Electric Power Systems Research 122 (2015) 1928

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Electric Power Systems Research

j o ur na l ho mepage: www.elsev ier .com/ locate /epsr

A mixed integer linear programming model for thmanagement problem of microgrids

Daniel TeLabPlan/UFSC, CampCEP 88040-900

a r t i c l

Article history:Received 30 JuReceived in re12 December 2Accepted 19 D

Keywords:Energy managMicrogridsMicroturbinesFuel cellsControllable loMixed linear integer programming

modeear plable ical as), wheratiThe p

loadd alo

d. The

1. Introduction

The modable electroEnergy Res(FCs), batteis currentlymanagemening, as weldistributionhighlight thtional elemmodern eleof a group ooperate eith

Despite inherent, suological ch

AbbreviatioFC, Fuel cells; turbine; RLD, r

CorresponE-mail add

operational issues of MGs is the energy management (EM) prob-

http://dx.doi.o0378-7796/ ern electrical energy industry addresses more afford-nic technologies. The integration of small Distributedources (DERs), such as Microturbines (MTs), Fuel Cellsries, wind and photovoltaic generators, is a trend that

in progress. The presence of the DERs and the demandt can reduce fossil fuel consumption, load peak shav-

l as postpone investments in new transmission and lines [1,2]. In this new paradigm, it is important toe Microgrids (MGs), which are emerging as an addi-ent to maintain the growth and sustainability of thectric energy industry. Roughly speaking, a MG consistsf DERs and controllable and uncontrollable loads thater synchronised with the main grid or autonomously.several advantages of MGs, the new challenges arech as those related to DERs. In this context, a method-allenge that supports the economic and technical

ns: CLD, curtailable load demand; DER, distributed energy resource;MG, microgrid; MILP, mixed-integer linear programing; MT, micro-eschedulable load demand; SOFC, solid oxide fuel cell.ding author. Tel.: +55 48 3721 9731; fax: +55 48 3721 7538.ress: tenfendaniel@gmail.com (D. Tenfen).

lem [3,4]. In general, solving this problem requires determining ageneration and a controllable load demand policy that minimises,over a planning horizon, an objective function subject to economi-cal and technical constraints. The policy is given by the on/off status,the respective output active power of each DER, the on/off sta-tus of the curtailable load demand (CLD) and the schedule of thereschedulable load demand (RLD). This strategy is used for the volt-age and frequency control in MG real-time operation. Therefore,because it is necessary to minimise an objective function subjectto constraints, the EM is usually performed based on the solutionof an optimisation problem, although there are other possibilities,such as fuzzy logic and expert systems [5,6] and hierarchical anddecentralised control [7,8].

The EM in [9] is performed for a MG with wind and thermal gen-erations, aiming for the minimisation of the total operation cost andconsidering the stochastic behaviour of the wind. In [10], the EMaddresses a MG composed of batteries and photovoltaic generationwith connected operation with the main grid, which allows for thepurchasing and selling of energy. In [11], the EM is performed fora MG composed of FC, wind, photovoltaic generation and batter-ies, considering a quadratic objective function for the cost. In [12],the EM is performed for a Microgrid with a MT, a FC, a diesel, awind, and a photovoltaic generator and a battery, with a multi-objective approach to minimise the cost of operation and reduce

rg/10.1016/j.epsr.2014.12.0192014 Elsevier B.V. All rights reserved.nfen , Erlon Cristian FinardiElectrical Systems Planning Research Laboratory, Federal University of Santa Catarina,

Florianpolis, Brazil

e i n f o

ne 2014vised form014ecember 2014

ement

ad demand

a b s t r a c t

This paper presents a mathematical (MG) by means of a mixed integer lindetermine a generation and a controlthe operation cost subject to econommicroturbines (MTs) and fuel cells (FCminimum up and downtime, and genadequately considered in literature. critical, reschedulable and curtailablethe proposed modelling, a MG is usegenerators connected to the main grie energy

us Universitrio, CP 476,

l for the energy management (EM) problem of a microgridrogramming approach. In the EM problem, the objective is toload demand policy that minimises, over a planning horizon,nd technical constraints. We propose a detail modelling forere the constraints associated with such factors as the ramps,on limits, represent various peculiarities that have not beenroposed model also considers a detailed representation of

s, which are important aspects in the MG concept. To analyseng with a MT, a FC, a battery bank, wind and photovoltaic

results indicate that the model is adequate for the MG EM. 2014 Elsevier B.V. All rights reserved.

20 D. Tenfen, E.C. Finardi / Electric Power Systems Research 122 (2015) 1928

Nomenclature

Index/setsa b c d e i

t

Variablesdpfbt

ebetpbcetpbdetpdcctpdddtpdetpextpfbtpgbtpgstptatrbetubet

ucct

uddt

ufbt

ugt

utat

ycct

yfbt

ytat

zfbt

ztat

ParameteATat, BTa

BPtCB CCc

CD CE CFbDB DCctDDTa

Dt forecast critical load demand in stage t (kW)DTa start-up cost of MT a (R$)EBeF nal energy of battery e (kWh)

maxindex related to MTs (a A) EBe

index related to SOFCs (b B)index related to CLDs (c C)index related to RLDs (d D)index related to batteries (e E)discretisation step index associated with the MTsand FCs ramps, RLD and CLDindex for the time step (t ND).

