Electrical Power and Energy Systems 44 (2013) 7–16

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Electrical Power and Energy Systems

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Hydro unit commitment and loading problem for day-ahead operationplanning problem

Erlon Cristian Finardi, Murilo Reolon Scuzziato ⇑LabPlan/UFSC, Electrical Systems Planning Research Laboratory, Federal University of Santa Catarina, Campus Universitário, CP 476, CEP 88040-900, Florianópolis, SC, Brazil

a r t i c l e i n f o

Article history:Received 31 October 2011Received in revised form 18 June 2012Accepted 5 July 2012Available online 9 August 2012

Keywords:Hydro unit commitment and loadingproblemHydropower functionLagrangian relaxationInexact Augmented Lagrangian

0142-0615/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijepes.2012.07.023

⇑ Corresponding author. Tel.: +55 48 3721 9731; faE-mail address: muriloscz@gmail.com (M. Reolon

1 The Brazilian ISO manages a mix of approximately170 plants, which are dispatched in 32 cascades. Atinstalled capacity in Brazil was nearly 85,000 MW.

a b s t r a c t

We describe a new model for the hydro unit commitment and loading (HUCL) problem that has beendeveloped to be used as a support tool for day-ahead operation in the Brazilian system. The objectiveis to determine the optimal unit commitment and generation schedules for cascaded plants with multipleunits and a head-dependent hydropower model. In this paper, we propose a new mathematical model forthe hydropower function where the mechanical and electrical losses in the turbine-generator areincluded. We model the HUCL problem as a nonlinear mixed 0–1 programming problem and solve it witha strategy that includes a two-phase approach based on dual decomposition. The computational toolallows the model to effectively schedule hydro units for the problem in the Brazilian regulatory frame-work. Application of the approach is demonstrated by determining a 24-time step HUCL schedule for fourcascaded plants with 4170 MW of installed capacity.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Brazil has a modern electricity industry that depends heavily onhydropower. Its power system has the largest capacity for waterstorage in the world and one of largest transmission networks, gi-ven the long distances between hydro plants and consumers. Inthe Brazilian regulatory framework, an Independent System Opera-tor (ISO) performs the system operation scheduling, and it conductsstudies extending from the long/medium-term to short-term prob-lems. This hierarchical approach is basically divided into three com-putational levels: (i) long-term planning, which covers ten yearswith monthly time-steps [1,2]; (ii) medium-term planning over atwo-month horizon with weekly time-steps [3]; and (iii) short-term optimization, which produces plant schedules for a day ahead[4–6]. In step (iii), the ISO performs every day a week-ahead studyto produce a day-ahead schedule for all plants to meet the systemload at minimum cost. However, the huge number of reservoirs pre-cludes the ISO from precisely taking into account the complex mod-elling associated with the hydro units.1 More specifically, in step(iii), the hydro units are modelled by a piecewise linear function,and the hydro unit commitment constraints are not taken into ac-count [7]. Therefore, an intrinsic nonlinear mixed-discrete modelling[8] is replaced by a linear continuous one.

ll rights reserved.

x: +55 48 3721 7538.Scuzziato).650 hydro units distributed in

the end of 2010, the hydro

As a result of the centralized dispatch aforementioned, each hy-dro power plant receives an hourly generation target for the day-ahead. As a consequence, the generation distribution among thegenerating units in the cascaded is a local decision. This task is de-fined in this paper as the hydro unit commitment and loading(HUCL) problem, which searches for the most economical sched-ules. In the Brazilian case, the most economical schedule is relatedto the efficient use of water and to minimize startups and shut-downs of hydro generating units as well. If any unit is committedto generate at an inefficient operating point, there is waste ofwater, which could be saved by more efficient generation. This sit-uation is particularly critical during drought periods. For instance,in Brazil, in the last energy rationing, in 2001, the regulatoryauthority launched a project aiming to incentive hydro plants toproduce more with less water. Furthermore, it is worth pointingout that all generating units need to comply with technical param-eters of efficiency declared to the regulatory authority, which isempowered to audit the performance of such generators.

From the point of view of hydro conversion efficiency, it is nec-essary to simulate all possible scenarios with respect the numberof hydro generating units in operation throughout the day in an at-tempt to find a better combination of units depending on the var-iation of the load. However, frequent unit switches reduce servicelife and increase maintenance costs [9]. Thus, the HUCL problem isformulated in a way that will minimize the startup/shutdown costsof the generating units in addition to the discharged outflow.

In real-life cascaded head-dependent hydro plants, a large num-ber of variables are involved in this problem and most of them areinteger, which are related to startup and shutdown decisions. Thus,

Fig. 1. Schematic diagram of a typical hydro generating unit.

8 E. Cristian Finardi, M. Reolon Scuzziato / Electrical Power and Energy Systems 44 (2013) 7–16

the optimization problem exhibits highly nonlinear, non-convex,and large-scale characteristics over mixed integer variables. Addi-tional complexity arises from the need to solve the HUCL problemover hourly time intervals. Operators must have access to informa-tion on which units should be operated when emergencies arise orhow to schedule around planned maintenance. Computer execu-tion time of the optimization model should allow rapid processingof schedule modifications so that plant operators do not experiencesignificant delays when receiving unit commitment schedulingupdates.

