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New approach to gas network modeling in unit commitment
Energy 36 (2011) 6243e6250
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New approach to gas network modeling in unit commitment Maziar Yazdani Damavandi, Iman Kiaei, Mohamad Kazem Sheikh-El-Eslami*, Hossein Seifi Faculty of Electrical and Computer Engineering, Tarbiat Modares University (TMU), Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 December 2010 Received in revised form 2 June 2011 Accepted 21 July 2011 Available online 24 August 2011
Expansion of gas fired units installation in power network results in more interdependency between gas and electric infrastructures. Thus new research has paid attention to the impact of gas network in power system studies. One such issue is the impact of gas network in a unit commitment problem. Many linear and nonlinear models are proposed to demonstrate this impact. However, the dynamics of these two infrastructures are different. In contrary to an electric network, the dynamic of a gas network lasts over a long period. In large scale networks, the impact of gas volume variations may remain for some hours or even for a whole day. Earlier research considered a gas network static model by which its dynamic behavior was not observed. In this paper, the quasi dynamic model is proposed for a gas network by which gas velocity and distances between gas areas may be considered. This model is assessed for a unit commitment problem on a test network and its capabilities are appreciated. 2011 Elsevier Ltd. All rights reserved.
Keywords: Combined cycle units Dual fuel units Gas network modeling Infrastructures planning Unit commitment
1. Introduction With the occurrence of energy crises in recent years and the importance of environmental issues, gas consumption has increased as a cheap and clean energy carrier. As a result of the advancements in gas turbine technology and utilization of combined cycle power plants, the installed capacity of these types of units is increased. In recent years, 98% of the constructed power plants in USA use gas fuel for generating the electrical power [1]. With ever increasing dependence on gas and power system infrastructures, the role of gas network in various power system studies has increased. Ref. [2] maximizes the profit of generating units, by simultaneously considering the gas and electricity contracts. Ref. [3] addresses market-based congestion management in electric power systems, taking into account the constraints of the electric power system and the natural gas system. The effects of gas network contingencies on power system security are discussed in Ref. [4]. Ref. [5] proposes a reliability model for multi-carrier energy systems with focus on the role of gas fuel as the most important energy carrier. The combined gas and electric infrastructure expansion planning is observed in Ref. [6]. In Ref. [7], a novel optimization model ‘eTransport’ is presented that takes into account both the topology
* Corresponding author: IPSERC e Tarbiat Modares University, Tehran, Iran. Tel./fax: þ98 21 88220121. E-mail addresses: [email protected] (M. Yazdani Damavandi), eiman. [email protected] (I. Kiaei), [email protected] (M. K. Sheikh-El-Eslami), seifi_ [email protected] (H. Seifi). 0360-5442/$ e see front matter 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2011.07.034
of multiple energy infrastructures such as gas network and the technical and economic properties of different investment alternatives. Furthermore, in Refs. [8,9], gas and electric markets are investigated in some countries. The effect of gas network on generating units maintenance scheduling is observed in Ref. [10]. Some research concentrates on the effect of gas network on unit commitment planning. The purpose of this planning is making an effort to specify unit participations in order to meet system constraints and to achieve higher social welfare. In Ref. [11], a linear model for gas network is proposed, including a set of nodes with predetermined inter-node transfer capability. The role of gas pressure is not considered and the gas transfer between nodes is assumed as SCF (Standard Cubic Foot). This model is extended in Ref. [12], and a nonlinear model for gas network based on the pressure difference between nodes is presented. This nonlinear model was already proposed in Refs. [3,13,14], for combined gas and electric network OPF (optimal power flow). The deficiency of these models is disregarding the impact of different dynamics of gas and electric networks. The power system dynamic is very fast and the variation in load levels is compensated in a short period so that the network reaches fast to the steady state condition. On the other hand, the slow velocity of natural gas results in remains its network dynamics even for some hours. The gas network infrastructure may affect both planning and operation of a power system. In terms of the planning phase, the generation units should be so allocated that their fuel requirements may be met through the gas network. The planning phase is not considered in this paper.
