Multiperiod optimal power flow using benders decomposition - Power ...

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Multiperiod Optimal Power Flow Using Benders Decomposition N. Alguacil, Student Membel; IEEE, and A. J. Conejo, Senior Membel; IEEE

Abstract-This paper addresses a multiperiod optimal power flow, properly modeling the start-up and shut-down of thermal units, the transmission network in terms of line capacity limits and line losses, and the constraints of hydroelectric plants integrated in river systems. The generalized Benders decomposition is used to solve this large-scale multiperiod problem. Spot prices are ohtained as a hypruduct of the decomposition procedure. A realistic case study based on the electric energy system of mainland Spain is analyzed and the results obtained are reported. Index Terms-Benders decomposition, multiperiod optiinnl power flow, power plant scheduling, spot pricing. 1. INTRODUCTION

T

HIS PAPER addresses a multiperiod optimal power flow (OPF), properly modeling the start-up and shut-down of thermal units, and the transmission network in terms of line capacity limits and line losses. Hydro plants integrated in river systems are also modeled in detail. The OPF [1]-151 considers a single time period (a snap shot in time) and determines the output of every on-line power plant so that a specified objective fuuctionis optimized. The transmission network is modeled in detail. Generating plants are considered either oil-line or off-line. Diverse objective functions are used depending upon the application under study. The unit commitment, or power system scheduling, problem [6]-[10] considers a multiperiod time horizon and determines the start-up and the shut-down schedules of thermal plants, as well as the production of thermal and hydro plants so as to optimize a specified objective. Thermal plauts and hydroelectric plants integrated in a river system are modeled precisely. However, the transmission network is typically not considered. This paper addresses simultaneously, the scheduling problem and the optimal power flow problem using the framework stated in [I I]. This results in a multiperiod optimal power flow. Network constraints are represented through a DC model [ 121 which incorporates losses through additional loads based on cosine approximations. The level of detail of this model is accurate enough to evaluate generator active power outputs, operating costs, active power spot prices 1131, and other economic quantities. References [9], [14]-[16] address, with different levels ofdetail, the multiperiod optimal power flow using Lagrangian relax-

ation. In the work reported in this paper, the generalized Benders decomposition [171-[19] i s used to solve this multiperiod problem. Benders decomposition is a natural way to decompose the problem because the 011 variable decisions are decoupled from continuous variable decisions. The inaster problem of this decomposition scheme fixes the start-up and shut-down schedules of thermal units, while the Benders subproblem solves a multiperiod optimal power flow. The subproblem sends to the master problem inargiiial information on the “goodness” of the proposed start-up and shut-down schedule, which allows the master problem to suggest an improved start-up and shut-down schedule. The procedure continues until some cost tolerance is met. The proposed decomposition shows good convergence properties for this application; the number of iterations required to attain convergence is typically low. Different types of operational constraints, such as preventive dispatch, can easily be incorporated into the framework of the methodology developed. The proposed procedure is intended to address short-term problems, e.g., a one day planning horizon hour by hour. However, it can also be used to address a longer time horizon, e.g., a one month time horizon using several power blocks per week to represent the evolution of the load. Readily available solution data provided by the model include power output and production cost per generator and time period, and spot price per bus and time period. The proposed technique improves currently available methodologies in the following respect: it considers the start-up and shut-down of thermal units, while simultaneously modeling precisely the transmission network, including line capacity limits and line losses. A large-scale case study based on the electric energy system of mainland Spain is analyzed and the results obtained are reported. This paper is organized as follows. The notation used throughout the paper is given in Section 11, the problem is formulated in Sectioii 111, the solution procedure is provided in Section IV, Section V presents a case study, and conclusions are stated in Section VI. The cosine approximation for line losses i s derived in the Appendix. 11. NOTATION The notation used throughout this paper is stated below Constants:

4 BnP

Start-up cost of power plant j subsceptance of line np

AI.OIIACI1. A N 0 CONBIO: MULTIPEI . Fixed COSt of power plant j constant to convert discharge volume into power Hi output of hydro plant i loss constant (conductance) of line np 1CnP spinning reserve requirement during period k R(k) RU,, resistance of line np maximum power output of thermal plant j T; minimum power output of thermal plant j 2 T last period index maximum water discharge of hydro plant i ui minimum water discharge of hvdro U; _ ” .~ d a n t i water intlow to rescrvoir i during period h. W(6,ik) - & & ( k ) )t &,(l

