new approach to gas network modeling in unit commitment

NEW APPROACH TO GAS NETWORK MODELING IN UNIT COMMITMENT Maziar Yazdani Damavandi, Iman Kiaei, Mohhamd Kazem Sheikh El Eslami Facultyof Electrical and Computer Engineering of Tarbiat Modares University Tehran, Iran [email protected], [email protected], [email protected]

Abstract - Natural gas plays impressive role in the world economy as one of the most important primary energy resources. Its proportionate advantages in contrary to the other energy resources result in increasing in the portion of gas fired units in generation expansion planning of power networks. This mushroom extension leads to more interdependency between gas and electrical networks. Therefore different surveys try to investigate the impact of the gas network on electrical infrastructure. Also, they proposed various gas network models to model this impact. But the critical point in all of these surveys is paying no attention to different dynamics of these two infrastructures. This paper makes an effort to propose a quasi dynamic model for gas network. This MILP model implements gas velocity and pipeline distances between gas areas for modeling gas network dynamic. Proposed model is tested on a test system and its proficiency is proved.

Keywords: Unit Commitment, Modeling, Infrastructures Planning.

Gas

Network

NUMENCLATURE Indices:

i

k s t

u w

Index of a generatingunit. The electric bus of a generatingunit. Gas area of a generatingunit. Time. Segment index for generation cost curves. Segment index for gas fuel consumption curves.

Parameters:

C

max i,k ,s,w

C

min i,k ,s,w

f smax ,s*

Maximum cost of a segment.

Fc

min g i , k , s ,u

Distance between gas areas.

lek ,t

Electric demand of a bus.

M

Pi, k , s, w

Very large positive number. Maximum generation of a segment for consumption curve. Minimum generation of a segment for consumption curve. Maximum generation of a segment generation cost curve. Minimum generation of a segment generation cost curve.

PLmax k ,k *

Transmission capability between buses.

Rkmin ,t

Minimum reserve requirement for a bus.

Rri , k , s

Ramp rate of a unit.

P

max i , k , s ,u min

Pi , k , s , u max

Pi , k , s, w min

S

g s ,t

Minimum cost of a segment. Gas volume transfer capability between gas areas.

fuel for for

Gas volume, deliverable to a gas area.

SDi ,k , s ,t Tion ,k ,s

Minimum up time of a unit.

Tioff ,k ,s

Minimum down time of unit.

Vsc Ve sg, s *

Gas reservoir capacity in a gas area. Gas volume, transferred to or from a reservoir. Gas velocity between gas areas.

X k ,k *

Line impedance between buses.

Vsr

k ,t

Voltage angel of bus. Time interval between gas areas.

Variables:

Ci , k , s ,t , w Cost i , k , s , t

Maximum fuel consumption of a segment. Minimum fuel consumption of a segment.

fuel

Cost of turning on of a unit during the period. Shut down indicator of a generating unit.

SUic, k , s

! s ,s *

max

Fc g i ,k ,s ,u

LPipe s , s*

Generation cost of a unit in each segment. Generation cost of a unit in each segment.

f sin, s*,t

Gas volume transferred between gas areas.

f sout , s*,t

Gas volume transferred between gas areas.

Fc g i ,k ,s ,t ,u Generation unit fuel consumption in each Fc

npp s ,t

segment.

Non power plant consumption.

Fc ipp,k ,s ,t 17th Power Systems Computation Conference

Generation unit fuel consumption.

Stockholm Sweden - August 22-26, 2011

Pi , k , s ,t

Decision variiable (1 if unit D u is availaable, o otherwise zeroo). D Decision variaable (1 if unit generation coost is inn segment u, otherwise o zeroo). T binary vaariable (1 if gas The g flow direcction is inlet otherw wise zero). T binary vaariable (1 if gas The g flow direcction is outlet otherw wise zero). T Three states dependent vaariable (1 if the f flow is from s to s* gas areeas; -1 if the flow f is in the oppossite direction otherwise o is 0). D Decision variaable (1 if unit fuel consumption is in segment u , otherwise zero). z G Generation levvel of a unit.

PLk , k *,t

P Power transferr between elecctric buses.

Ri , k , s ,t

R Reserve level of a unit.

Re s,g t

R Required gas volume v in a gaas area.

I i , k , s ,t

Ici ,k ,s ,t ,u I f in s , s*,t I f out s , s*,t I f ds , s*,t

I si ,k ,s ,t ,u

SU i ,k , s ,t

Transfer capaability of the reserve betw T ween a areas. S Start up indicaator of a generrating unit.

vs , t

S Stored gas vollume in a gas area.