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On the other hand, a unit commitment problem is an operation issue in a power system with its time scale of several hours to a week or so. Its aim is finding out the units combinations to meet the hourly load variations in such a way that both technical and non-technical constraints are met. A generation unit operation is, typically, dependent on fuel (here, natural gas) availability. The fuel availability itself, depends on the gas network infrastructure. The gas network static model is suitable for the planning like gas and electric networks common OPF in one snapshot or limited and long term snapshots planning like gas and electric networks expansion. But the Refs. [15,16] determine these nonlinear equations are only valid in steady state condition and in unit commitment planning with hourly time snapshots the gas network is in transient condition. This paper proposes a MILP quasi dynamic model for gas network considering the gas velocity and distances between nodes. On the contrary of nonlinear models, the linear models are simple and the linear solvers are very efficient and fast. If a static model is used for this network, the unit commitment problem may result in improper results; implying either higher or lower cost than the actual that may be achieved in practice. That is why a dynamic gas network modeling is a must in unit commitment problem. This issue is addressed in this paper. The paper is organized as follows. In Section 2, the unit commitment problem is discussed. In Section 3, the dual-fuel and combined cycles units are modeled. The proposed gas network model is illustrated in Section 4. Numerical results are investigated in Section 5. Concluding remarks are provided in Section 6.
2. Unit commitment model The unit commitment problem may be formulated as an optimization problem. The objective function and various constraints of this problem are discussed as follows.
Minimization of system operation costs is the most important objective of unit commitment problem. The major portion of the system operation costs is power plants costs which consists of the units generation and start up costs. Therefore the proposed objective function consists of two terms. The first term is the generation costs which is derived from units cost functions. Due to the important role of dual fuel units in a power network, these units are modeled in this paper. Therefore two cost function curves are namely, one for natural gas and the other for alternative fuel considered. The second term is the start up cost, calculated from multiply of single cold startup cost to number of startup.
t
s
k
i
Costi;k;s;t þ
XXXX s
k
i
c SUi;k;s *SUi;k;s;t
(1)
t
The generation cost curves of the units are assumed to be as piecewise linear functions. Eqs. (2) and (3) show the operation cost of each unit in terms of the unit output power. Eqs. (4)e(6) model the linear segments limitations of output power and operation cost for each segment. Eq. (7) demonstrates the total cost of each generation unit for a specified time scale. The relationship between units availability and linearized segment indicators is represented by Eq. (8). max min C i;k;s;u C i;k;s;u min * P P i;k;s;t i;k;s;u max min P i;k;s;u P i;k;s;u min þ C i;k;s;u þ 1 Ii;k;s;t;u *M
Ci;k;s;t;u
(3)
Ii;k;s;t;u *ðMÞ Ci;k;s;t;u Ii;k;s;t;u *ðMÞ
(4)
max Pi;k;s;t Ii;k;s;t;u *P i;k;s;u þ 1 Ii;k;s;t;u *M
(5)
min Pi;k;s;t Ii;k;s;t;u *P i;k;s;u þ 1 Ii;k;s;t;u *ðMÞ
(6)
Costi;k;s;t ¼
X
Ci;k;s;t;u
(7)
u
X
Ii;k;s;t;u ¼ Ia
(8)
u
The dual-fuel unit generation cost consists of two terms. The first term shows the generation cost of gas fuel (modeled by Eqs. (10) and (11)) and the second term shows the generation cost based on alternative fuel (modeled by Eqs. (12) and (13)). Eqs. (14) and (15) limit operation cost of units considering the value of the dual fuel indicator.