absolute power output difference between t 1 andt stages of SOFC b (kW)energy of battery e (kWh) in step tpower charge of battery e (kW) in stage tpower discharge of battery e (kW) in stage tpower of CLD c (kW) in stage tRLD d in stage t (kW)system decit (kW) in stage texcess generation (kW) in stage tpower output of SOFC b (kW) in stage tpower purchased from the grid (kW) in stage tpower sold to the grid (kW) in stage tpower output of MT a (kW) in stage treserve of battery e (kW) in stage tbinary variable that indicates whether battery e isdischarging (ubet = 1) in stage tbinary variable that indicates whether CLD c is on(ucct = 1) or off (ucct = 0) in stage tbinary variable that indicates whether RLD d starts(udct = 1) in stage tbinary variable that indicates if an SOFC is on (uft = 1)or off (uft = 0) in stage tbinary variable that indicates whether the MG isimporting energy (ugt = 1) in stage tbinary variable that indicates whether MT a is on(utat = 1) or off (utat = 0) in stage tauxiliary binary variable for indicating the start ofthe load shedding in stage t of CLD cSOFC b auxiliary binary variable for the start-upramp rate in stage tMT a auxiliary binary variable of start-up ramp ratein stage tSOFC b auxiliary binary variable for the shutdownramp rate in stage tMT a auxiliary binary variable of shutdown ramprate in stage t.

rst constants associated with the fuel consumptionfunction of MT a operating with a xed ambienttemperature, in (R$/h) and (R$/kWh), respectivelyenergy purchase price in stage t (R$/kWh)battery e charge step ramp (kW)incremental cost during one hour of load shedding(R$/kWh) of CLD cload decit incremental cost (R$/kWh)system excess energy incremental cost (R$/kWh)incremental operating cost of SOFC b (R$/kWh)battery e discharge ramp (kW)forecast CLD c in stage t (kW)number of stages of MT as shutdown ramp rate

EBemin

EBeI

EBeL

ED EFbEPV

EPW FDDd

FTaGFbH IDDdMFCbMTCaNCcmax

ND NDCcst

NFbst

NTast

Pamax

PDDidPFbmax

PFbmin

PFUbi

PGBtmax

PGBtmin

PGStmax

PGStmin

PTmaxat

PTamin

PTDai

PTUai

PVtPWtRB SPtTca

UDDdUDFbUDTaaebc

ebd

b

the greenhthe EM is pand a battecost of opeenergy capgrid with thbattery e maximum energy (kWh)battery e minimum energy (kWh)initial energy of battery e (kWh)energy lost in one time step of battery e (kWh)error associated with the demand (%)start-up cost of FC b (R$)

error associated with the photovoltaic generation(%)error associated with the wind generation (%)nal stage where the RLD d load has to be fully sup-pliedshutdown cost of MT a (R$)shutdown cost of FC b (R$)horizon time (h)initial stage where the RLD d could be turned onmaintenance incremental cost of FC b (R$/kWh)maintenance incremental cost of MT a (R$/kWh)maximum number of stages of load shedding forCLD cnumber of stages in the planning horizonmaximum number of load shedding for CLD cmaximum number of start-ups of SOFC bmaximum number of start-ups allowed of MT anominal maximum output power of MT a (kW)forecast RLD d in stage i (kW)maximum output power of SOFC b (kW)minimum output power of SOFC b (kW)output power in stage i of the SOFC b start-up ramprate (kW)grid maximum power purchase in step t (kW)grid minimum power purchase in step t (kW)grid maximum power sell in step t (kW)grid minimum power sell in step t (kW)maximum output power of MT a operating with anambient temperature of Tt (kW)minimum output power of MT a (kW)output power in stage i of MT as shutdown ramprate (kW).output power in stage i of MT as start-up ramp rate(kW)forecast photovoltaic power in step t (kW)forecast wind power in step t (kW)number of time steps to the reserve of the systemenergy selling price in stage t (R$/kWh)ambient temperature where the maximum outputpower of MT a decreases (C)number of stages in which RLD d is onnumber of stages in the SOFC b start-up ramp ratenumber of stages of MT a start-up ramp rateconstant for MT a (slope of the line, in kW/C)charge efciency of battery e (%)discharge efciency of battery e (%)incremental cost associated with the cycling opera-tion of SOFC b (R$/kWh).

ouse gas emissions, such as NOx, SO2 and CO2. In [13],erformed for a MG with thermal and wind generatorsry, with a multi-objective approach to minimise theration and reduce the greenhouse gas emissions andacity reserve. In [14], the EM is performed for a Micro-ermal, wind and photovoltaic generators with batteries,

D. Tenfen, E.C. Finardi / Electric Power Systems Research 122 (2015) 1928 21

4.00

4.50

5.00

5.50

6.00

6.50

7.00

7.50

8.00

01

Co

st (

R$

/h)

0 C 10 C 20 C 30 C

Fig. 1. MT oppower.

consideringused for thof batteriesnected and CLD. In [17]the EM modpower for a

It is esseelling to achgeneral, MTmal units, i.variation bea unit mustever, FCs annot been ad

For instentire powenecessary tdown and rramp rate, MT achievebecause thishutdown rtion of a Munit, and, duishing the swhich is nothe maximuthe ambien

Regardinrequiremenent temperwork, we arhigh tempereaching thstart-up ramis decreasestarvation.