~

HiUi(k) iEI

PER,

- cos(hp(k) -

&(k)))

I’tlL,

= U , ( k ) :& ( k )

Vn

E N ,V k t R

(2)

I

where d,(k) are the dual variables (spot prices) associated with the above constraints. The loss constant (conductance) of line rip is

ICnp=

R,1,

(3)

+ x$

The expression for the loss term is derived in the Appendix. A spinning reserve constraint per period is written below Power output of plant j at period k . water discharge of hydro plant i at period k 011 variable which is equal to I when plant j is committed i n period k constant values of variables U; ( k ) fixcd by Benders’ master problem at iteration vu volume of reservoir i at the beginning of period k O/I variable which is equal to 1 when plant j is started-up at the beginning of period k total operating cost at iteration I / continuous variable which approximates operating costs i n Benders master problem at iteration U angle of bus n, in period k dual variables provided by the subproblem which are associated to the decisions of fixing variables ‘ u j ( k ) at constant values spot price of bus 71 at period k ,

Set of indexes of all hydro plants. set of indexes of all thermal plants set of period indexes set of indexes of all buses set of indexes of the power plants at bus 71 set of indexes of buses connected to bus 71 sct of indexes of upstream reservoirs of reservoir i set of iteration indexes.

il;llj(k)

The multiperiod optimal power flow problem is stated below. It should be noted that lower case letters represent optimization variables whereas upper case letters reprcsent constants.

Iliiii itZ

2

U,(k)+ll(k)

V k €IC.

(4)

n€N

More complex constraints to express specific spinning reserve requirements are easily incorporated. Plant power output is limited above and below as stated below

c ’ u j ( k )5 I j ( k ) 5 T ; v ; ( k )

Q jt

9,V k t IC.

(5)

Transmission capacity limits of lilies are stated below -cnp

5 nnPi6,ik) - 6n(k)) V ~ tX N , Q p t Cl,, V k E IC. 5 C,,,

(6)

Hydraulic continoity equations are formulated as follows

Q ~ E ZV ,k E K Vi€Z,Vk€K rj(O)=Xiir,zi(T+l)=Xif VitZ.

(7)

Xi 5 z i ( k ) < X i

(8)

(9)

The following two sets of equations preserve the logic of running, start-up, and shut-down (e.g., a running plant cannot be started-up) :yj(k)

111. PROBLEM FOllMULAlON

+

jELi

2 o ; ( k ) - v j ( k - I)

.;(k), U j ( k ) E {U, I}

K:

(10)

ER.

(11)

V j t 3 ,V k E V jE

9,V k

It should be noted that minimum up- and down-time constraints are not considered. However, they can be easily incorporated in the formulation as additional linear constraints.

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’

I

I

1 8 0 L M TFE SUBPROBLEM

Vj(k)

Fig. 1.

Dccornposition structure,

IV. SOLUTION PROCEDURE The multiperiod optimal power ilow problem formulated in the previous section is a large-scale mixed integer nonlinear optimization problem. This problem is solved using the generalized Benders decomposition method [17], [I 81. This technique decomposes the original problem into a 011 mixed integer linear master problem and a nonlinear subproblem. The subproblem is a multiperiod optimal power flow with the 0/1 variables fixed to given values. The master problem determines the running, start-up and shut-down schedule of the plants, For the schedule fixed by the master problem, the Benders subproblem determines total operating cost, properly enforcing transmission capacity limits and taking into account line losses. The solutions to the subproblem provide marginal information on the goodness of the scheduling decisions made at the master problem. This information enables the master problem to propose a refined running, start-up, and shut-down schedule. This iterative procedure continues until some cost tolerance is met as it is illustrated in Fig. 1 .

= V3P 1 J ( k ) : y ( k )

vj t

9 , V k E IC. (19)

The above subproblem formulation is similar to the original problem formulation (Section 111) once the 011 variables are fixed to given values. Therefore the description of equations is not repeated below. The last block of constraints, [see (19)], which enforces the running, start-up and shut-down schedule, fixed in the master problem, deserves special mention. The dual variables Xj”)(k) associated with this block of constraints provide the master problem with relevant dual information to improve the current schedule. If no hydro constraints are considered and ramp constraints are not binding, the above subproblem decomposes by time period. This decomposition facilitates its solution. If hydro and ramp constraints are to be considered, two alternative solution procedures are possible: * to solve the single nondecomposable subproblem directly, to decompose the subproblem using Lagrangian relaxation or augmented Lagrangian techniques. The first alternative is used in the work reported in this paper.