Ci;k;s;t;u ¼ Cgi;k;s;t;u þ Cmi;k;s;t;u
(9)
max min C gi;k;s;u C gi;k;s;u min min þ C gi;k;s;u Cgi;k;s;t;u max * P P i;k;s;t i;k;s;u min P i;k;s;u P i;k;s;u þ 1 Ii;k;s;t;u *M
(10)
max min C gi;k;s;u C gi;k;s;u min * Pi;k;s;t P i;k;s;u max min P i;k;s;u P i;k;s;u þ 1 Ii;k;s;t;u *ðMÞ
(11)
max min C mi;k;s;u C mi;k;s;u min min þ C mi;k;s;u * P P Cmi;k;s;t;u max i;k;s;t i;k;s;u min P i;k;s;u P i;k;s;u þ 1 Ii;k;s;t;u *M
(12)
max min C mi;k;s;u C mi;k;s;u min min þ C mi;k;s;u * P P Cmi;k;s;t;u max i;k;s;t i;k;s;u min P i;k;s;u P i;k;s;u þ 1 Ii;k;s;t;u *ðMÞ
(13)
Cgi;k;s;t;u
2.1. Objective function
XXXX
max min C i;k;s;u C i;k;s;u min min * Pi;k;s;t P i;k;s;u þ C i;k;s;u max min P i;k;s;u P i;k;s;u þ 1 Ii;k;s;t;u *ðMÞ
Ci;k;s;t;u
min
þ C gi;k;s;u
1 Isi;k;s;t *ðMÞ Cgi;k;s;t;u 1 Isi;k;s;t *ðMÞ
(14)
Isi;k;s;t *ðMÞ Cmi;k;s;t;u Isi;k;s;t *ðMÞ
(15)
The number of startup and shut down of the units is defined by Eqs. (16) and (17).
SUi;k;s;t SDi;k;s;t ¼ Iai;k;s;t Iai;k;s;t1
(16)
SUi;k;s;t þ SDi;k;s;t 1
(17)
(2)
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2.2. Constraints The constraints to be observed can be categorized as: a) System constraints b) Dual-fuel and combined cycle units constraints c) Gas network constraints
2.2.1. System constraints 2.2.1.1. Line flow. DC load flow equations are defined as follows:
XX s
Pi;k;s;t lek;t þ
X
PLk;k* ;t ¼ 0
(18)
PLk;k* ;t ¼
1 Xk;k*
(19)
(20)
2.2.1.2. System adequacy. To achieve system reserve adequacy, a minimum reserve for each bus is considered. Eqs. (21)e(24) are implemented to prevent the reserve capture in electric buses. Eq. (21) shows that, the reserve in each bus should be greater than a specific amount with regard to transfer between that bus and the others. The reserve level and its transfer capability are described by Eq. (22).
XX
Ri;k;s;t þ
X
RLk;k* ;t Rmin k;t
(21)
k*
i
RLk;k* ;t PLmax k;k* PLk;k* ;t
(22)
RLk;k* ;t ¼ RLk* ;k;t
(23)
Ri;k;s;t 0
(24)
2.2.1.3. System security. The N1 criteria for transmission lines, is considered as the system security constraint. 2.2.1.4. Power plants generations. The generation and reserve of each power plant are restricted by Eqs. (25) and (26):
(25)
min Pi;k;s;t Iai;k;s;t *Pi;k;s :
(26)
Also, Eqs. (27) and (28) are applied to define the resultant reserve and output power from gas fueled units.