Concernas critical, C

1 Due to intesteps of few mcharacteristic the EM proble

of being able to be allocated across a range of time. The CLD mayhave the power supply cut, as a non-priority load, if necessary.

To analyse the inuence of the proposed modelling, in this paperwe use a MG with a MT, a FC, batteries, and wind and photovoltaicgenerators supplying a CLD, RLD and critical load demand. The EMcan be performed for both islanded and connected operation withthe main grid. In this paper, the EM is obtained by solving a deter-ministic mixed-integer linear programming problem, where the

ng hos fold mo

relatmentsions

rotu

s Mi

opecycle

powcy in

turalimilh funst du

ATat

inat c

of(Rth

his pe coteriscy vaffec

maxig. 2,tput

= Pma0251

Power (kW)

erating cost as a function of the ambient temperature and output

the power distributions lines limits. The EM in [15] ise planning of the MG to determine the optimal sizing. The EM in [16] has different objectives for the con-stand-alone (island) operation of the MG and considers, a review is presented regarding the state-of-the-art forelling and planning of combined cooling, heating and

MG.ntial to have precise DERs and controllable load mod-ieve a high-quality operational strategy for the EM. Ins and FCs can be modelled similarly to the large ther-e. it is necessary to represent maximum rates of powertween two consecutive stages, the minimum time that

be on or off, output generation limits, etc. [18,19]. How-d MTs possess peculiarities in their modelling that hasequately considered in the literature.ance, a typical MT (a few dozen kW) may vary ther range in a few dozen seconds. Consequently, it is noto represent operating ramp rate (also known as ramp-amp-up rate) in the EM problem. However, the start-upwhich depends on an auxiliary power source until thes nominal operation, must be accurately representeds process may take a few minutes.1 In addition, theamp rate also has a peculiar feature; initially, the rota-T is decreased to a specied value to cool down thering this time, the power decreases linearly before n-

hutdown cycle. Another particularity of MT modelling,t presented in the literature, concerns the variation ofm power and the electrical efciency as a function of

t temperature.g FC, this type of DER has the same modelling

planninised ademanmodelexpericonclu

2. Mic

2.1. Ga

Thenamic outputefcienthe nations, s

Eaction coas:

cptat =where

a ATat, BT

ptat

In ttenanccharacefcienis also the MT

In FMT ou

PTmaxat

where

ts aforementioned, with the exception of the ambi-ature inuence and the shutdown ramp rate. In thise interested in Solid Oxide FC (SOFC), which works at arature and requires constant power consumption untile temperature of nominal generation. As a result, thep rate is very slow. In addition, the lifetime of a SOFC

d with load cycling, the number of start-ups and fuel

ing load demand, it can be classied by priority and typeLD and RLD [20]. The RLD has a particular characteristic

rmittent generation, it is crucial to discretise the planning horizon ininutes [21], with 1 min considered for this paper. Thus, any operationalof DERs that takes more time than this must be precisely modelled inm.

PTmaxat men

Pamax noTca am

ofa co

Regardinup ramp of

2 In fact, theand ambient pdepend on the

3 The ofciarizon is 24 h with 1-min time steps. This paper is orga-lows: in the next section, we detail the MT, FC and loaddelling; then, in Section 3, we present the optimisationed to the EM problem; in Section 4, some computationals are presented; nally, Section 5 provides the primary

of this paper.

rbine, fuel cell and load demand modelling

croturbine

ration of a gas MT is based on the Brayton thermody- [22]. The MT electrical efciency depends on electricaler (kW) and ambient temperature (C)2 [23]. With theformation provided by the turbine manufacturer and

gas price, it is possible to construct a set of cost func-arly to those curves presented in Fig. 1.ction in Fig. 1, which represents the MT fuel consump-ring a one-hour period for the cpta in R$,3 can be written

+ BTat ptat, (1)

dex related to MTs (a A);onstants associated with the fuel consumption function

MT a operating with a xed ambient temperature, in$/h) and (R$/kWh), respectively;e power output of MT a (kW) in stage t.

aper, we also consider start-up, shutdown and main-sts, as detailed in Section 3. Another important MTtic is related to the electrical power limits. Because thearies with the temperature, the maximum output powerted by this parameter. Fig. 2 presents the behaviour ofimum power as a function of the temperature [24].

when the temperature T is higher than Tc, the maximumpower can be expressed as:

ax a (Tt Tca), (2)

aximum output power of MT a operating with an ambi-t temperature of Tt (kW);minal maximum output power of MT a (kW);bient temperature where the maximum output power

MT a decreases (C);nstant for MT a (slope of the line, in kW/C).

g constraints, in general, a typical MT possesses a start-a few minutes since, before starting the generation, the

electrical efciency also depends on the natural gas inlet pressureressure, but these parameters can be considered constant since they

installation site.l Brazilian currency since 1994 is the Brazilian real (sign R$, code BRL).

