presented. Two approaches, one based on G S I D E S and the other on S E G - G S I D E S are discussed. The G S I D E S is suitable in cases where exhaustive decomposition can be performed. When decomposition is stopped at a predetermined threshold probability, S E G - G S I D E S is more suitable. Finally, case studies are presented for three area and four area systems, showing some comparative CPU times. These case studies assumed identical areas and tie lines. This is done only for the sake of data but is not a limitation on the methodology. The methodology is valid for any configuration and any composition of the system. Similarly although units in the examples are committed in a cyclic order, this is again done for convenience. The units can be committed in any order. An interesting feature of SIDES and the approache:~ based on it is that the number of load states does not significantly affect the computation

VIII.

Acknowledgement

References

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Decomposition approach for multi-area production costing: C. Singh and N. Gubbala

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