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18 th Power Systems Computation Conference Wroclaw, Poland – August 18-22, 2014 A Decomposition Scheme for Short Term Hydrothermal Scheduling Problems suitable for Parallel Processing Tiago Norbiato Federal University of Rio de Janeiro Rio de Janeiro, Brazil [email protected] André Diniz Electric Energy Research Center Rio de Janeiro, Brazil [email protected] Carmen Borges Federal University of Rio de Janeiro Rio de Janeiro, Brazil, [email protected] Abstract This paper presents the application of the multi-stage Benders decomposition methodology to solve the Short Term Hydrothermal Scheduling problem with a smart definition of stages. The transmission network is represented in detail by a DC-linear model with losses, where the voltage phase angles are included directly in the optimization problem. The decomposition approach allows the application of parallel processing. Two cases are pre- sented: the first one based on the IEEE 118 buses system, in order to assess the consistency of the methodology, while the second one is the actual Brazilian System, for perfor- mance analysis. The CPU time is reduced with the pro- posed methodology as compared to traditional Benders decomposition, while obtaining the same optimal solution for system operation. Keywords: Power generation dispatch, Benders de- composition, DC power flow, Parallel Processing. I. INTRODUCTION The operation planning of hydrothermal systems, usually called Hydrothermal Coordination (HTC), is a very complex optimization problem. Decisions to be made are coupled in time, as future reservoirs storages depend on the previous operation of the system. Genera- tions of hydro and thermal plants must be coordinated, not only because of system constraints such as satisfac- tion of demand and reserve, but also because of plant operation characteristics, such as hydro plants in cas- cade. In addition, uncertainties of both demand and hydrological conditions have to be managed. The HTC problem is usually solved by decomposi- tion of the original problem into long, medium and short term problems [1] each one considering the appropriate aspects for its time step and horizon of study. In general, uncertainties are modeled accurately in the long run, while system constraints are more detailed in the short term horizon. Coordination among the models can be done either by setting targets [2] or by giving economic signs [1] to the downward models, in order to guarantee a proper system optimization. The focus of this work is the Short Term Hydro- thermal Scheduling (STHTS), which in general com- prises a horizon ranging from 1 day to 1 week and may have different levels of detail in the representation of the electrical network [3], [4], [5]. Extensive bibliographical surveys of this problem can be found in [6], [7], for example. In the specific problem considered in this paper, the transmission system is represented not only by major interchanges among areas, as is in the mid-long term planning, but also by the detailed electrical net- work within each system area. In [8] an approach to consider DC-power flow in STHTS was presented, where voltage phase angles are represented directly as additional variables of the optimization problem, which is solved as an overall multi-stage Linear Program (LP). One of the most used tools to solve the STHTS prob- lem, especially in predominantly hydro systems, is a deterministic version of the multi-stage Benders decom- position (MSBD) approach proposed in [9], also known in the power systems literature as Dual Dynamic Pro- gramming (DDP). The usual definition of stages in the DDP approach is to assign one stage for each time step, although more general decompositions can be used [10]. This paper proposes the use of MSBD for the short term hydrothermal scheduling problem, with the smart definition of stages presented in [10] to solve the net- work constrained hydrothermal scheduling problem, and including transmission losses following the approach proposed in [8], where the problem is solved by a single linear program and bus phase angles in the electrical network are represented directly in the optimization problem. In the proposed methodology, the stages can be solved simultaneously by a parallel approach, in order to take advantage of parallel processing to solve the deterministic STHTS problem, what is not usual in the literature. In the parallel approach, coordination between the stages is not performed within each MSBD iteration, but between successive iterations, where a Benders cut built in a given stage and iteration is sent to previous stage in the next iteration. In a similar way, final system condi- tions for each time step and iteration is set as initial condition for the subsequent time step in the next itera- tion. Details of the algorithm, as well as the results ob- tained, will be presented in the next sections, where it will be noted that valid lower and upper bounds to as- sess convergence are obtained by the proposed algo- rithm.