In this scenario, improvements in modelling are an importantissue because studies show that significant gains are obtained withan efficient generation allocation among hydro units [10]. Becausethe hydro plants possess different characteristics (diverse kinds ofconduits, turbines, generators), an important challenge in theHUCL problem is to represent a precise model that takes into ac-count these peculiar characteristics; in this case, the most impor-tant modelling issue is related to hydropower function.2

There are different approaches presented in the literatureregarding the hydropower function. For example, Yang et al. [11]model this characteristic in terms of discharge and reservoir vol-ume. Li et al. [12] represent hydropower as a piecewise quadraticfunction of the discharge, while Guan et al. [13] adopt a single qua-dratic function. Nilsson and Sjelvgren [14] represent this model interms of most efficient points, while Conejo et al. [15] model it bymeans of piecewise linear functions. Diniz [16] and Borghetti [17]present models that take into account the head effects on powerproduction and use a linearization technique for represent thehydropower function in the problem. In [18], Catalão et al. use aquadratic function to consider the hydropower model as a functionof discharge and the head.

Despite the aforementioned differences, all of these papers andmost works addressing the optimal HUCL and similar problems ne-glect (or represent in a simplified manner) the influence of headvariations on the power production function, hydraulic lossesand, in particular, hydraulic efficiency and mechanical losses inthe turbine, as well as mechanical and electrical losses in the gen-erator. An accurate model of these parameters plays an importantrole in the HUCL problem, in particular for the Brazilian hydro-dominated system, where every tenth of a percentage increase inthe energy conversion efficiency is welcome.

In this paper, we initially propose a new mathematical modelfor the hydropower, which considers hydraulic losses in the con-duits and in the turbine suction tube, nonlinear forebay and tail-race functions, hydraulic efficiency and mechanical losses in theturbine, and mechanical and electrical losses in the generator.We do not know of a detailed model such as that proposed herein the literature, and we will show that some simplifications ofthese issues affect the hydro plants turbine outflow and, conse-quently, the system storage. In the Brazilian case, the reservoir vol-umes are modelled as state variables in the long/medium-termoptimization models, and thus, a detailed hydro production func-tion is important for using the generation resources efficiently overtime.

Mathematically, the HUCL problem is formulated to minimize,in each plant and time stage, the discharged outflow necessary tomeet the hourly generation target defined by the ISO. As will beshown, the HUCL is modelled by means of a mixed 0–1 NonlinearProgramming (NP) problem, and, to solve it efficiently, we proposea two-phase decomposition approach.

The method applied in this paper is similar to [19]. Firstly, weapply the Lagrangian Relaxation (LR) [20,21] to obtain a primal

2 The hydropower function has been referred to by different names in theliterature, such as water rate curves, characteristic curves and hydro productionfunctions.

point, which solves a certain convexified form of the HUCL prob-lem. Subsequently, we use this solution as a starting point in aninexact Augmented Lagrangian (AL) [22] to obtain a primal feasible(or near-feasible) solution. Although this two-phase approach isnot new, the results section shows how modelling simplificationscan be used to improve the performance of the overall solutionstrategy, which is another contribution of this paper and can be amotivation for use in similar problems that require dual decompo-sition based in LR and/or AL algorithms.

There are different approaches when LR is used to solve a prob-lem, especially regarding with the primal recovery phase. Forexample, in [23–25] Lagrangian heuristics are used to obtain a fea-sible solution from LR results. On the other hand, in [13] a nonlin-ear network flow algorithm is used to obtain a feasible solution,sometimes in combination with a heuristic method. In turn, in[26] integer programming is used to refine Lagrangian-based solu-tions to find a feasible or near-feasible primal solution. We havechosen inexact AL in the primal recovery phase because of the fol-lowing characteristics: (i) the structure of the dual problem is ex-actly the same of the LR phase; (ii) the same algorithms developedto the LR phase are applicable in the primal recovery phase, and(iii) the dual problem associated with the AL is differentiable,which allows us to use techniques with low computational cost,such as the gradient method.

This paper is organized as follows: in Section 2, we present thenew model for the hydropower function; the optimal HUCL prob-lem formulation is shown in Section 3; in Section 4, we focus onthe solution strategy of HUCL problem; and, in Section 5, the testresults are provided. Finally, in the last section, we present themain conclusions.

2. The hydropower function

Initially, consider Fig. 1.The power variables (MW) shown in Fig. 1 include the

following:

hp available hydraulic power, where G3 is a constant, q is theunit discharge (m3/s), and gh is the gross head (m):

3 Thiswater dGRAV =consequ

hp ¼ G � gh � q; ð1Þ

rp mechanical power transmitted through the coupling of therunner and the turbine shaft:

rp ¼ G � g � nh � q; ð2Þ

where g is the turbine hydraulic efficiency and nh is the net head(m), i.e., the part of the gh that is available for the turbine;

sp mechanical power delivered by the turbine shaft, assigningto the hydraulic machine the mechanical losses of the relevantbearings and shaft seals [27];gp power output at the generator terminals.

constant depends on the local value of gravitational acceleration (GRAV),ensity (RHOW) and the unit system considered. We consider in this paper9.7904 m/s2, RHOW = 997 kg/m3 and MW as the power unit. As aence, G = 10�6 GRAV RHOW = 9.76102�10�3 kg/(m2s2).

E. Cristian Finardi, M. Reolon Scuzziato / Electrical Power and Energy Systems 44 (2013) 7–16 9

Because the hydropower function depends on state and controlvariables, we describe the heads, losses, and efficiencies as a func-tion of these variables next.

2.1. Gross head

The gross head gh is given by the difference between the fore-bay, fbl(�), and tailrace, trl(�), levels:

gh ¼ fblðvÞ � trlðQ ; sÞ; ð3Þ

where v is the reservoir volume (hm3), Q is the plant dischargedoutflow (m3/s), and s is the spillage (m3/s).