g Pi;k;s;t
Ri;k;s;t Rri;k;s
(31)
2.2.1.6. Min up time and min down time. Eqs. (32) and (33) describe the minimum up time and minimum down time of the units, respectively. off
tþTi;k;s X off off Ti;k;s * Iai;k;s;t Iai;k;s;tþ1 þ Iai;k;s;h Ti;k;s
þ
Rgi;k;s;t
max Iai;k;s;t Isi;k;s;t *Pi;k;s
Iai;k;s;t Isi;k;s;t
(32)
h ¼ tþ2
on Ti;k;s *
tþTi;k;s X on Iai;k;s;tþ1 Iai;k;s;t þ 1 Iai;k;s;h Ti;k;s
(33)
h ¼ tþ2
3. Dual-fuel & combined cycle units As a result of the growth in the price of natural gas, the installation of high efficiency units such as dual-fuel and combined cycle units has increased. Some research models these units in unit commitment problem. 3.1. Dual-fuel units Ref. [17] is the first paper which used seasonal switching of dualfuel units for minimizing greenhouse gases. The dual fuel and fuel blending units are modeled in the unit commitment problem in [11,18,19]. g If such a unit is on the gas fuel, the gas resultant power (Pi;k;s;t ) will be equal to (Pi;k;s;t ). Otherwise, if it is on the alternative fuel, the gas consumption will be zero while the electric power is still produced. Eqs. (34)e(36) are the constraints to be applied to these types of the units:
if
i˛Dual fuel units
if
g Isi;k;s;t ¼ 00Pi;k;s;t ¼ Pi;k;s;t
(34)
if
g ¼ 0 Isi;k;s;t ¼ 10Pi;k;s;t
(35)
h ¼X tþTSi;k;s
jIsi;k;s;h Isi;k;s;h1 j NSi;k;s :
(36)
h ¼ tþ1
max Pi;k;s;t þ Ri;k;s;t Iai;k;s;t *Pi;k;s
g Pi;k;s;t
(30)
on
* qk;t qk* ;t
max PLmax k;k* PLk;k* ;t PLk;k* :
s
min Pi;k;s;t1 Pi;k;s;t Rri;k;s *Iai;k;s;t1 þ Pi;k;s * 1 Iai;k;s;t1
k*
i
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min : *Pi;k;s
Isi;k;s;t Isi;k;s;t1 Zi;k;s;t
(37)
Isi;k;s;t1 Isi;k;s;t Zi;k;s;t
(38)
(27) h ¼X tþTSi;k;s
(28)
2.2.1.5. Ramp rates. The output power variation of each generation units is limited by its ramp rates as follows:
min Pi;k;s;t Pi;k;s;t1 Rri;k;s *Iai;k;s;t þ Pi;k;s * 1 Iai;k;s;t1
Eq. (36) is linearized to Eqs. (37)e(39) as follows:
(29)
Zi;k;s;t NSi;k;s :
(39)
h ¼ tþ1
3.2. Combined cycle units Combined cycle power plants consist of a set of gas turbines and a set of steam units. The output power of steam unit depends on the
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output power of the gas turbine units. Refs. [20e22] model combined cycle units, considering their roles in the unit commitment problem. These units are modeled based on CCM (Combined Cycle Modes) in Ref. [21]. In this model, each mode is considered as virtual generation units which can generate only one virtual unit in every moment. Also, transition between modes is limited to specific states. Fig. 1 demonstrates this state transition diagram. Inaccuracy and complexity of CCM model lead to propose CCC (Combined Cycle Component) model in Ref. [17]. The steam and gas turbine units are separately modeled there. The steam production of gas turbine units, which determines the limitation of steam unit maximum power, dependents on their output powers (Fig. 2). The principle of energy conservation is applied in this paper. The proposed model is a linear simplification of CCC model. The output power limitation of steam unit is determined based on units efficiency and output power of gas turbine units; as shown by Eqs. (40) and (41).