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18th Power Systems Computation Conference Wroclaw, Poland – August 18-22, 2014

A Decomposition Scheme for Short Term Hydrothermal Scheduling Problems

suitable for Parallel Processing

Tiago Norbiato

Federal University of Rio de Janeiro

Rio de Janeiro, Brazil

[email protected]

André Diniz

Electric Energy Research Center

Rio de Janeiro, Brazil

[email protected]

Carmen Borges

Federal University of Rio de Janeiro

Rio de Janeiro, Brazil,

[email protected]

Abstract – This paper presents the application of the

multi-stage Benders decomposition methodology to solve

the Short Term Hydrothermal Scheduling problem with a

smart definition of stages. The transmission network is

represented in detail by a DC-linear model with losses,

where the voltage phase angles are included directly in the

optimization problem. The decomposition approach allows

the application of parallel processing. Two cases are pre-

sented: the first one based on the IEEE 118 buses system,

in order to assess the consistency of the methodology, while

the second one is the actual Brazilian System, for perfor-

mance analysis. The CPU time is reduced with the pro-

posed methodology as compared to traditional Benders

decomposition, while obtaining the same optimal solution

for system operation.

Keywords: Power generation dispatch, Benders de-

composition, DC power flow, Parallel Processing.

I. INTRODUCTION

The operation planning of hydrothermal systems, usually called Hydrothermal Coordination (HTC), is a very complex optimization problem. Decisions to be made are coupled in time, as future reservoirs storages depend on the previous operation of the system. Genera-tions of hydro and thermal plants must be coordinated, not only because of system constraints such as satisfac-tion of demand and reserve, but also because of plant operation characteristics, such as hydro plants in cas-cade. In addition, uncertainties of both demand and hydrological conditions have to be managed.

The HTC problem is usually solved by decomposi-tion of the original problem into long, medium and short term problems [1] each one considering the appropriate aspects for its time step and horizon of study. In general, uncertainties are modeled accurately in the long run, while system constraints are more detailed in the short term horizon. Coordination among the models can be done either by setting targets [2] or by giving economic signs [1] to the downward models, in order to guarantee a proper system optimization.

The focus of this work is the Short Term Hydro-thermal Scheduling (STHTS), which in general com-prises a horizon ranging from 1 day to 1 week and may have different levels of detail in the representation of the

electrical network [3], [4], [5]. Extensive bibliographical surveys of this problem can be found in [6], [7], for example. In the specific problem considered in this paper, the transmission system is represented not only by major interchanges among areas, as is in the mid-long term planning, but also by the detailed electrical net-work within each system area. In [8] an approach to consider DC-power flow in STHTS was presented, where voltage phase angles are represented directly as additional variables of the optimization problem, which is solved as an overall multi-stage Linear Program (LP).

One of the most used tools to solve the STHTS prob-lem, especially in predominantly hydro systems, is a deterministic version of the multi-stage Benders decom-position (MSBD) approach proposed in [9], also known in the power systems literature as Dual Dynamic Pro-gramming (DDP). The usual definition of stages in the DDP approach is to assign one stage for each time step, although more general decompositions can be used [10].

This paper proposes the use of MSBD for the short term hydrothermal scheduling problem, with the smart definition of stages presented in [10] to solve the net-work constrained hydrothermal scheduling problem, and including transmission losses following the approach proposed in [8], where the problem is solved by a single linear program and bus phase angles in the electrical network are represented directly in the optimization problem. In the proposed methodology, the stages can be solved simultaneously by a parallel approach, in order to take advantage of parallel processing to solve the deterministic STHTS problem, what is not usual in the literature.

In the parallel approach, coordination between the stages is not performed within each MSBD iteration, but between successive iterations, where a Benders cut built in a given stage and iteration is sent to previous stage in the next iteration. In a similar way, final system condi-tions for each time step and iteration is set as initial condition for the subsequent time step in the next itera-tion. Details of the algorithm, as well as the results ob-tained, will be presented in the next sections, where it will be noted that valid lower and upper bounds to as-sess convergence are obtained by the proposed algo-rithm.