2.2. Net head

In this paper, the net head is modelled as:

nh ¼ gh� ðk0q2 þ k1q2Þ: ð4Þ

The term between parentheses is the total head loss (m), wherek0 and k1 are constants (s2/m5). The function k0q2 is the head lossassociated with the penstock water friction, and k1q2 is the loss re-lated to the hydraulic energy dissipated between the tailrace leveland the low pressure reference section of the turbine.

2.3. Turbine hydraulic efficiency

The turbine hydraulic efficiency represents the transfer effec-tiveness to the runner of the available power in water that flowsthrough it [27]. It depends on nh and rp and is furnished by the tur-bine manufacturer as a set of triples (g, rp, nh). We use the follow-ing mathematical model to represent the hydraulic efficiency:

g ¼ c0 þ c1 � qþ c2 � nhþ c3 � nh � qþ c4 � q2 þ c5 � nh2; ð5Þ

where c0, . . . ,c5 are efficiency coefficients. To estimate these coeffi-cients, we take the set (g, rp, nh) and use (2) to compute q; as a result,we can generate a new set of triples4 (g, q, nh). The coefficients of (5)are determined with a regression technique applied in the set (g, q, nh).

2.4. Turbine mechanical losses

The turbine mechanical efficiency is the ratio of the poweravailable at the turbine shaft to the power transmitted from thewater to the runner [27], that is,

sprp¼ spðspþ tmlÞ : ð6Þ

Here, tml is the turbine mechanical loss (MW) associated withthe power consumed by the mechanical friction in the guide bear-ings, thrust bearing and shaft seals. By means of unit field tests[28], a set of points (tml, gp) can be obtained, and we can adjusta polynomial function, where b0, b1, and b2 are constants:

tml ¼ b0 þ b1 � gpþ b2 � gp2: ð7Þ

2.5. Generator global losses

The generator efficiency is defined as the ratio of the poweravailable at the generator terminals to the mechanical power deliv-ered by the turbine shaft, that is:

gpsp¼ gpðgpþ gglÞ : ð8Þ

4 In some cases, the turbine manufacturer also supplies (g, q, nh), and thecomputation of q is not necessary.

In (8), ggl is the global losses, i.e., the mechanical and electriclosses in the generator. As the turbine mechanical losses, by meansof field tests, a set of points (ggl, gp) can be obtained, and we adjustthe following exponential function, where a0 and a1 are constants:

ggl ¼ a0 � ea1 �gp: ð9Þ

2.6. Hydropower function

Observing Fig. 1, it is easy to conclude that:

gp� rpþ tmlþ ggl ¼ 0: ð10Þ

As a consequence, the hydropower function is modelled as anonlinear equality constraint instead of a single function, as oftenshown in the literature.

The problem variables are q, s, v, Q and gp. Thus, it is importantto express (10) in terms of these variables because they must beused in the optimization problem formulation. First, notice in (3)that gh = f(v,Q,s) and nh = f(gh,q) = f(v,Q,s,q). Second, by (5),g = f(nh,q) = f(gh,q) = f(v,Q,s,q). Finally, considering (2), (7) and(9), the nonlinear equality constraint related to the hydropowerfunction proposed in this paper has the following mathematicalstructure:

gp� G � gðv ;Q ; s; qÞ � nhðv ;Q ; s; qÞ � qþ ðb0 þ b1 � gpþ b2 � gp2Þþ ða0 � ea1 �gpÞ ¼ 0: ð11Þ

In addition to (11), it is necessary to take into account the unitoperating characteristics such as forbidden zones,5 generator out-put and discharged outflow capacities. An illustration of these char-acteristics is given in Fig. 2 (Hill-Diagram), where we can also see thecomplex behaviour associated with g, rp, nh, and q. In Fig. 2, for nethead values smaller than the so-called nominal value (41 m), theturbine is unable to make the generator attain its nominal power(120 MW). This variable maximum discharged outflow must be in-cluded in the model as a function of the net head. On the other hand,for values higher than 41 m, there is a power limit imposed by thegenerator capabilities because the turbine could effectively reachpower levels beyond 120 MW. As a consequence, it is necessary toinclude this limit on generator output. Finally, Fig. 2 shows a forbid-den operation zone ranging from 80 to 90 MW.

In the next section, we show mathematically how these operat-ing issues are considered in the HUCL problem.

3. The problem formulation

The HUCL optimization problem related to this paper is de-scribed below. The main result is a set with the commitment stateof each generating unit and generation level over a planninghorizon.

minH ¼XT

t¼1

XR

r¼1

Qrt þXnrt

i¼1

qir½uirtð1� uir;t�1Þ þ uir;t�1ð1� uirtÞ�( )

;

ð12Þ

s:t: :Xnrt

i¼1

gpirt ¼ Lrt ; ð13Þ

gpirt � rpðv rt ;Q rt; qirt; srtÞ þ tmlirtðgpirtÞ þ gglirtðgpirtÞ ¼ 0; ð14Þ

5 A forbidden zone is a useful concept to describe the dynamic transitions andoperating characteristics of a hydro unit precisely. These zones result from mechan-ical vibrations in the turbine shaft (which cause an oscillation in the power output),cavitation phenomena and low efficiency levels [29].

Fig. 2. Illustrative hill diagram.

6 These limits depend on the net head and consequently the limits depend on theway of the conveying the water to the turbines. As described in Section 2.6, the nethead depends on the variables vrt, Qrt, srt and qirt.