-

B. Master Problem The master problem is stated below. The objective function includes an underestimate of total operating costs in all periods (variable a ) ,fixed costs, and start-up costs.

A. Subproblem The subproblem at iteration I / is formulated below. Minimize

Minimize

+2

a(’,)

{Fjt1j(k)

+ Ajyj(k)}.

(20)

k € h jt.7

Subject to: The constraints helow are the Benders cuts. These cuts provide a lower estimate of total operating costs in the Benders suhproblem (as a function of the scheduling variables which are the variables of the master problem). An additional cut is added in every iteration

Subject to:

E j ~ j ( k 5) t j ( k ) 5 Tjvj(k)

V jE

Y,V k € t i

(14)

v u t T.

(21)

ALGUACII. A N 0 CONEJO: MULTlPWlOO OWIMAL POWER FLOW USING BENDERS DECOMPOSITION

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The next two sets of equations, as in the original problem, enforce the logic of running, start-up and shut-down of the power plants ! / j ( k ) 2 V j ( k ) - W j ( k - I)

v j ( k ) , y j ( k ) E (0, 1)

v j E 9 ,V k E x : V j E 1,V k E

(22)

1. (23)

The following equations are called feasibility cuts and force the master problem to generate solutions that satisfy the load and reserve requirements. These constraints enforce the feasibility of the subproblem

TjU3(k) + 3c.7

1

9

17

25

33

41

49

57

65

BUS

Fig. 2. Space distribution of spat prices (I US 5

-

. .. . . .,... ... ... , ... .. 73 81 89 97

150 Pta).

* losses of every line per period,

IfiTii

spot price per bus and period. The spot price at bus n in period k is the dual variable of the power balance equation of bus n in period k , once the Benders procedure has converged. See block of equations (2). *

iEZ

V. CASESTUDY

5

Dn(k)

V k EX.

(25)

ntN

It should be noted that the only real variable in the above problem is a(’),all other variables are 011 integer. This 0/1 mixed integer master problem is solved using a commercial branch and bound algorithm. No convergence problems have been experienced. If binary variables are relaxed to belong to the [O, 11 interval, the solution time decreases substantially, but many solution results do not change significantly (see Section V). C. Stopping Criterion

The iterative Benders procedure stops when the operating cost computed through the inaster problem and the operating cost computed through the subproblem are close enough that is, when the equation below holds

where f is a per unit cost tolerance. It should be noted that start-up and fixed costs are not taken into account in the stopping criterion above because they affect identically both the master problem and the subproblem. Convergence troubles are usually not encountered if the total operating costs over all periods as a function of the scheduling variables (the optimization variables of the master problem) is a nonincreasing function [20:1-[231. This requirement is met in the multiperiod optimal power flow problem because the larger the number of committed units is, the lower the total variable operating costs are over a11 periods.

D. Output Data Readily available solution data include: * on/off status of evely plant per period, * power output of every plant per period, production cost of every thermal plant per period. * voltage angle of every bus per period, * power flow of every line per period,

A case study based on the electric energy system of mainland Spain is presented. The generation system includes 71 thermal plants modeled in detail (start-up cost, shut-down cost, and quadratic operating cost), and 8 cascaded hydro plants in a river system. The transmission network consists of 104 buses and 160 lines representing the 400 kV ‘aud the 220 kV networks. A 24 h time horizon is considered. The master problem has 2208 binary variables and 1 continuous one, and approximately 1 160constraints. It is solved using CPLEX [24] under GAMS [25]. The subproblem includes 4344 continuous variables and 12 577 constraints. It is solved using MINOS 1261 under GAMS. General results are as follows. The total demand is 357.5 GWh, thermal generation is 334.4 GWh, hydro generation is 25 GWh, and losses are 1.9 GWh. It should also be noted that there is an important transmission overcapacity in this case study. Losses are therefore small. Fig. 2 shows the space (bus) distribution of spot prices at the peak load. Note that the vertical axis does not start at 0. It should be noted the relevant variation of spot prices throughout the transmission network. Fig. 3 shows the time evolution of i) the total power production and ii) the spot price time evolution in an important bus of central Spain. It should be noted the correlation among the time evolution of total generation and the time evolution of spot prices. It should also be noted that the “spike” in price that occurs for period 15 is due to the start-up of a peaker in that period. The total CPU time required to carry out this case study was about 1/2 b on an SGI Origin 2000 computer. The number of iterations required was 7, for a cost tolerance of 0.001%. If 0/1 integer variables are relaxed to belong to the real [O, 11 interval, solution rimes are reduced by a factor of approximately 2. However, total cost is changed by less than 0.08%. This is illustrated in Table 1.