22 D. Tenfen, E.C. Finardi / Electric Power Systems Research 122 (2015) 1928

MT consumes power from the system (or from an internal battery).The shutdown ramp rate takes more time, but still around a fewminutes to nish the cool down. After starting the shutdown ramprate, the power is decreased abruptly in a few seconds and, afterthis, the power decreases linearly. From the middle to the end ofthe shutdown ramp rate, the MT might have power consumption[25,26]. The equations that represent the constraints associatedwith the output power limits, as well as the start-up and shutdownramp rate, can be written as:

PTmina (

utat UDTai=1

yta,ti+1 DDTai=1

zta,t+i

)+

UDTai=1

PTUai yta,ti+1 +DDTai=1

PTDai zta,t+D

PTmaxat [utat

UDTai=1

yta,ti+1 DDTai=1

zta,t+i

]

UDTai=1

PTUai yta,ti+1 DDTai=1

PTDai zta,t+

ytat ztat utat + uta,t1 = 0, (5)UDTai=1

yta,ti+1 utat 0, (6)

DDTai=1

zta,t+i utat 0, (7)

ytat +UDTa+

i=

utat, ytat, zt

where

utat bior

ytat Mst

ztat Mst

DDTa nuPTamin mPTDai ou

(kPTUai ouUDTa nu

Eqs. (3)start-up an

20

22

24

26

28

30

32

0

Po

wer

(k

W)

Fig. 2. MT ma

Eqs. (5)(8) represent integrality constraints. The modelling of con-straints (3)(8) is based on [27], with the difference that theremay be power consumption during the ramps. Fig. 3 illustrates thebinary variables and start-up and shutdown ramp rates consideringUDT = 3 and

Finally, (NTast) musfollowing c

NDt=1

ytat N

2.2. Fuel Ce

technlassi

this presnsidds oncpfb, on:

CFb

inpoin

30], i for culathe pclingtion,nal c

bt,

fb,t

pfb,

abstpoinSO

strai, resphavioic mse isDDTa1

1

zta,ti+1 1, (8)

at {0, 1}, (9)

nary variable that indicates whether MT a is on (utat = 1) off (utat = 0) in stage t;T a auxiliary binary variable of start-up ramp rate inage t;T a auxiliary binary variable of shutdown ramp rate inage t;mber of stages of MT as shutdown ramp rate;inimum output power of MT a (kW);tput power in stage i of MT as shutdown ramp rateW);tput power in stage i of MT as start-up ramp rate (kW);mber of stages of MT a start-up ramp rate.

and (4) refer to the minimum, maximum andd shutdown output power constraints, respectively.

6 11 17 22 28 33 39 44

Ambient temper ature (C)

Maximum p ower

Tc

ximum output power as a function of the ambient temperature.

FC mon ccells. Inmodelthat codepencosts, equati

cpfbt =where

b pfbtCFb

In [ciationin modble if tThe cygeneraadditio

b dpfpfbt ppfbt +where

dpfbt

pfbtb

Conpowerthe bedynamresponDTai+1 ptat 0,(3)

DDTai+1 + ptat 0,(4)

DDT = 6.due to the warranty issues, the number of start-upst be limited over a dened period; thus, we include theonstraint in the optimisation model:

Tsta , (10)

ll

ology presents nearly zero emissions. The most com-cation is related to the type of electrolyte used in the

paper, we are interested in Solid Oxide FC (SOFC). Theented here is based on the dynamic modelling [28,29]ers a 100 kW power capacity. The FC fuel consumption

the output power and, consequently, the operatingin R$/h, can be modelled by means of the following

pfbt, (11)

dex related to SOFCs (b B);wer output of SOFC b (kW) in stage t;cremental operating cost of SOFC b (R$/kWh).

t is noted that the SOFC unit can incur additional depre-yclic operation costs. In this context, a FC is consideredion if the output power changes in a stage, and it is sta-ower remains identical to that of the previous stage.

operation, resulting from modulation and non-stable reduces the SOFC lifecycle. To prevent SOFC cycling, anost (12) is considered as follows:

(12)

1 dpfbt 0, (13)

t1 dpfbt 0, (14)

solute power output difference between t 1 and tages of SOFC b (kW);wer output of SOFC b (kW) in stage t;cremental cost associated with the cycling operation ofFC b (R$/kWh).

nts (13) and (14) compute the increase and decrease ofectively, during two consecutive times-steps. To modelur of SOFC in terms of operating ramp rate, the sameodel shown in [28,29] was implemented. The signal

presented in Fig. 4.

D. Tenfen, E.C. Finardi / Electric Power Systems Research 122 (2015) 1928 23

d shu

0

20

40

60

80

100

0

Pow

er (k

W)

This mo(the FC modto the cells,Fig. 4, the Soperation sthe behaviothe ramp-dation, as thoutput powtime, but itof oscillatinFig. 5, the SOup ramp ra(1720) and

PFminb [ufb

+UDFbi=1

PFU

PFmaxb [u

UDFbi=1

PFU

yfbt zfbt

,ti+1

ufbDFb

i=1zf

t N

bt, zf

biofSOFig. 3. MT binary variables and start-up an

0

20

40

60

80

100

50 250

500

750

Cur

ren

t (A

)

Time (s)

Output Power Inpu t Sign al

Fig. 4. SOFC command input signal and output power.