RLk , k *,t

1 INTR RODUCTION N Cheapness and proporttional stabilitty of natural gas price result in more insttallation of gaas fired unitss in power netw works. On thhe other hand, this kind of generation expansion e leaads to consum mption of more m natural gas and a more interdependency between gas and electric netw works. Thereffore differentt surveys tryy to investigate thhe impact of the gas netw work on electrrical infrastructuree. [1] maximizzes the generaation units proofits with consiidering gas and electtrical contraacts simultaneoussly. The gas network conttingency impaacts are investigaated in [2].A Also, [3] propposes a comm mon expansion pllanning for gaas and electricc networks. [4- 5] have surveyeed the jointlyy gas and eleectric marketss in different couuntries. Finallyy, [6] observess the influencee of gas network on maintenaance schedulinng of generattion units. On contrrary of diffferences in details, variious investigationns have propposed two major m classes of models for gas networkk, namely linnear models and nonlinear moodels. In lineear models, thhe impact of gas areas pressurre is neglecteed and the model m works only o base on trannsmitted gas volume v betweeen gas areas [67]. On the otther hand, the nonlinear moodels implemeents gas areas pressure to deetermine the transmitted gas volume betw ween areas. [88- 9] utilize a nonlinear moodel for jointly gas g and electrric networks OPF. Also, [10] [ uses this moddel in UC plannning. Neverthelless both of thhese classes don’t d consider the different dynnamics of gaas and electric networks. The T power system m dynamic is very fast andd the variationn in load levels is compensaated in short period and the network reacches to steadyy state conditiion. On the otther hand the sloow velocity of natural gaas results in the impact of gaas consumptioon variation remains for soome 17th Power Systems Computation Conference

urs in gas neetwork. In thee previous mo odels the gass hou nettwork has beeen considereed as static model. m Somee plaanning like UC U planning consist of tim me snapshotss tan ndemly. Thesee closest tim me snapshots and a the slow w dyn namic of gas network leadd to the depen ndency of gass nettwork variablles in differeent snapshotss. This paperr pro oposes a quasii dynamic moodel for gas neetwork in unitt com mmitment plaanning which is considered d gas velocityy and d distances beetween gas areeas to determiine time delayy forr conveying thhe impact of ggas volume vaariations from m onee gas area to another one. The paper is organized ass folllows. In section 2, the ggas and electtric networkss interaction is discussed. T The unit com mmitment iss mo odeled in secttion 3. The gas network quasi q dynamicc mo odel is propossed in sectionn 4. Numericcal results aree inv vestigated in section 5. Concluding remarks aree pro ovided in sectiion 6.

2

GAS AND ELECT TRIC NETW WORKS INTERAC CTION The gas andd electric netw works are in ndependent inn mo ost sections. The T only interraction points of these twoo inffrastructures are a gas fired units. Figuree 1 illustratess this structure. The T natural gas is conveerted throughh diffferent technology to electrric energy. A second orderr currve is implem mented to deemonstrate th he conversionn ratio of these unnits in differennt output poweers. The curvee is divided to soome segmentss for linearzin ng the MILP P mo odel. It is shoown in figuree 2. (1) and (4) show thee plaacement of fuel f consumpption from reelated outputt pow wer. G G G G Fig gure 1: Gas annd electric netwoorks interaction n. max

Fc g i , k , s ,u

min

Fc g i , k , s , u max

min

P i ,k ,s ,u

P i ,k ,s ,u

Fig gure 2: Input gas g to output poower curve.

if

i " Gas fuuel units max

Fcgi ,k , s ,t ,u $ # Fc

min g i , k , s ,u

min

Fcc gi ,k , s ,u % Fc gi ,k , s ,u P

max i ,k , s ,u

%P

min i ,k , s ,u

min m

* ( Pi ,k ,s ,t % P i ,k ,s ,u )

(1)

# (1 % I si ,k , s ,t ,u ) * M

Stockholm Sweden - August 22-26, 2011

max

# Fc

min g i , k , s ,u

(I

min

Fc gi ,k ,s ,u % Fc gi ,k ,s ,u

Fcgi ,k ,s ,t ,u &

max

min

P i ,k ,s ,u % P i ,k ,s ,u

min

* ( Pi ,k ,s ,t % P i ,s ,u )

(2)

# (1 % I si ,k ,s ,t ,u ) * (% M )

si ,k ,s ,t ,u

' I i , k , s ,t

(3)

u

i , k , s, t

Fc i ,k ,s ,t ' ( Fc g i , k , s , t , u

(4)

u

The limitations of output power and fuel consumption of generation units in each segment are demonstrated by (5) and (6). min

max

I si ,k ,s ,t ,u * Pi,k ,s,u $ Pi,k ,s,t $ I si ,k ,s ,t ,u * Pi ,k ,s,u min

max

3.1 The objective function: Minimizing the operating cost of generation units is the objective function in assumed optimization problem. In (7) the first term is units generation costs and the second term is units start up costs.

