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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


    Andr Diniz

    Electric Energy Research Center

    Rio de Janeiro, Brazil


    Carmen Borges

    Federal University of Rio de Janeiro

    Rio de Janeiro, Brazil,


    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.


    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.

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


    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






    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) titik










    i IVSQSQVi

    1 (2)

    (3) tititititi SQVVfphaGH ,,, 1 (3) Constraints (4) represent generation limits for all

    thermal plants.

    (4) 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


    add constraints


    with violations

    Calculate the

    line flows

    and verifies



    Solve next stage

    Stage s

    Errors within

    tolerance ?




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


    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