UDFbi=1

yfb

zfb,t+1

yfbt +U

NDt=1

yfb

ufbt, yf

where

ufbt

yfbt

del considers SOFC in nominal operating temperatureel needs hours of constant heating, to prevent damage

before starting operation [31]) and, in 50 s, as seen inOFC starts. It was sent some input signals to change theet point of the SOFC at 250 s, 500 s and 750 s, to analyseur of the ramps for the SOFC. The output responses toown (250 sec) and ramp-up (500 sec) in nominal oper-e shutdown (750 s), are very fast. The stability of theer, after a different set point operation, may take some

is not considered in our case due to the small amountg power. As a result of the simulation presented inFC constraints of output power limits, as well as start-

te, (15) and (16), the logical auxiliary binary variables the maximum number of start-ups (21), are given by:

t UDFbi=1

yfb,ti+1 zfb,t+1

]

bi yfb,ti+1 pfbt 0, (15)

tbt UDFi=1

yfb,ti+1 zfb,t+1

]

bi yfb,ti+1 + pfbt 0, (16)

ufbt + ufb,t1 = 0, (17)

inzfbt SO

raNFbst mPFbmax mPFbmin mPFUbi ou

(kUDFb nu

2.3. Load d

The nonthe EM propartially sh[10,11,14]. load shift wing bids anOther impothe EM is pfor the loadthe MG [1,2three kindsand CLD.

Naturallsibility of suan expensivapproach, bThe equatio

CCc pdcct, tdown ramps.

ufbt 0, (18)

t 0, (19)

b,ti+1 1, (20)

Fstb , (21)

bt {0, 1}, (22)

nary variable that indicates if an SOFC is on (uft = 1) orf (uft = 0) in stage t;FC b auxiliary binary variable for the start-up ramp rate

stage t;FC b auxiliary binary variable for the shutdown rampte in stage t;aximum number of start-ups of SOFC b;aximum output power of SOFC b (kW);inimum output power of SOFC b (kW);tput power in stage i of the SOFC b start-up ramp rateW);mber of stages in the SOFC b start-up ramp rate.

emand

-controllable load is the most common load model forblem, where in some works there is the possibility ofed or shifted loads using the energy storage systemOn the other hand, some authors consider the directhile others consider the load demand as an agent offer-d taking priorities in account; for instance, see Ref. [7].

rtant review for the modelling approaches to loads toresented in [17]. In our paper, the proposed modelling

demand is to t the concept of load as a resource within0,32]; therefore, the paper includes simultaneously the

of load aforementioned, i.e., critical load demand, RLD

y, the critical load is the most important, and the impos-pplying this type of demand is modelled in Section 3 ase ctitious generator. We model the CLD using an on/offecause these loads are usually controlled by a switch.ns for the CLD are given by:

(23)

24 D. Tenfen, E.C. Finardi / Electric Power Systems Research 122 (2015) 1928

pdcct ucct DCct DCct, (24)

pdet + ucct [Dt +

Cc=1

(DCct) +D

d=1(max(PDDid))

] Dt +

Cc=1

DCct

+D

d=1(ma

ucc,t1 ucNDt=1

ycct N

NDt=1

ucct

0 pdcct

ucct, ycct

where

c ind inpdcct popdet syucct bi

orycct au

shC, D nuCCc in

CLDt foDCct foNCcmax mND nuNDCcst m

Eq. (23)used to seused to prany situati(1-on to 0quency of d

respectivelycut and the

4 The sum wof the (25) is jubigger value.

Now, in the next equations, we show the modelling for eachRLD.UDDdi=1

PDDdi udd,ti+1 pdddt = 0 for IDDd t FDDd, (31)

pdd = 0 for FDDd < t < IDDd, (32)DDd

d

ud

{0, 1

RLbi(uninfonu

(31)ary e thed w

varicouldle anles.

imis

EM racted. Inn froith Md phootovitiontors ad dermorittenad defocuming

min f =

Ta z

SPt

A

=1pta

xt x(PDDid)), (25)

ct ycct 0, (26)

DCstc , (27)

ND + NCmaxc , (28)

DCct, (29)

{0, 1}, (30)

dex related to CLDs (c C);dex related to RLDs (d D);wer of CLD c (kW) in stage t;stem decit (kW) in stage t;nary variable that indicates whether CLD c is on (ucct = 1)

off (ucct = 0) in stage t;xiliary binary variable for indicating the start of the loadedding in stage t of CLD c;mber of CLDs and RLDs, respectively;cremental cost during 1 h of load shedding (R$/kWh) ofD c;recast critical load demand in stage t (kW);recast CLD c in stage t (kW);aximum number of stages of load shedding for CLD c;mber of stages in the planning horizon;aximum number of load shedding for CLD c.

is the cost due to the discontinuity. Eq. (24) ist DCct to pdcct when ucct is off, while (25)4 isevent the decit before any load demand shed inon. Eq. (26) is the logical to the change of status-off). Eqs. (27) and (28) are the maximum fre-iscontinuity and the maximum time of discontinuity,

. Eqs. (29) and (30) are the limits of the load demand associated binary variables, respectively.

ithin the brackets, which is multiplied by ucct , and in the right sidest to guarantee that it will a big number, although it could be used a

dt

FDDdUt=IDD

uddt

where

pdddtuddt

FDDdIDDdPDDidUDDd

Eq. the binoutsiddemanbinaryelling divisibvariab

3. Opt

Thecal chainsertehorizoMG wies, anand phacquisgenerative loFurtheintermand loto the Progra

ND

t=1

{A

a=1

[(ATat utat + (MTCa + BTat) ptat) H/ND + DTa ytat + F

+ EFb yfbt + GFb zfbt] +C

c=1

[(CCc pdcct) H/ND

]+ [BPt pgbt

s.t. :a

pedt = 1, (33)