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18th Power Systems Computation Conference Wroclaw, Poland – August 18-22, 2014

II. SHORT TERM HYDROTHERMAL SCHEDULING

In this work, the STHTS problem is formulated by a Linear Program, the hydroelectric plants are considered individually and the transmission system is represented by a linear DC model. The goal is to provide a genera-tion scheduling solution for hydro and thermal plants that minimizes total operation costs.

A. Problem Formulation The objective function (1) comprises fuel costs for

thermal generation and future costs related to hydro generation deficit, which are evaluated by a Future Cost Function (FCF) provided by a medium term model, based on the system state at the end of the week [1] . In (1) T is number of time step, nt is number of thermal plants, CTj is thermal fuel costs, GTj

t is the generation of

thermal plant and T is the future cost.

(1) TT

t

nt

j

t

jjGTCTfob

1 1

(1)

The major hydro constraints of this problem are the water balance equations (2), where Vi

t is the storage in

the reservoir, Qit is the turbine outflow, Si

t is the spil-

lage, Iit is the natural water inflow to reservoir, and i is

the set of upstream plants to plant i. Hydro generation GHi

t is modeled as a piecewise linear function of sto-

rage, turbined outflow and spillage (3), as described in [11]. Both constraints are defined for each plant i and time step t.

(2) t

i

t

i

k

t

k

t

k

t

i

t

i

t

i IVSQSQVi

1 (2)

(3) t

i

t

i

t

i

t

i

t

i SQVVfphaGH ,,, 1 (3)

Constraints (4) represent generation limits for all thermal plants.

(4) t

j

t

j GTGT , (4)

(5) where x along the text indicates an upper bound on

variable x.

B. DC Power Flow

The load balance is considered for each bus of the electrical network through a linear DC model (6), where P is the vector of bus generations, D is a vector with bus loads, B is the susceptance matrix of the electrical net-

work and is the vector of bus angles.

(6) BDP (6)

Equation (7) represents the power flow fkm from bus k to bus m (negative values in the opposite direction),

where km is the susceptance and km is the angle differ-ence between buses k and m, while equation (8) represents the power flows limits in the electrical net-work.

(7) kmkmkmf (7)

(8) kmkmkm fff (8)

C. Transmission Losses

The consideration of electrical network losses pro-vides a more realistic dispatch, especially for hydro-thermal system operation with a large transmission sys-tem. In the linear DC model of the electrical network [8], line losses are given by:

(9) 2

kmii gl (9)

where: li are the losses at line i and gi is the conductance of line i. Instead of representing the nonlinear function, we use an iterative piecewise-linear approximation pro-cedure, leading to a set of linear constraints [12]:

(10) ikmhihii nchbal ,...1,, (10)

D. Iterative approach to consider losses and flow limits

The procedure to include the transmission losses di-rectly in the optimal scheduling problem is to perform an iterative approach, as shown in Fig. 1, where the losses cuts (9) are built and included until the differenc-es between real losses and the losses calculated by the model are sufficiently close.

Any violations?

Calculate real line

loss and compare

losses obtained for

each line by model

Solve STHTS

add new linear

cut to the loss

model of each

line

add constraints

associated

with violations

Calculate the

line flows

and verifies

line

capacities

Solve next stage

Stage s

Errors within

tolerance ?

Yes

Yes

NoNo

Fig. 1: Algorithm of DC power flow with losses in STHTS

III. SOLUTION STRATEGY BY MSBD

The Multi-stage Benders decomposition (MSBD) methodology uses a time decomposition to obtain S nested sub-problems defining Future Cost Functions for every stage. However, unlike the traditional dynamic programming approaches, where an a priori discretiza-tion of state variables is required, each FCF is approx-imated iteratively by applying Benders Cuts. Each itera-tion to solve the problem by MSBD consists in:

a forward path, from stage 1 to S, where initial conditions for each stage are obtained from the so-lution of the previous stage;

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18th Power Systems Computation Conference Wroclaw, Poland – August 18-22, 2014

a backward path from stage S-1 to 1, with the same initial conditions as in the latest forward path, and adding to the constraints of sub-problem for stage s-1 a new Benders cut obtained from the solution of stage s at current iteration.