(((( Cost s

k

i ,k , s ,t

i

(7)

# (((( SU ic,k ,s * SU i ,k ,s ,t s

k

i

t

The generation cost versus output power of units is based on second order curves. This curve is linearized through dividing to w linear segments. Ci ,k ,s ,t ,w $ min i ,k , s , w

#C

min

max i ,k , s , w

min i ,k , s ,w

P

%P

min

* ( Pi ,k ,s ,t % P i ,k ,s,w )

(8)

# (1 % I ci ,k ,s ,t ,w ) * M

Ci ,k ,s ,t ,w & min i ,k , s , w

max

C i ,k , s , w % C i ,k , s , w

C

max i ,k , s , w

%C

max

min i ,k , s , w min

P i ,k , s , w % P i , k , s , w

i , k , s ,t , w

(( R

*(

k ,t

%

k * ,t

)

(18) (19)

# ( RLk ,k * ,t & Rkmin ,t

(20)

k*

i

RL k , k * ,t $ PL max % PL k , k * ,t k ,k *

(21)

RLk ,k* ,t ' %RLk * ,k ,t

(22)

Ri,k ,s,t & 0

(23) - Generation units limitations (24) and (25) demonstrate the generation limit of each unit. (24) Pi , k , s , t # Ri , k , s , t $ I i , k , s , t * Pi ,max k,s (25)

Pi ,k ,s ,t & I i ,k ,s ,t ) Pi ,min k ,s .

- Ramp rate Ramp rate of generation units are described by (26)- (28). Pi ,k ,s ,t % Pi ,k ,s ,t %1 $ Rri ,k ,s * I i ,k ,s ,t (26) # Pi ,min k , s * (1 % I i ,k , s ,t %1 ) Pi,k ,s ,t %1 % Pi,k ,s ,t $ Rri,k ,s * I a i ,k ,s ,t %1

* ( Pi ,k ,s ,t % P

min i ,k , s , w

' Costi , k , s ,t

(27)

# Pi,min k ,s * (1 % I i ,k ,s ,t )

)

(9)

Ri , k , s ,t $ Rri ,k , s .

(28)

- MUT and MDT Min up time and min down time of generation units are defined by (29) and (30).

(10)

t #Tioff ,k , s

( Ici,k ,s,t ,w ' Ii,k ,s,t

(11)

w

(12)-(14) model the operation constraints of output power and operation cost in each segment. I ci ,k ,s ,t ,w * ( % M ) $ Ci ,k ,s ,t ,w $ I ci ,k ,s ,t ,w * ( M ) (12) max

Pi , k , s , t $ I ci ,k ,s ,t ,w * P i , k , s , w # (1 % I ci ,k ,s ,t ,w ) * M min i ,k , s ,w

# (1 % I ci ,k , s ,t , w ) * ( % M )

(13) (14)

(15) and (16) define the numbers of start up and shut down of each generation unit. (15) SU % SD 'I %I i , k , s ,t

X k ,k *

i , k , s ,t

s

w

Pi ,k ,s ,t & I ci ,k , s ,t , w * P

1

- System security Transmission lines N-1 criteria is implemented for system security. - System adequacy Minimum generation reserve for each bus is considered. (20)– (23) are utilized for preventing of reserve capture in electric buses.

# (1 % I ci ,k ,s ,t ,w ) * (%M )

(C

(17)

k*

% PL max $ PL k , k * ,t $ PL max . k ,k * k ,k *

(6)

3 UNIT COMMITMENT The unit commitment planning is an optimization program which is consisted of two main parts: the objective function and the constraints.