}, (34)

D d in stage t (kW);nary variable that indicates whether RLD d startsddt = 1) in stage t;al stage where the RLD d load has to be fully supplied;itial stage where the RLD d could be turned on;recast RLD d in stage i (kW);mber of stages in which RLD d is on.

is the constraint to set the PDDdi to pdddt consideringvariable uddt of the start. Eq. (32) is to set zero to pdddt

range of RLD d. Meanwhile, (33) ensures that the loadill be turned on only once, and (34) is the associatedable. If the RLD could be interruptible, the same mod-

be used for the each period where the load could bed some constraints for the sequence add, with the binary

ation model

optimisation problem is strictly related to the physi-ristics and the regulatory framework in which a MG is

this paper, a MG can buy or sell energy in a day-aheadm the main grid. Additionally, the model considers aTs, SOFCs, critical load demand, CLDs, RLDs, batter-tovoltaic and wind generators. The modelling of windoltaic generators does not consider the cost due to the

as well as operational cost. Thus, wind and photovoltaicare included in the energy balance constraints as nega-mands, whose values as supplied by a forecast model.e, it is assumed that there is a (good) forecast for thet generation, as well as for the ambient temperaturemand. Accordingly, the optimisation problem related

s of this paper is a deterministic Mixed-Integer Linear (MILP) [33] model, as follows:

tat] +B

b=1

[((MFCb + CFb) pfbt + b dpfbt

) H/ND

pgst + CD pdet + CE pext] H/ND} (35)

t +B

b=1pfbt +

Ee=1

(pbdet pbcet) + pgbt pgst + pdet

D

d=1pdddt

Cc=1

(ucct DCct) = Dt PVt PWt, (36)

D. Tenfen, E.C. Finardi / Electric Power Systems Research 122 (2015) 1928 25

Aa=1

[PTmaxat

(utat

UDTai=1

yta,ti+1 DDTai=1

zta,t+i

)+ ptat

]

UDTai=1

PTUai yta,ti+1 DDTai=1

PTDai zta,t+DDTai+1

+B

b=1

[PFmaxb

(ufbt

UDFbi=1

yfb,ti+1

)+ pfbt

]

UDFbi=1

PFUbi yfb,ti+1 E

e=1(rbet +

+ pdet + E t,

ebe,t+1 eb

ebe1 = EBIe,

ebet HND

0 pbcet

pbdet ubet

PGBtmin p

pgbt ugt + pgst P

pext + utat

+C

c=1DCc

pext + ufbt

+C

c=1DCc

MT consstraints (24

All the Nomenclatu

The objeoperationalCLD costs, atransactionpower geneference betthe cost of imframework

The deelled in thHowever, itties to al-lo

5 Additional

hey d an

thaentientaive irgy p

to [1ation

req the

frome, whrgy l

lossand nimund m

for c to prEq. (4hile

the essacess 6) fol

form the vm th

put

preseatterRLD

cent to ts andal mliable imardin

in Fi genrt-up, exce

7 shonerain tim

forurcha

maxnimucess D (

Cc=1

(ucct DCct) +D

d=1pdddt

) ED Dt EPV PVt EPW PW

et +(

pbdet

bde bce pbcet

) HND

= EBL, (38)

ebeND = EBFe , (39)RBi=1

rbe,t1+ibde

EBmine , ebet EBmaxe , (40)

CBe, 0 pbdet rbet DBe, (41)

DBe 0, ubet CBe + pbcet CBe, ubet {0, 1}(42)

gbt PGBtmax, PGStmin pgst PGStmax, (43)

PGBmaxt 0, ugt PGSmaxtGSmaxt , ugt {0, 1} (44)

[Dt +

Cc=1

(DCct) +D

d=1(max(PDDid))

] Dt

t +D

d=1(max(PDDid)) , (45)

[Dt +

Cc=1

(DCct) +D

d=1(max(PDDid))

] Dt

t +D

d=1(max(PDDid)) , (46)

traints (3)(10), SOFC constraints (13)(22), CLD con-)(30), RLD constraints (31)(34).index/sets, variables and parameters are dened inre.ctive function (35) is composed of the MT and SOFC5

costs (fuel, maintenance, start-up and shutdown) ands well as costs and prots associated with grid energys and the articial variables for decit and excessiveration. The customer electricity cost is given by the dif-ween the revenue associated with exported energy and

ported energy by the MG. Depending on the regulatory, it is necessary to replace prices by tariffs in (35).cit and the excess generation, in each stage t, are mod-e optimisation problem similarly to slack variables.

is necessary to include in the objective function penal-w nonzero values of decit and excess generation only

ly, there is a term necessary to avoid SOFC cycling.

when tthe loaenergyaforemincremexpensor enesimilar

Equreserveside ofcriteriabalancon eneenergyinitial the mistep) apower(42) isstep t. grid, wgrid inare necthe exand (4

Thegrid, ifproble

4. Com

To a FC, bCLD, a

Thesignalstrollerthe locered re

SomReg

shownThe

ing stawhere

Fig.tent gein 1-m

The(sell/p

Thethe mithe ex pbcet) pgst + pgbt

(37)

are necessary (decit, when there is no power to supplyd excess generation when there is more intermittentn the MG can consume and/or export). The penaltiesoned are given by the decit and excess generationl costs. The decit should be greater than the mostncremental value of the energy resources available (DERrices), while the excess of generation cost is modelling1].