In each forward path, the upper bound Zsup for the op-timal solution of the problem is updated. Total cost for the first stage at the end of each backward path gives an updated lower bound Zinf. The solution procedure stops when the relative difference is within a certain tolerance

.

In the traditional approach, each sub-problem is composed by one time step, thus the number of stages is equal to the number of periods. Therefore, in the STHTS problem within a week-horizon hourly based, we have 168 stages. A drawback of such a large number of stages is a longer iterative process due to constraints with strong temporal coupling. An alternative approach is to aggregate in a same stage two or more time steps [10], but in this case the size of LP problem becomes larger. Therefore there is a tradeoff between the number of stages and the size of the LP subproblem for each stage. For example, Fig. 2 illustrates the case where 8 time steps are evenly divided into 4 stages. In order to apply parallel processing, we consider in this paper several stages, as will be seen in the next section.

t=1t=1 t=2t=2 t=3t=3 t=4t=4 t=5t=5 t=6t=6 t=7t=7 t=8t=8

s=1 s=2 s=3 s=4

Fig. 2: Example of multi-period stage definition for MSBD ap-

proach

IV. PARALLEL APPROACH

In order to use parallel processing in the proposed approach, we take into account the following important points:

In the problem formulated in section ii, storage in the reservoirs, which are state variables, does not vary much during the time horizon considered;

There is an exact recourse function of each stage, which consists in the future cost for a given set of state variables, and is independent of the way sys-tem states are visited during the MSBD approach;

The Benders Cuts yield a lower approximation of the resource function.

The decomposition scheme presented in section iii is adopted, where all stages can be solved simultaneously, one in each CPU.

The solution obtained in the previous iteration is used as initial condition for the constraints that are coupled in time, as for example the water balances equation (2). At each iteration, stage s builds a new Benders cut for stage

s1, and an initial condition is sent to stage s+1. We note that, although the initial conditions of the stages have been obtained from the previous iteration, all Benders Cuts are valid, since they yield a lower approx-imation of the recourse function.

Fig. 3 compares the sequence in which the stages are solved in the forward and backward paths in the tradi-tional approach described in section iii as well as in the parallel methodology proposed in this paper. In this example, the master problem was divided in four stages, the iterative process converges in two iterations in the traditional methodology and in five iterations in the proposed approach.

The red lines represent the forward paths and the sending of the initial conditions from stage s to stage s+1. The blue lines represent the backward paths and the

sending of Benders Cuts from stage s to s1.

1 2 3 4

1 2 3

2 3 4

1 2 3

2 3 4

1 2 3 4

1 2 3

2 3 4

1 2 3

2 3 4

1

1

4

4

(a) Serial processing (b) Parallel Processing

Proc. 1 Proc. 1 Proc. 2 Proc. 3 Proc. 4

Backward Path

Forward Path

Fig. 3: Comparison between serial and parallel approach

We note that, in the parallel approach, in each itera-tion a new approximation for the recourse function is obtained. However, the operation may not be feasible, because the initial condition to stage s was obtained in a previous iteration. For this reason, the value of the upper bound Zsup needs to be obtained in a sequential simula-tion, where the initial conditions, of all stages, are ob-tained in the same iteration. In the proposed approach the value of Zsup is calculated by summing the operation costs on “diagonal paths” that appear during the course of the algorithm, without the need to perform extra simulations. Therefore, in Fig. 3, the value of Zsup in iteration i, is composed by the solution of the following stages:

stage 1 in iteration i-3;

stage 2 in iteration i-2;

stage 3 in iteration i-1;

stage 4 in iteration i.

V. CASE STUDY

Two cases were used in this work in order to validate the alternative MSBD methodology proposed in this paper for the STHTS problem, with a detailed represen-

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18th Power Systems Computation Conference Wroclaw, Poland – August 18-22, 2014

tation of the electrical network. The IEEE 118 buses system was used for a consistency analysis and the real Brazilian system for an analysis of performance. These cases were processed in a cluster, composed by 42 blades, each one with 2 Intel Xeon X5355 Quad Core 2.66GHz and 16GBytes RAM, with Linux Operating System (CentOS) and used MPI (Message Passing Inter-face) for communication between processors.