10 E. Cristian Finardi, M. Reolon Scuzziato / Electrical Power and Energy Systems 44 (2013) 7–16

mrt � mr;t�1 þ c1 Q rt þ srt �X

m2Rupr

ðQm;t�smr þ sm;t�smr Þ

24

35 ¼ c1yrt ; ð15Þ

mminr 6 mrt 6 mmax

r ; ð16Þ

Xnrt

i¼1

qirt � Qrt ¼ 0; ð17Þ

uirt � qminir ðmrt ;Q rt; srt; qirtÞ 6 qirt 6 uirt � qmax

ir ðmrt;Q rt; srt ; qirtÞ; ð18Þ

XUir

k¼1

zirtk � gpminirtk 6 gpirt 6

XUir

k¼1

zirtk � gpmaxirtk ; ð19Þ

uirt ¼XUir

k¼1

zirtk;XUir

k¼1

zirtk;� 1; zir0k ¼ zirk; zir0k and zirtk 2 f0;1g ð20Þ

where T is the number of stages (h); t the index of stages, so that t=1,T; R the number of hydro plants; r the index of reservoirs, so thatr = 1,R; c1 the conversion factor of (m3/s) in (hm3); Lrt the plant loadrequirement of plant r and stage t (MW); nrt the number of units inoperation in the plant r and stage t; i the hydro unit index, so thati = 1, nrt; vr

min(max) the minimum (max.) volume of the reservoir r(hm3); m the index of reservoir upstream of reservoir r; smr thewater travelling time between hydro plants m and r (h); r

up theset of reservoirs immediately upstream of reservoir r; Uir the num-ber of non-forbidden zones of the unit i and plant r, so that k = 1,Uir; gpikrt

min(max) the minimum (max.) power of unit i, zone k, plantr and stage t (MW); qir the startup/shutdown penalty of unit i andplant r; vrt the volume of the reservoir r and stage t (hm3); Qrt thedischarged outflow in the reservoir r and stage t (m3/s); srt the spill-age of the reservoir r and stage t (m3/s); yrt the incremental inflowin the reservoir r and stage t (m3/s); qirt the discharged outflow ofthe unit i, reservoir r and stage t (m3/s); gpirt the electric power ofthe unit i, reservoir r and stage t (MW); uit the binary variable thatindicates if unit i is operating (uit = 1) or not (uit = 0) during stage t;zirtk the binary variable that indicates if the unit i of the reservoir r is

operating (zikrt = 1) or not (zikrt = 0) in the non-forbidden zone k dur-ing stage t; zirk the initial condition of unit i, reservoir r and zone k;and qir

min(max)(�) is the minimum (max.) discharge of the unit i andplant r as a function of the net head.

By means of the objective function (12), the model seeks tominimize the total discharged outflow necessary to meet thehourly generation targets defined by the ISO over the planninghorizon. Additionally, when necessary, the model must minimizethe number of startups and shutdowns of the units, which is animportant aspect of preventing fast degradation of the units. Weare proposing a general formulation, where it is possible to attri-bute a penalty for each startup and each shutdown of a unit indi-vidually. In practical terms, the company generation can use realstartup/shutdown costs (which can be different among the unitsin the same plant [9]) or adjust switch penalties to analyse howthe overall hydro efficiency is affected as a function of the numberof unit switches.

The constraint set (13) is the plant load requirement. Each con-straint (14) is the hydropower function. Constraint (15) representsthe stream-flow balance equations so that the change in volume ofa reservoir is equal to the sum of the total outflow from upstreamreservoirs (considering water travelling time) and incremental in-flow minus the total outflow to downstream reservoirs. Constraint(16) describes the volume limits, (17) represents the penstockwater balance in each reservoir, and (18) expresses the units dis-charged outflow limits.6 Constraint (19) is related to the power lim-its for each non-forbidden operating zone. Finally, Eq. (20) representthe integrality constraints and the initial operating condition of theunits.

4. Solution strategy

The solution strategy was chosen to yield accurate and practicalresults, and, hence, it can be used to assist operators in the gener-

E. Cristian Finardi, M. Reolon Scuzziato / Electrical Power and Energy Systems 44 (2013) 7–16 11

ation allocation. We propose a two-phase decomposition approach,where, in the first phase, we apply a Lagrangian Relaxation (LR) toobtain a dual (lower) bound of the HUCL optimal solution and, inthe second phase, we use an inexact Augmented Lagrangian (AL)to find a feasible solution. This second phase is necessary becausethe LR method usually yields an infeasible primal solution due todifference between the optimal values of the primal and dualobjective functions. When solving non-convex problems, as is thecase with the HUCL problem, the inexact AL method may obtaina local optimizer or even any feasible solution [30,31] withoutmeasure of its quality. In this case, the lower bound given by theLR method (Phase I) can be used as a quality measure of an HUCLsolution supplied by the AL. In this scenario, it is important to startthe inexact AL with the primal and dual solutions found in the LRlast iteration because the inexact AL is a local search approach,which is strongly dependent on the initial point. In [32], bothmethods were compared computationally, and in general, AL wasslower than LR but provided feasible solutions. Thus, we believethat this two-phase approach is advantageous, and we see the LRmethod as an approach to be used together with the AL methodwithin a two-phase framework.