(36) is the energy balance constraints, and (37) is theuirements. Notice in (37) that PGBtmax is not on the right

equation, which makes the reserve consider the N-1 the main grid. Equation (38) is the battery e energyere the energy level for the next stage t + 1 depends

evel, charge or discharge considering the efciency and in the stage t. Eq. (39) are the battery e constrains for thenal energy, respectively. Eq. (40) are the equations form (considering the reserve requirements for RB timesaximum energy, while (41) are the battery e maximumharge and discharge (also considering the reserve). Eq.event the charge/discharge of the battery e in the same3) represent the limits of energy transactions with the(44) prevent the import (export) energy from (to) thesame step t. Regarding (45) and (46), these constraintsry to shut down all MTs and FCs generating units whenof generation is present. The logical modelling in (45)low the same logical presented in (25).ulation considers a connected operation with the mainalues of PGB(S)tmin(max) are set to zero, the optimisationen represents an island operation of the MG.

ational experiments

nt the numerical examples, we use a MG with a MT,ies, and wind and photovoltaic generators supplying aand critical load demand. The MG is presented in Fig. 5.ral controller is responsible for the EM, sending controlhe DERs and controllable loads through the local con-

receiving information through the local controller andeters. The telecommunication infrastructure is consid-e and represented by the dotted line in Fig. 5.portant data of the MG are presented in Table 1.g AT and BT for the MT, we consider a set of curves asg. 6.eration and energy consumption (negative values) dur-

and shutdown of the MT and FC are shown in Table 2,pt for PFU, each column represents 1 min.ws the forecast values of load demand and the intermit-tion, where the planning horizon H is 24 h, discretisede-steps (therefore, ND = 1440).

ecast temperature, as well as the energy pricesse), is detailed in Fig. 8.imum power grid interchange is 15 kW in all stages, andm is null. The decit incremental cost is 10 R$/kWh, andgeneration incremental cost is 2 R$/kWh. The forecast

26 D. Tenfen, E.C. Finardi / Electric Power Systems Research 122 (2015) 1928

ration

Table 1MT, FC, batter

MT

DDT 10DT 0FT 0MTC 0NTst 2PTmax 30PTmin 6Tc 17UDT 3 0

3

4

5

6

7

8

9

10

11

5

Cos

t (R$

/h)

errors of theand 5% for t

Then, coing EM provariables (1tional modeexecuted onthe MILP so

Table 2Generation an(kW).

Minu

1

PTU 1 PTD 3 PFUa 1.5a For all the

25

30

er (k

W)

WinFig. 5. MG schematic congu

y and load demand data.

FC Battery CLD/RLD

CF 0.198 CB 7.5 CC 0.8.020 0.1 DB 15 NCmax 3

st

6

7

8

ower

(k

W).080 EF 0.8 RB 10 NDC 240

.006 GF 0.0 EBL 0 IDD 180 MFC 0.008 EBF 10 FDD 1240

NFst 1 EBI 10 UDD 240 PFmax 9 EBmax 15.22 PFmin 7 EBmin 3

UDF 60 bc 0.8.3204 bd 0.85

10 15 20 25 30Power (kW)

T= 5 CT= 10CT= 15 CT= 20 CT= 25 CT= 30 CT= 35 CT= 40 CT= 45 C

Fig. 6. MT operating cost functions.

intermittent generation are 10% for the wind and solarhe demand.nsidering the data presented in this section, the result-blem is a MILP optimisation problem with 34,5608,720 continuous and 15,840 binary). The computa-l is implemented in MATLAB 2011b, and the tests were

an Intel quadcore i7 2.80 GHz CPU. We use Gurobi aslver.

d energy consumption during start-up and shutdown of MT and SOFC

te

2 3 4 5 6 7 8 9 10

1.5 31 0.63 0.25 0.13 0.5 0.88 1.25 1.63 2

period considered.

0

1

2

3

4

5

1 121

Inte

rmitt

ent g

ener

atio

n p

Fig. 7.

20

22

24

2628

3032

34

3638

40

1 12

Tem

pera

ture

(C

)

F

4.1. Compu

The resuwith the daby 2; (iii) reby 4 and 5, rfor the MT contrast, cais to analystively.

6 The classicfor FC and MTtaking into acc.

35d Pho tovolt aic Critical L oad De mand RLD CLD0

5

10

15

20

241 361 481 601 721 841 961 1081 1201 1321

Load

de

ma

nd

pow

Time step

Load demand and intermittent generation power input data.

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

1 24 1 361 481 60 1 72 1 841 961 1081 1201 132 1Pr

ice

(R$/k

Wh)

Time step

Ambient Temperature Sell/ Purchase Price

ig. 8. Ambient temperature and energy price input data.

tational results and analyses

lts are divided into four cases: (i) base, which is relatedta presented previously; (ii) all load demands multipliednewable photovoltaic and wind generation multipliedespectively; and (iv) base, with the classical modelling6

and FC. Case (i) and (iv) represent normal operation; inses (ii) and (iii) are extreme ones, where the objectivee the MG with decit and excess generation, respec-

al modelling does not consider the ramps in start-up and shutdown. Additionally, the ambient temperature inuence in the MT is notount.