% lek,t # (PLk,k* ,t ' 0

i

PLk ,k * ,t '

(5)

I si ,k ,s ,t ,u * Fc g i ,k ,s ,u $ Fcg i ,k ,s ,t ,u $ I si ,k , s ,t ,u * Fc g i ,k ,s ,u

#C

((P s

pp

t

3.2 Constraints: Different constraints are considered in this paper. - DC load flow DC load flow equations are described by (17) and (18). Also (19) shows the transmission limits of transmission lines.

i , k , s ,t

i , k , s ,t

i , k , s ,t

SU i ,k , s ,t # SD i ,k , s ,t $ 1 17th Power Systems Computation Conference

(16)

off i ,k , s

T

* ( I i ,k ,s ,t % I i ,k ,s ,t #1 ) #

(I

i ,k ,s ,t

$ Ti ,offk ,s

(29)

h't #2 t #Tion ,k ,s on i ,k , s

T

* ( I i ,k ,s,t #1 % I i ,k ,s,t ) #

((1 % I

i ,k ,s ,t

) $ Ti ,onk ,s

(30)

h't #2

4

GAS NETWORK QUASI DYNAMIC MODEL Two major models has been proposed for gas network, the linear model which is independent from gas nodes pressure and the nonlinear model which is dependent to gas nodes pressure. There are three clusters of constraints in nonlinear model: * The gas flow equations:

Stockholm Sweden - August 22-26, 2011

The sum of inlet and outlet gas flow to each gas node considering their direction equals to zero. o The amount of inlet gas to a gas pipeline equals to gas outlet from other side of pipeline in each time snapshot of planning. * The gas pipelines flow equation based on pressure difference between pipelines two sides. * The pressure limits for each gas node. o

in

0 $ fsin,s*,t $ I f s,s*,t * fsmax ,s* out

$ I f s,s*,t * fsmax 0 $ fsout ,s* ,t ,s*

v s ,t % v s ,t %1 $ V sr

(42)

v s ,t %1 % v s ,t $ V sr

(43)

5 NUMERICAL RESULTS The modeling in this paper is based on MILP model. The six buses IEEE test system is implemented to survey the accuracy of proposed model. The gas and electric networks data is determined in appendix. The Figure 3 shows the system load in study period. Load (MW)

300 270 240 210 180 150 0

6

Retg,s % ( f sout # ( f sin,s* ,t # vs ,t # vs ,t %1 ,s* ,t s*

i g s,t

s

s

Re $ Ssg,t

(32) This paper considers the impact of gas network dynamic on generation units operation. Therefore, the transferred gas between two areas is divided to two parts; inlet gas to the area and outlet from it. The threed

level indicator ( I f s, s* ,t ) is implemented to demonstrate the gas flow direction in pipelines. This indicator is derived by (33) and (34). I f ds , s* ,t ' I f in % I f out s , s * ,t s , s * ,t in

(33)

out

I f s , s * ,t # I f s , s * , t $ 1

(34) (35) and (36) consider gas velocity and distances between gas areas to determine time delay ( ! s , s* ) for conveying the impact of gas volume variations from one gas area to another one. ! s,s* +

LPipe s, s*

G11 Output Power Located in 1st Gas Area G21 & G22 Output Power Located in 2nd Area

I f s ,s* ,t ' I f s* ,s ,h

(37)

' f sin* ,s ,h f sout , s* ,t

(38)

Transfer limitation between demonstrated by (39) and (40).

gas

areas

is

G31 & G32 Output Power Located in 3rd Area Output Power (MW)

(36) The relationship between adjacent areas with regard to their time delay is denoted by (37) and (38). in

24

Unit commitment program is considered in four conditions. Case1) UC considering gas network static model. Case2) UC considering gas network quasi dynamic model. Case3) UC considering gas network static model and 150KSCF/h non power plant consumption in period 1821(h). Case4) UC considering gas network quasi dynamic model and 150KSCF/h non power plant consumption in period 18-21(h). G21 and G22 are the cheapest units so they generate base load. On the contrary, G31 and G32 have higher generation cost than prior units so they generate in medium level load. Also, the G11 is the most expensive generation unit therefore it only generates in peak hours. The Figure 4 depicts the units generation in study period.

h ' t # f sd, s * ,t *! s , s *

out

18

Figure 3: System load in study period.

(35)

Vesg, s *

12 Time (h)

(31)

npp % (( Fcipp ,s ,t % ( Fcs ,t ' 0

(40)

(41)-(43) show the reserve restriction in gas areas. 0 $ v s ,t $ Vsc (41)

In all of these constraints the independent variables are gas pipelines flows and calculated based on gas consumption and supply. On the other hand the gas nodes pressures are the dependent variables in the model and are derived from gas pipelines flows. In linear models (independent from pressure), the calculation of gas network pressures are denied. On the contrary of nonlinear models, the linear models are simple and the linear solvers are very efficient and fast. The proposed gas network model consists of some gas areas which are connected by gas pipelines. (31) describes the gas volume constancy in each gas area. This equation is composed of four parts; required gas, reserved gas, transferred gas and power plant and non power plant consumption. s*

(39)

150 100 50 0 0

6

12

18

24

Time(h)

Figure 4: The output power of generation units.