To build the LR dual function, a new set of ‘‘copy’’ constraints isintroduced in the original problem. This copy strategy is known inthe literature as Lagrangian Relaxation with Variable Splitting(LRVS) and has several advantages over the classical LR. For exam-ple, every constraint in the original problem appears in one of thesubproblems, avoiding the need to choose between the various setsof structured constraints. Additionally, the bounds from LRVS canbe tighter than those for classical LR. The solution strategy is fur-ther detailed in the next two sections. As it will be shown below,the basic idea of the employed decomposition is to separate thetime and space linking constraints (15) from the nonlinear mixed0–1 constraints so that the primal problem can be reduced to rel-atively easier subproblems that can be solved by existing solutionmethods. This basic strategy is commonly suggested in nonlinearmixed-integer programming, as described in [33].

4.1. Phase I: decomposition by Lagrangian relaxation

In the LRVS context, we initially include the artificial variablesvart, Qart, and sart in (12), (15), and (16) in the following way:

min H ¼XT

t¼1

XR

r¼1

Qart þXnrt

i¼1

qir½uirtð1� uir;t�1Þ þ uir;t�1ð1� uirtÞ�( )

;

ð21Þ

s:t: : vart � var;t�1

þ c1 Qart þ sart �X

m2Rupr

ðQam;t�smrþ sam;t�smr Þ

24

35

¼ c1yrt; vminr 6 vart 6 vmax

r ; ð22Þ

ð13Þ; ð14Þ; ð17Þ—ð20Þ; ð23Þ

v rt ¼ vart;Qrt ¼ Qart ; srt ¼ sart: ð24Þ

Dualising the artificial constraints (24) above, by means of the La-grange multipliers krt, prt and brt, leads to:

U0 ¼ minH�XT

t¼1

XR

r¼1

½krtðv rt � vartÞ þ prtðQ rt � QartÞ þ brtðsrt � sartÞ�

s:t: : ð22Þ; ð23Þ:ð25Þ

The dual function above can be evaluated by means of:

U1 ¼ minXT

t¼1

XR

r¼1

ðkrtvart þ prtQart þ brtsartÞ

s:t: : ð22Þ;ð26Þ

and

U2 ¼minXT

t¼1

XR

r¼1

Xnrt

i¼1

qir ½uirtð1� uir;t�1Þ þ uir;t�1ð1� uirtÞ�(

� krtv rt � prtQrt � brtsrt

)s:t: : ð13Þ; ð17Þ � ð20Þ:

ð27Þ

Subproblem (26) is a set of Linear Programming (LP) problems,where each LP represents an individual cascade. On the other hand,Subproblem (27) can be solved by R uncoupled mixed 0–1 NPproblems, each one with variables and constraints related to hydroplant r. We implement a specific Dynamic Programming approach,which exploits the feasible set characteristics to decrease the statespace associated with the binary variables. In general terms, theplant load requirement constraint reduces the number of feasibleunits’ combinations in each stage substantially, and a highly struc-tured graph can be optimized in a backward recursion.

To sum up our LRVS description, we implemented a Bundlealgorithm [34] to update the Lagrange multipliers to find a dualoptimum efficiently. Today, Bundle methods are the most promis-ing methods in nonsmooth optimization. Their origin is the classi-cal cutting plane method, and they are based on the piecewiselinear approximation of the dual function U0. They are also calleddiagonal variable metric methods because a stabilizing quadraticterm is added to accumulate some second order information aboutthe curvature of U0. Given that U0 is concave, the model approxi-mation construct by the Bundle method U0 is an upper estimativeof U0, and a nonnegative linearization relative error (U0 �U0)/U0

measures how good the approximation is in relation to the originalproblem U0. Thus, this error can be used as a stopping criterion; formore details, see [34] and references therein.

4.2. Phase II: primal recovery by Augmented Lagrangian

Due to the non-convexities in the HUCL problem, solving thedual problem by LRVS can only be viewed as a first step towardsolving (12)-(20). It is therefore followed by a second phase thataims to find schedules while realizing a good compromise betweenminimizing the objective function and satisfying the relaxed artifi-cial constraints. The method proposed is based on AL, where a qua-dratic stabilization term is added to the LRVS dual function.Because the quadratic term destroys the decomposability property,a ‘‘partial linearization’’ as employed by the Auxiliary ProblemPrinciple (APP) concept [22] is applied. The AL dual function hasthe same formulation as (25) except that it is necessary to includequadratic terms as follows:

U3¼minH�XT

t¼1

XR

r¼1

½crtðv rt�vartÞþjrtðQrt�QartÞþxrtðsrt� sartÞ

þ 12lðv rt�vartÞ2þ

12lðQrt�QartÞ2þ

12lðsrt� sartÞ2�

s:t: : ð22Þ;ð23Þ:ð28Þ

Above, l > 0 is the penalty parameter, and crt, jrt, and xrt arethe Lagrange multipliers associated with the AL function. Toachieve the same separability obtained by LRVS, the APP approxi-mates each quadratic term of (28) as follows. Suppose that, atiteration it, we have calculated vait

rt , v itrt , Qait

rt , Qitrt , sait

rt and sitrt . Using

the APP at iteration it + 1, we approximate:

Fig. 3. Schematic diagram of the cascaded system.

Table 1Reservoir data.

Plant v0 (hm3) vmin/max (hm3) Nominal net head (m) y (m3/s)

H1 1400 1320/1477 182 559H2 3800 2711/4905 152 463H3 2800 2283/3348 108 475H4 4700 4300/5100 100/105 669

8 The fbl and trl are modeled as a0 + a1 �v + a2 �v2 + a3 �v3 + a4 �v4 andb0 + b1�(Q + s) + b2�(Q + s)2 + b3�(Q + s)3 + b4�(Q + s)4, respectively.