D. Tenfen, E.C. Finardi / Electric Power Systems Research 122 (2015) 1928 27

-16

-11

-6

-1

4

9

14

19

241

Pow

er (k

W)

MT FC Batt ery Main grid

-16

-11

-6

-1

4

9

14

19

24

1

Pow

er (k

W)

-16

-11

-6

-1

4

9

14

19

24

1

Pow

er (k

W)

Figs. 91the power ihorizon for

In Fig. 9,online becaDERs. The Mgrid is highnotice that ambient tem

The enerthe system DERs have could turnemight increforecast is pwith levels

The CLDgeneration by the own

25

30

1

241

481

721

961

1201

1441

MT FC Batt ery Main grid

xcess gLD cas241

481

721

961

1201

1441

Time step

Fig. 9. Microgrid EM case (i).

MT FC Batt ery Main grid

-15

-10

-5

0

5

10

15

20

Pow

er (k

W)

3

4

5

6

) -ca

se (ii

i)

ER241

481

721

961

1201

1441

Time step

Fig. 10. Microgrid EM case (ii).

241

481

721

961

1201

1441

Time step

MT FC Batt ery Main grid

Fig. 11. Microgrid EM case (iii).

2 present the operation of the MT, FC, and battery andnterchange with the main grid in the 24-hour planning

cases (i), (ii), (iii) and (iv), respectively. Case (i), the FC is turned on in the rst minute and staysuse it possesses the lowest incremental cost among theT is on only when the price of energy from the main

er than the price to produce via the MT. It is possible tothe MT cannot reach the maximum power due to theperature inuence in the modelling.

gy forecast will inuence on the amount of reserve thathave to deal in (37). If the forecast error increases someto provide a great reserve. In some scenarios the DERsd on just to not violate this constraint. This consequencease the EM total cost. To quantify the benet of a betterossible consider the system without forecast errors, orof errors and compare the results.

in cases (i), (iii) and (iv) are not shed due to enoughto supply it, with the cost lower than that stipulateder to shed this load. In case (ii), there are load sheds in

0

1

2

1

Pow

er (k

W

Fig. 13.

two differet = 1143, as rst load ppeak there

Regardincase: t = 260t = 290 in camoment wilocation wathe intermiFig. 13, whe

The EM the proposeFC. Graphicto the modewhere therand more psystem primon in differoptimal EM(iv).

In TableNotice thatthe ambiencost due to t

Table 3Optimal cost a

Objective fuComputatioTime step

Fig. 12. Microgrid EM case (iv).

0

2

4

6

8

10

12

241 481 721 961 1201

Pow

er (k

W) -

case

(ii)

Step time

ener. - case (iii) RLD case (iii) Deficit - case (ii)e (ii) CLD case (ii)

RLD, CLD, decit and excess generation for cases (ii) and (iii).

nt steps times, in t = 655 until t = 677 and t = 962 untilpresented in Fig. 13. The rst shed in case (ii) is in theeak, where there is not decit, although in the secondis load shed and also decit.

g the RLD, there is a different step time to start in each

in case (i), t = 223 in case (ii), t = 687 in case (iii) andse (iv). In case (iii), the RLD is clearly dislocated to theth more intermittent generation. However, this RLD dis-s not enough to consume all the power provided byttent generation, causing excess generation as seen inre it is possible to compare the RLD start for case (ii).

case (iv) in Fig. 12 is to illustrate the difference betweend modelling and the classical modelling for the MT andally it is possible to verify the differences to the EM duelling for the FC, as the lack of the pre heating, for the MT,e are more power dispatch than case (i) when operatingower export to the grid. Another concern is due to theary reserve, which forces the MT, FC and RLD to be

ent time steps. Additionally, there are difference to the cost, as present in for Table 3 for cases (i), (ii), (iii) and

3 it is possible to compare the costs of cases (i) and (iv). the difference is 7.5%, mostly due to the inuence oft temperature in the MT (5.9%). Case (ii) has the highesthe increase of the load demand. On the other hand, Case

nd computational time.

Case

(i) (ii) (iii) (iv)

nction (R$) 92.88 405.30 11.49 85.94nal time (s) 28 29 27 25

28 D. Tenfen, E.C. Finardi / Electric Power Systems Research 122 (2015) 1928

(iii) provides a negative cost, or revenue, due to the large amountof renewable generation. Table 3 also presents the computationaltime to solve the EM problem, and it was less than the stipulatedstep time of 1 min, even in extreme situations of decit and excessof generation considering the misbehaving of the temperature, loadand intermittent generation forecasting.

5. Conclusions

This paper presents a mathematical model for the EM problem ofa MG, which could be used for the connected or island MG operationmode. It is possible to measure the technical and economic impactsof the MG on the main grid using the proposed MILP model. Thisapproach proved to be effective even in extreme situations withthe misbehintermittenparticularitthey will aftemperaturof the operaimum genethe new mothe SOFC, bresource (Rwithin the problem, alrelated to o

Acknowled

The authPrograma dde Energia regulated bimpactos dtemas de diCientco e

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A mixed integer linear programming model for the energy management problem of microgrids1 Introduction2 Microturbine, fuel cell and load demand modelling2.1 Gas Microturbine2.2 Fuel Cell2.3 Load demand

3 Optimisation model4 Computational experiments4.1 Computational results and analyses

5 ConclusionsAcknowledgmentsReferences