A. Case I: IEEE 118 buses

The methodology was applied to the IEEE 118 buses system with 186 lines, 9 hydroelectric plants in cascade (8 storage reservoirs and 1 run-of-the river plant) and 5 thermal plants, for a 1-week horizon discretized in hour-ly time steps. TABLE I shows the basic data for the thermal plants.

TABLE I: THERMAL PLANTS DATA

Name Max Generation (MW) Cost ($/MWh)

Lc. Prestes 572.0 130.55

Juiz de Fora 237.0 150.00

Alegrete 240.0 546.40

Charqueada 162.0 135.83

Termope 382.8 70.16

The problem was decomposed in 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84 and 168 stages and the results were compared with the resolution of the problem as a single LP. In order to properly assess the consistency of the methodology, we used a very small tolerance: 10

-8

for the general convergence of the MSBD approach () and 10

-3% or 10

-3MW for the losses approximation.

TABLE II shows the CPU time, number of iterations and the optimal total cost, defined by sum of present cost, given by the thermal generation cost, and by future cost evaluated by a Future Cost Function, for all runs of the model.

TABLE II: RESULTS OF IEEE 118 BUSES SYSTEM

# stages # iter Total Cost (103$)

Single LP - 31,161,835

2 5 31,161,835

3 7 31,161,835

4 7 31,161,835

6 8 31,161,835

7 8 31,161,835

8 9 31,161,835

12 8 31,161,835

14 8 31,161,835

21 9 31,161,835

24 9 31,161,835

28 9 31,161,835

42 9 31,161,835

56 9 31,161,835

84 9 31,161,835

168 12 31,161,835

As expected, in all cases the total cost is the same, which shows that the proposed approach can correctly calculate the optimal solution. Fig. 4 shows the decrease

of CPU time when the number of stages increases. The reduction can be as large as 85% with 84 stages.

Fig. 4: Analysis of CPU time of Case I.

TABLE III compares the generation of thermal plants obtained by solving the problem either through a single LP or with 168 stages. We note that the values are equal for both cases. This same behavior occurs for the results obtained with any number of stages.

TABLE IV presents the final storage in the hydro

plants and the water value (), i.e., the reduction of the future cost due the storage in the reservoirs, obtained by both solutions. It can be seen that similar values are obtained by both approaches, what demonstrates the accuracy of the proposed approach.

TABLE III: GENERATION OF THERMAL PLANTS

Name

Generation (MWh) Cost (103$)

Single

LP S=168 Single LP S=168

Lc. Prestes 216.8 216.8 4,754.94 4,754.94

Juiz de Fora 0.0 0.00 0.00 0.00

Alegrete 0.0 0.00 0.00 0.00

Charqueada 43.7 43,7 997.21 997.21

Termope 113.1 113,1 1,342.53 1,342.53

TABLE IV: FINAL STORAGE IN THE RESERVOIR AND WATER

BENEFITS ().

Name

Vol. (hm3) ($/MWh)

Single

LP S=168

Single

LP S=168

Emborcação 10,793.5 10,793.5 91.84 91.84

A. Vermelha 4,744.7 4,744.6 63.63 63.63

Marimbondo 5,260.0 5,260.0 47.54 63.17

Segredo 388.0 388.0 55.57 75.81

Slt. Santiago 3,560.0 3,560.0 109.45 109.45

Sobradinho 24,182.6 24,182.6 88.91 88.91

Itaparica 3,273.5 3,273.5 102.66 102.66

Tucuruí 31,959.5 31,959.5 40.41 40.42

B. Case II: Brazilian System

This is a real case of the Brazilian System, with 144

hydroelectric plants and 126 thermal plants. The elec-

trical network is composed by over than 6800 lines and

4900 buses. The study horizon is discretized into 28

time steps (4 per day).

The Brazilian System is divided into 4 sub-systems

which can transfer energy among each other through

major interchange lines as show in Fig. 5. Fig. 6 shows

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18th Power Systems Computation Conference Wroclaw, Poland – August 18-22, 2014

the total load of the Brazilian system and TABLE V

presents the cost and the capacity of thermal generation

for each sub-system.