9 The maximum unit discharged outflow is given by q0 + q1�nh + q2�nh2 + q3�nh3.The minimum value is zero.

10 Typically, a Bundle method defines iterates by maximizing a classical cutting-plane model of the LR dual function. In order to prevent the well-known cutting-planeoscillations, a stabilization term is added to the model function to cause the nextiterate to be near a ‘‘good’’ point, the last serious iterate in the bundle terminology. In

12 E. Cristian Finardi, M. Reolon Scuzziato / Electrical Power and Energy Systems 44 (2013) 7–16

ðvart � v rtÞ2 ¼ vart �ðv it

rt þ vaitrtÞ

2

� �2

þ v rt �ðv it

rt þ vaitrtÞ

2

� �2

; ð29Þ

and the other quadratic terms in (28). Using this approximation, thedual function (28) can be evaluated by:

U4 ¼ minXT

t¼1

XR

r¼1

fðcrtvart þ jrtQart þxrtsartÞ

þ 12l

vart �ðv it

rt þ vaitrtÞ

2

� �2

þ 12l

Qart �ðQ it

rt þ QaitrtÞ

2

" #2

þ 12l½sart �

ðsitrt þ sait

rtÞ2

�2�

s:t: : ð22Þ;

ð30Þ

and

U2 ¼minXT

t¼1

XR

r¼1

Xnrt

i¼1

qir ½uirtð1�uir;t�1Þþuir;t�1ð1�uirtÞ�(

�crtv rt�jrtQrt�xrtsrtþ1

2l½vrt�

ðv itrtþvait

rtÞ2

�2

þ 12l½Q rt�

ðQitrtþQait

rtÞ2

�2þ 12l½srt�

ðsitrtþ sait

rtÞ2

�2�

s:t: : ð13Þ;ð14Þ;ð17Þ�ð20Þ:

ð31Þ

The iterative process in the inexact AL can be summarized asfollows. Initially, in the first AL iteration, (i.e., it = 0),va0

rt ;v0rt ;Qa0

rt ;Q0rt; sa0

rt and s0rt must be equal to the respective values

obtained in the LR last iteration. In the same way, at iteration it = 0,we must set crt = krt, jrt = prt and xrt = brt.

To solve the dual problem, we must decrease l over the itera-tions to force the primal feasibility. At the same time, it is neces-sary to maximize the dual function, and we use an inexactgradient-like method as follows:

cnþ1 ¼ cn þu � vnrt � van

rt

kvnrt � van

rtk1;jnþ1

¼ jn þu � Q nrt � Qan

rt

kQnrt � Qan

rtk1;xnþ1 ¼ xn þu � sn

rt � sanrt

ksnrt � san

rtk1: ð32Þ

where u is the step size and n is the AL phase iteration index. Thealgorithm stopping criteria is defined by an absolute error tolerancee between each original variable and its associated duplicated var-iable; for more details, see [35] and references therein.

5. Computation results

The computational model was implemented in LabVIEW 9.0, andthe tests were executed in an Intel Quadcore i7 2.80 GHz. Initially,we describe important plant data, whose cascade configuration ispresented in Fig. 3. In this figure, we show the plants’ capacity, watertravelling time (in square brackets), and number of units in eachplant (in parentheses). All 14 units possess Francis turbines.

Table 1 shows other reservoir data, such as initial and limit vol-umes, nominal net head, and the incremental inflows.7

7 The incremental inflows are constant over the planning horizon.

The polynomial coefficients of the forebay and tailrace levels8

are shown in Table 2. Except for H4, all plants possess identical unitswith a single non-forbidden zone. In Table 3, we can see the non-for-bidden zones, the head loss constants and the polynomial coeffi-cients of the function, which represents the unit’s maximumdischarged outflow.9

H4 possesses two groups of different units. The first one is com-posed by units 1 and 2, and the remaining belongs to the secondgroup. The only difference among these groups is the turbinehydraulic efficiency function. Table 4 depicts the function coeffi-cients related to hydraulic efficiency and turbine mechanical lossesas well as the generator global losses.

Finally, Fig. 4 shows the load requirement of each plant.

5.1. Base Case analysis

All data described up to this subsection were used to composethe Base Case, where the startup/shutdown penalty values are con-sidered null. Initialising the Lagrange multipliers equal to 0.1 givesthe LRVS iterative process shown in Fig. 5. The optimal dual LRVSvalue, 51,780 m3/s, is found in 75 iterations, where the 23 itera-tions detailed in Fig. 5 represent the serious steps10 performed bythe Bundle algorithm.

The primal solution is infeasible because the subgradient vectornorm in the last LRVS iteration is 3668.1. Using the Lagrange mul-tipliers and the primal solution supplied by the LRVS in iteration22, we apply the inexact AL, where the initial penalty parameteris 500, and its update is performed as follows:

litþ1 ¼ 0:5 � lit if l > 0:05 and litþ1

¼ 0:95 � lit otherwise: ð33Þ

In Fig. 6, we present the inexact AL performance. The algorithm pre-sents convergence in 20 iterations, and the optimal value is51869.6 m3/s. We use as the stopping criterion ||AC||2 6 0.05, where||AC||2 represents the Euclidian norm of the relaxed artificial con-straints detailed in (22). As a result, a high precise feasible solutionis supplied by the AL because||AC||2 is 0.04729 at the last iteration.