v1. CONCI.USIONS This paper addresses a multiperiod optimal power flow model, properly modeling the start-up and shut-down of

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Per unit active power at the sending end is given by a

pa =

&

c~s((R,,~,) -

R,,cos(QnP+ 6, "n " p

-

6,).

Per unit active power at the receiving end is given by

Considering vn p,i.

M

up w 1, per unit losses in line n p are:

= P, - P I

1

- 7 [a cos(B,,,) ~

- 2 cos(O,,)

cos(6, - 6P)].

4 t p

Taking into account that cos(OnP) = Rnn/Z,,,, losses become:

4

3

5

B

1

1,

$3

15

17

18

21

23

And finally,

Pericd [h]

Fig. 3. Time evolution of the tottl power production and the spat pricc ill an important bus of central Spain.

ACKNOWLEDGMENT

TABLE 1 EFFECT OP RELAXING011 VARIABLES TO BELONGTO THE 10.11 INTERVAL

The authors would like to thank Prof. J. L. de la Fuente for support and help.

REFERENCES [ I ] H. W. Dommel and W. E Tinney, "Optimal power flow solutions," IEEE Tran,wzctionron Power Apparatus and Sysremr, vol. PAS-87, pp.

Time Is1

Master

I

9864-

thermal units, the constraints related to hydroelectric plants of river systems, and the transmission network in terms oi line capacity limits and line losses. The generalized Benders decomposition method is used to solve this large-scale multiperiod problem. This method shows good convergence ~. properties for this application. The methodology _.developed improves currently available approaches in the following respect: the start-up and shutdown of thermal units are considered simultaneously with a precise model of the transmission network including line capacity limits and line losses. For cost and energy calculations, the O/1 variables of the master problem call be relaxed to belong to the real interval [O, I ] without significant loss of accaracy. This results in an important decrease in computing time. A large-scale case study based on the electric energy system of mainland Spain is analyzed and the results obtained are reported. ~~

APPENDIX

A

COSINE LOSS APPROXIMATION Line n p is modeled using a per unit series impedance denoted by Z,LpLO,P or by RnP+jX,,. Per unit voltage at the sending end is denoted by vrl L6,, and per unit voltage at the receiving end by wpLbP.

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ALGUACIL AND CONEIO: MULTIPBRIOD OPTIMAL POWER FLOW USING BBNDFRS DECOMI'OSITION

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N. Algiiacil (S'97) received the Ingcnielo en lnfarmatica degree from thc U n versirlad de Malaga, Malaga, Spain, in 1995. She is currently working toward thePh.D. d e g m i n power system operations planning. Herresearch interests it)clude operalions, planning, and ecoiioinics of electrical energy systems, as well 21s optimimtian and parallel computation,

A. J. Concjo (S'86-M9I-SM'98) receivcd the B.S. dcgree from the Universidird P. Cumillas, Madrid, Spain, in 1983, the M.S. degreee from MIT, Cambridge, MA, in 1987, and the PI1.D. degree from the Royal Institute of Technology, Stockholm, Sweden, in 1990, all in electrical engineering. He was II visiting Engineer a1 MIT, Cambridge, MA, and a visiting Lcctwer at the Royal lnstitutc of Technology, Stockholm, Sweden. He is currently aProfeessuraf Electrical Engincciing at the Universidad de Castilla, La Msncha, Ciudad Real, Spain. His research inlercsts include cmtml, operations, planning, and economics of electric energy systems, as well as optimization theory and its applications.