N NE

SE

S

Fig. 5: Scheme of the Brazilian subsystems

Fig. 6: Total load per period of Case II

TABLE V: CAPACITY AND COST OF THERMAL PLANTS

Sub-

System

Max.

(MW)

Cost ($/MWh)

Average Minimum Maximum

SE 8461.9 283.86 0.01 1047.38

S 1893.2 267.30 56.6 780.00

NE 3848.3 588.81 70.16 1066.18

N 318.4 574.47 574.47 574.47

The tolerance for MSBD () was set to 10-3

%, which is the value that has been used by the Independent System Operator (ISO) for the medium term operation planning of the Brazilian System. The losses approxima-tion was set to 0.5% or 0.5MW of the actual losses. TABLE VI shows the CPU time, number of iterations and the optimal cost for all runs of the model.

TABLE VI: RESULTS OF THE BRAZILIAN SYSTEM

# stages # iter Total Cost (1000$)

Single LP - 44,898,142

2 6 44,898,494

4 10 44,898,701

7 13 44,898,582

14 22 44,898,850

28 29 44,900,649

The largest difference among the values of Total Cost was 0.005%, which shows the consistency of the results of this case for all number of stages. The CPU Time was reduced in 90% when 28 stages were used, as shown in Fig. 7, what shows the good performance that can be obtained by applying the proposed approach.

Fig. 7: Analysis of CPU time of Case II

TABLE VII compares the total losses computed a posteriori (based on the DC power flow), with the losses obtained by the optimization problem through the itera-tive piecewise-linear approximation procedure of [12]. Note that 90% of the actual losses are considered in the problem with nearly 25,000 losses cuts. TABLE VIII and TABLE IX show the total generation of hydro plants and thermal plants, where it can be seen that the values are similar for all number of stages.

TABLE VII: TOTAL LOSSES CONSIDERED IN THE PROBLEM

# stages Actual losses Losses #losses cuts

Single LP 56711 50700 22,708

2 56740 50811 23,878

4 56867 50972 24,610

7 56873 51021 25,167

14 56746 50956 25,167

28 56873 51096 25,151

TABLE VIII: TOTAL GENERATION OF HYDRO PLANTS

(106MWH)

# stages SE S NE N

Single LP 5.13 1.37 1.33 0.35

2 5.13 1.38 1.33 0.35

4 5.12 1.39 1.33 0.35

7 5.11 1.40 1.33 0.35

14 5.13 1.38 1.33 0.35

28 5.12 1.38 1.33 0.35

TABLE IX: TOTAL GENERATION OF THERMAL

PLANTS(106MWH)

# stages SE S NE N

Single LP 1.19 0.28 0.26 0.00

2 1.19 0.28 0.26 0.00

4 1.19 0.28 0.26 0.00

7 1.19 0.28 0.26 0.00

14 1.19 0.28 0.26 0.00

28 1.19 0.28 0.26 0.00

Fig. 8 and Fig. 9 show the Cumulative Distribution Function of the differences of generations calculated by the single LP and by 28 and 4 stages, respectively, where only differences higher than zero are included. In both situations, 80% of the values are the same.

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18th Power Systems Computation Conference Wroclaw, Poland – August 18-22, 2014

Fig. 8: Cumulative distribution of differences of Generations be-

tween single LP and 28 stages.

Fig. 9: Cumulative distribution of differences of Generations be-

tween single LP and 4 stages.

The same cases have been solved in a parallel processing environment, and exactly the same results of the sequential processing environment have been ob-tained. TABLE X and TABLE XI show the CPU time, number of iterations and the optimal cost for some pre-liminary runs for the IEEE 118 buses and the real Bra-zilian systems. As it can be noted, the CPU time is re-duced with the parallel processing, although the effi-ciency is still low. Further improvements on the parallel implementation are being developed in order to enhance this performance.