In terms of the primal solution, it is important to show the losspattern in the HUCL problem. In this sense, Fig. 7 shows the sum ofthe losses in all units for each stage, i.e., gglir + tmlir + thlir + chlir,where thlir = rpir (1/gir � 1) is the turbine hydraulic loss andchlir = G (ghr � nhir) qir is the loss associated with the head lossesin the conduits and in the low pressure section of the turbine. Asshown in Fig. 7, on average, the losses represent nearly 10% ofthe total plant load requirements in each stage.

this method, a trial direction is obtained by performing quadratic programming and aline search is then performed along the trial direction, generating a serious step if theLR dual function value is improved or a null step otherwise.

Table 2Forebay and tailrace data.

Plant fbl (a0/a1/a2/a3/a4) trl (b0/b1/b2/b3/b4)

H1 242.9/1.074/�1.1 � 10�3/5.2 � 10�7/�9.2 � 10�11 470.1/1.0 � 10�2/�5.6 � 10�6/1.7 � 10�9/�2.0 � 10�13

H2 540/4.5 � 10�2/�8.1 � 10�6/9.3 � 10�10/�4.4 � 10�14 480/1.7 � 10�3/�4.6 � 10�8/0/0H3 401.2/5 � 10�2/�1.6 � 10�5/3.3 � 10�9/�2.9�10�13 371.2/1.9 � 10�3/�8.5 � 10�5/2.4 � 10�12/�2.6 � 10�17

H4 335.1/6.8 � 10�3/0/0/0 263.5/9.2 � 10�4/�6.7 � 10�9/0/0

Table 3Unit operational data.

Plant Zone k0 + k1 qmaxj (q0/q1/q2/q3)

H1 172–293 1.307 � 10�4 2.58 � 103/�4.87 � 101/0.32/�6.76 � 10�4

H2 136–236 1.401 � 10�4 1.87 � 103/�4.13 � 101/0.33/�8.30 � 10�4

H3 223–380 7.038 � 10�6 1.50 � 104/�4.87 � 102/5.39/�1.97 � 10�2

H4 200–290 1.939 � 10�5 5.93 � 103/1.95 � 102/2.21/�8.21 � 10�3

Table 4Unit efficiency data.

Plant g (c0/c1/c2/c3/c4/c5) tml (b0/b1/b2) ggl (a0/a1)

H1 3.59 � 10�1/5.54 � 10�3/1.99 � 10�3/1.05 � 10�5/�2.73 � 10�5/�9.43 � 10�6 �0.336/3.78 � 10�3/�2.62 � 10�6 1.99/1.735 � 10�3

H2 3.59 � 10�1/5.85 � 10�3/2.37 � 10�3/1.33 � 10�5/�3.04 � 10�5/�1.34 � 10�5 �0.266/3.00 � 10�3/�2.08 � 10�6/ 1.58/1.377 � 10�3

H3 6.90 � 10�2/3.01 � 10�3/5.56 � 10�3/5.84 � 10�6/�4.64 � 10�6/�3.64 � 10�5 �0.435/4.90 � 10�3/�3.40 � 10�6 2.58/2.248 � 10�3

H14

0.245/2.886 � 10�3/6.66 � 10�3/1.87 � 10�5/�9.18 � 10�6/�5.74 � 10�5 �0.332/3.74 � 10�3/�2.59 � 10�6 1.97/1.716 � 10�4

H24

3.59 � 10�1/3.23 � 10�3/3.44 � 10�3/1.07 � 10�5/�9.26 � 10�6/�2.84 � 10�5 �0.332/3.74 � 10�3/�2.59 � 10�6 1.97/1.716 � 10�4

Fig. 5. LR dual optimization.

E. Cristian Finardi, M. Reolon Scuzziato / Electrical Power and Energy Systems 44 (2013) 7–16 13

To observe the participation of gglir, tmlir, thlir and chlir in the to-tal loss, we analyse the results of H4 at stage 10. The total loss in thiscase is equal to 148.04 MW. As shown in Fig. 8, the turbine hydrau-lic loss represents the major part in the total loss value. We high-light also that almost 12% of the total loss is due to the ggl andtml; therefore, the results justify these modelling issues in the HUCLproblem. In the next section, we address this question again.

5.2. Mechanical and electrical losses analysis

Now we evaluate the impact of the tml and ggl functions on theresults. In this sense, we eliminate these functions of the Base Casedetailed in the last subsection. If tml and ggl functions are not takeninto account, the new optimal value is 51077.1 m3/s, i.e., approxi-mately 1.55% lower than the previous optimal solution. We presentin Fig. 9 the difference between the plant’s total discharged out-flow of the Base Case and the present case, where the mechanicaland electrical losses are neglected.

As a consequence of the discharged outflow difference pre-sented in Fig. 9, the final volumes at the end of the planning hori-

Fig. 4. Hydro plant load requirement over the planning horizon.

zon are altered in relation to the Base Case. The system storagedifference between the cases analysed in stage 24 is 1.07 hm3.We are also interested in the impact of the tml and ggl functionsin terms of different operational conditions. In Table 5, we presentfour different cases where the initial useful volumes are modifiedin relation to the Base Case. All the remaining data are the same.Below, Qdif = Q1 � Q0, where, Q1 (Q0) is the optimal objective func-tion when the tml and ggl functions are neglected (included) in theHUCL problem. In addition, vdif is the total volume difference at theend of the planning horizon associated with Qdif.

The difference in the volumes at the end of horizon is more con-siderable when low values of the initial volumes are available. Inthese cases, due to the low net head, it is necessary to use morewater to generate the same plant load requirements.

5.3. Solution strategy improvements

It is well known that the solution strategy performance of dualmethodologies can be drastically improved when good starting

Fig. 6. AL dual optimization.