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Transferred Gas Between 1st & 2nd Gas Areas

1600

1600 1200 800 400 0 0

6

12

18

24

Time(h)

Figure 7: Inlet gas to the 2nd and the 3rd gas areas considering gas network quasi dynamic model.

800 400

Transferred Gas Between 1st & 2nd Areas

0 6

12

18

Transferred Gas Between 2nd & 3rd Areas

24

Figure 5: The transferred gas between gas areas considering gas network static model.

In Case3, the non power plant consumption is occurred when the load is in its lowest amount. Therefore, the transfer amounts between gas areas exactly arise only in supposed period. On the other hand, in Case4 with considering gas network quasi dynamic model the non power plant consumption is shifted and situated on peak of electric load. Hence, the transmitted gas between areas arise their maximum capacities. So the generation of G11 (most expensive unit) which is located in the 1st gas area rises to compensate this shortage. Figures 8- 11 illustrate these results. Outlet Gas from 1st to 2nd Gas Area Outlet Gas from 2nd to 3rd Gas Area 1600 1200 800

Transferred Gas (KSCF/h)

0

Time(h)

Transferred Gas (KSCF/h)

Intlet Gasto 3rd from 2nd Gas Area

1200

1600 1200 800 400 0 0

6

12

18

24

Time(h)

Figure 8: The transferred gas between gas areas considering gas network static modeland non power plant consumption.

6 CONCLUSION In this paper the gas network quasi dynamic model is proposed for modeling the impact of gas velocity and distances between areas in unit commitment planning. The application of MILP model is demonstrated by some case studies. The numerical results show the impact of gas velocity and distances between areas. It also declares that pay no attention to these factors can result in significant error in results.

400

Outlet Gas from 1st to 2nd Gas Area

0

Outlet Gas from 2nd to 3rd Gas Area 0

6

12

18

24

Time(h) st

nd

Figure 6: Outlet gas from the 1 and the 2 considering gas network quasi dynamic model.

gas areas

Transferred Gas (KSCF/h)

Transferred Gas(KSCF/h)

Transferred Gas Between 2nd & 3rd Gas Areas

Intlet Gas to 2nd from 1st Gas Area Transferred Gas (KSCF/h)

As a result of these generations, the transfer curves between gas areas in Case1 and Case2 are illustrated in Figures 5-7. As can be observed, the patterns of these two cases are alike but in quasi dynamic model because of gas low velocity the required gas is flown some hours in advance.

1600 1200 800 400 0 0

6

12

18

24

Time(h)

Figure 9: Outlet gas from the 1st and the 2nd gas areas considering gas network quasi dynamic model and non power plant consumption.

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Table 3: Data of the generators (Gas Consumpton Coeffeission)

Inlet Gas to 2nd from1st Gas Area Transferred Gas (KSCF/h)

Inlet Gas to 3rd from 2nd Gas Area 1600

Bus No.

1200

G11 G21 G22 G31 G32

800 400 0 0

6

12

18

Figure 10: Inlet gas to the 2nd and the 3rd gas areas considering gas network quasi dynamic model and non power plant consumption. G11 Output Power Considering Static Model

Output Power (Mw)

G11 Output Power Considering Quasi Dynamic Model 90 60 30 0 6

12

18

24

Time (h)

Figure 11: G11 generation considering gas network static and quasi dynamic models.

APPENDIX

Figure 12: IEEE 6 Bus diagram and the gas network. Table 1: Data of the generators(Cost Coefficients)

Bus No. G11 G21 G22 G31 G32

a

b

c

($/MW2.h)

($/MW.h)

($/h)

0.004683 0.021575 0.001534 0.00473 0.00578

12.5629 5.92653 7.10787 10.7154 9.28

520.15 245.894 279.781 351.0288 336.7752

Table 2: Gas area interaction data. Pipeline No.

From

1 2

1 2

To

Flow Limit (KSCF/h)

2 3

1500 900

Length (mile)

c (KSCF)

0.00078 0.00288 0.00022 0.00068 0.00083

2.0938 0.7902 1.0154 1.5308 1.3257

86.691 32.787 39.969 50.147 48.11

Gas Average Velocity

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(mile/h)

85 75

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b (KSCF/MW)

24

Time(h)

0

a (KSCF/MW2)

40 35 Stockholm Sweden - August 22-26, 2011