TABLE X: RESULTS OF IEEE 118 BUSES SYSTEM -

PARALLEL PROCESSING

# stages # iter CPU Time (s) Total Cost (103$)

Single LP - - 31,161,835

2 9 14,076 31,161,835

3 15 9,253 31,161,835

4 18 5,407 31,161,835

TABLE XI: RESULTS OF EQUIVALENT OF THE BRAZILIAN

SYSTEM - PARALLEL PROCESSING

# stages # iter CPU Time (s) Total Cost (1000$)

Single LP - - 44,898,142

4 21 24,364 44,898,675

VI. CONCLUSIONS

The Short term hydrothermal scheduling problem is a very complex optimization problem, where the decisions are coupled in the time and the system is represented in details. When the detailed electrical network is consi-dered by a DC power flow with losses, it is advanta-geous to represent voltage phase angles directly as addi-tional variables of the optimization problem solved as a single Linear Program [8].

This paper proposes an alternative multi-stage bend-ers decomposition approach to solve the problem, using the model of the electrical network presented in [8]. This approach allows application of parallel processing to reduce the CPU time even for a deterministic prob-lem. The methodology was tested in two cases: the IEEE 118 buses test system, to show the consistency of the approach and a large case for the real Brazilian system, for performance analysis. The proposed approach ob-tained the same total costs for both cases with any num-ber of stages, with a reasonable reduction in CPU time. However, future work aims to improve the efficiency of the parallel processing scheme.

REFERENCES

[1] M. E. P. Maceira et al., “Chain of optimization mod-els for setting the energy dispatch and spot price in the Brazilian system,” in Proc. Power System Com-putation Conf. (PSCC’02), Sevilla, Spain, Jun. 24–28, 2002

[2] N. Tufegdzic, R. J. Frowd and W. O. Stadlin, “A coordinated approach for real-time short term hydro scheduling”, IEEE Trans. Power Syst., vol. 11, no. 4, Nov. 1996, pp. 1698-1704.

[3] X. Bai and S. M. Shahidehpour, “Hydro thermal scheduling by tabu search and decomposition me-thod,” IEEE Trans. Power Syst., vol. 11, no. 2, pp. 968-974, Aug. 1996.

[4] S. Al-Agtash, “Hydrothermal scheduling by aug-mented Lagrangian: consideration of transmission constraints and pumped-storage units,” IEEE Trans. Power Syst., vol. 16, no. 4, pp. 750-756, Nov. 2001

[5] N. Alguacil, A. J. Conejo, “Multiperiod optimal power flow using Benders decomposition,” IEEE Trans. Power Syst., vol. 15, no. 1, pp. 196-201, Feb. 2000.

[6] N. P. Padhy, “Unit commitment-a bibliographical survey”, IEEE Transactions on Power Systems, v. 19, n. 2, pp. 1196-1205, May 2004.

[7] H. Yamin, “Review on methods of generation sche-duling in electric power systems”, Electric Power Systems Research, v.69, n.2-3, pp. 227-248, May 2004.

[8] T. N. Santos and A. L. Diniz, “Alternative approach-es to consider DC-power flow with losses in a linear program for short term hydrothermal scheduling”, Transmission and Distribution: Latin America Con-ference and Exposition (T&D-LA), 2012 Sixth IEEE/PES, Montevideo, Uruguay, pag.1-6

[9] J.R. Birge, “Decomposition and partitioning me-thods for multistage stochastic linear programs”, Op-erations Research, v.33, n.5, pp. 989-1007, 1985.

[10]T. N. Santos and A. L. Diniz, “A new multi-period stage definition for the multi-stage benders decom-position approach applied to hydrothermal schedul-

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18th Power Systems Computation Conference Wroclaw, Poland – August 18-22, 2014

ing,” IEEE Transactions Power Systems, Vol. 24, No3, 2009

[11]A.L. Diniz, M.E.P. Maceira, , “A four-dimensional model of hydro generation for the short-term hydro-thermal dispatch problem considering head and spil-lage effects”, IEEE Transactions Power Systems, v. 23, n.3, pp. 1298-1308, Aug. 2008.

[12]T. N. Santos, A. L. Diniz," A Dynamic Piecewise Linear Model for DC Transmission Losses in Op-timal Scheduling Problems", IEEE Trans. Power Systems, vol. 26, no. 2, May. 2011, pp. 508-519.