Fig. 7. Loss pattern over the planning horizon.

Fig. 8. Loss sharing in the HUCL problem.

Fig. 9. Differences in the discharged outflow of plants.

Table 5Different initial conditions.

Case Initial volume (%) Qdif (m3/s) vfdif (hm3)

I 30 852.9 1.13II 40 823.3 1.10III 60 775.4 1.06IV 70 766.4 1.07

Fig. 10. Warm-start strategies.

Table 6Warm-start procedure performance (time in minutes).

Case Simplified model HUCL Problem Total time

Q� Iter. Time Q� Iter. Time

BC – 51869.6 43 5.0 50(a) 52327.7 19 1.2 51879.3 23 1.6 2.8(b) 52753.3 36 1.5 51868.6 22 1.5 3.0I – 53200.8 42 4.2 4.2(a) 53905.1 21 1.1 53201.5 25 1.8 2.9(b) 54072.8 37 1.3 53201.8 23 1.6 2.9IV – 50885.6 42 4.0 4.0(a) 51269.0 14 0.8 50888.7 24 1.5 2.3(b) 51706.6 23 1.0 50883.1 23 1.4 2.4

Table 7Number of unit switches versus penalty.

Switchpenalty (m3/s)

Number ofswitches

Total plant dischargedoutflow (m3/s)

Objectivefunction (m3/s)

0 36 51869.4 51869.610 34 51884.6 52224.720 28 52001.8 52561.950 26 52048.6 53348.675 22 52245.2 53895.3

100 22 52245.7 54445.7120 22 52245.1 54885.1

14 E. Cristian Finardi, M. Reolon Scuzziato / Electrical Power and Energy Systems 44 (2013) 7–16

points are available. In this sense, a possible strategy consists ofusing, as a warm start, the optimal Lagrange multipliers and primalvariables obtained from the HUCL problem solution of a previousday. However, this option can be inefficient if the operational condi-tions in 24 h are altered because, for instance, there are considerablechanges in the incremental inflow forecast and plant load require-ment. For this reason, we propose an alternative warm-start proce-dure, which consists of using a solution of the HUCL simplifiedmodel. This simplified model possesses the same mathematical for-mulation as (12)-(20), except for the following modelling issues:

Fig. 11. Number of switches in the H3 plant over the planning horizon.

E. Cristian Finardi, M. Reolon Scuzziato / Electrical Power and Energy Systems 44 (2013) 7–16 15

(i) the tml and ggl functions are removed;(ii) the limits of unit’s discharged outflow are constant;

(iii) the units’ hydraulic efficiency is considered constant;(iv) the fbl functions are linearized using the initial volumes.

Therefore, we simplify some complex functions of the originalHUCL problem, solve the resulting problem and feed the originalone with the corresponding primal and dual solutions. In thetwo-phase solution strategy proposed in this paper, there are sev-eral ways to use the simplified solution as a starting point. Below,we highlight two of them as follows:

(a) Utilize only the Lagrange multipliers obtained in the LRVSoptimization of the simplified problem, kLR0, as a warm startto the original HUCL and

(b) Utilize (a) plus the primal variables obtained in the LRVS-ALoptimization of the simplified problem, XAL0.

Fig. 10 shows the warm-start strategies proposed.In Table 6, we summarize the main results obtained with the

warm start procedures considering three operational conditions,where Q� is the objective function optimal value. Specifically, thefirst operational condition is the Base Case (BC), and the remainingare related to the Cases I and IV shown in Table 5.

In the nine cases above, the warm-start strategies (a) and (b) re-duce on average 40% of the total time necessary to solve the HUCLwithout any simplification. The tests also show a slightly bettercomputational burden of (a) in comparison to (b), despite the opti-mal value differences are quite small (both strategies suppliespractically the same solution). Therefore we can conclude thatthe resolution of the simplified model supplies good Lagrange mul-tipliers as starting points for the Bundle method.

5.4. Inclusion of the startup/shutdown penalties

Now we analyse the inclusion of the startup/shutdown penal-ties in the Base Case. Table 7 shows the number of unit switchesas a function of the switch penalty qir.

As shown in Table 7, when qir P 75, the number of switches isminimized, although it is necessary to increase the discharged out-flow of the total plants over the planning horizon. The followingfigure illustrates the number of switches obtained in the Base Caseby considering two different penalty parameter values.

Fig. 11 is related to the H3 plant, which possesses three identicalunits. The vertical axis shows the number of units, where, in moststages, it is possible to utilize two or three units to meet the load plantrequirement. In the HUCL problem optimization without penalties,the number of the switches is equal to 12. On the other hand, usinga 100 (m3/s) penalty, the number of switches is decreased to six.

6. Conclusions

This paper presented a model and a solution strategy to theHUCL problem with cascaded and head-dependent reservoirs. Anew mathematical model for the hydropower function is proposedwhere head variations, hydraulic losses and, in particular, hydrau-lic efficiency and mechanical losses in the turbine as well asmechanical and electrical losses in the generator are included.Additionally, an integrated methodology combining several opti-mization techniques was utilized to solve the problem. The modeland the solution strategy proposed are suitable for cascaded sys-tems that require unit-based scheduling. The expected benefitsfor the HUCL problem include the improvement in the basin oper-ation, ease of scheduling the unit in advance (e.g., maintenancestudies) and prevention of forbidden zone generation to reducewear on turbines.

Acknowledgement

The authors gratefully acknowledge the financial support inpart by CAPES.

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