Pi;k;s;t
Pi0 ;k;s;t * 1 Ei0 ;k;s Ei0 ;k;s
Pi;k;s;t þ Ri;k;s;t
*Ei;k;s
Ei00 ;k;s
(40)
Pi0 ;k;s;t þ Ri0 ;k;s;t * 1 Ei0 ;k;s
þ
þ
! Pi00 ;k;s;t * 1 Ei00 ;k;s
Ei0 ;k;s
! Pi00 ;k;s;t þ Ri00 ;k;s;t * 1 Ei00 ;k;s
*Ei;k;s
Ei00 ;k;s
(41)
4. Gas network quasi dynamic model Two major models have been so far proposed for a gas network; a linear model which is independent from the gas nodes pressure and a nonlinear model which depends to it. The constraints in nonlinear models are classified as follows: The gas flow equations: B The sum of inlet and outlet gas flow to each should balance. B The inlet gas to a sending end of a pipeline equals to the gas outlet from its receiving end in each time snapshot of the problem. A gas pipeline flow equation is proportional to the pressure difference between its two sides. The pressure limits for each gas node should be observed. In all of these constraints, the independent variables are gas pipelines flows, calculated based on gas consumption and supply. On the other hand, the gas nodes pressures are the dependent variables derived from gas pipelines flows. In a linear model, the
Fig. 2. The relationship between gas turbines and steam unit of a combined cycle power plants.
calculations of gas network pressures are disregarded. The linear models are simple which can be solved fast and efficiently. Both linear and nonlinear models are of static type in which the gas velocity is ignored. In practice, due to this velocity, a delay, depending on the distance between gas nodes, would appear in dynamic conditions, which should be properly modeled and taken into account. In the proposed model, gas network is considered as set of areas, capable of supplying gas for non-power plant consumption, power plant consumption, reservation, and interaction with other areas. Difference between the gas input volume and non-power plant consumption demonstrates the amount of gas supply volume in each area. Also it is assumed that the gas reservoir in each area consists of the estimated gas volume within its connected pipelines, because in reality if an interruption occurs in the supply side, available gas in pipelines could supply gas for some hours. Interactions between areas are independent from gas pressure and only considered as SCF (Standard Cubic Feet). Gas network dynamics and the continuous impact of gas volume variation ahead of time imply that a proper time scale should be used to treat the trend of these variations. Unlike the time scale of a power system dynamics, the time scale of a gas network study is considered as a fraction of hour. This ratio is described by b parameter. Since gas fired power plants are the connection point of the gas network and the power system infrastructures, time scale adjustment in reserve and output power of power plants are implemented by Eqs. (42)e(44).
if
ðt 1Þ*b l t*b0
(42)
0GT , 0ST g
g
(43)
g
g
(44)
Pi;k;s;t ¼ Pppi;k;s;l 1GT , 0ST
2GT , 0ST
Ri;k;s;t ¼ Rppi;k;s;l
Quasi dynamic model considers gas flow velocity and distance between areas, with regard to gas flow direction. Thus Eqs. (45) and (46) model gas flow direction in pipelines. In addition, time delay between gas areas is shown by Eqs. (47) and (48). 1GT , 1ST
2GT , 1ST
Fig. 1. Combined cycle units operating modes.
Ifd
s;s* ;l
¼ Ifin * Ifout* s;s ;l
s;s ;l
(45)
M. Yazdani Damavandi et al. / Energy 36 (2011) 6243e6250
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Fig. 3. The relationship between a gas pipeline variables.
Ifin * þ Ifout* 1 s;s ;l
(46)
s;s ;l
Fig. 5. The output power of gas fueled units located in the 3rd gas area for both static and quasi dynamic models.
Pipe
as;s* y
Ls;s*
(47)
g
Ves;s*
h ¼ l þ Ifd
s:s* ;l
*as;s*
(48)
Eqs. (49)e(52) describe the gas flow interactions between areas. Transfer limitation between gas areas is demonstrated by Eqs. (49) and (50). Also the relationship between adjacent areas involving time delay is denoted by Eqs. (51) and (52). The Fig. 3 depicts the gas pipeline parameters and indicators. in in max 0 fs;s *fs;s * ;l If * *
(49)
out out max 0 fs;s *fs;s * ;l If * *
(50)
Ifout* ¼ Ifin *
(51)
out in fs;s * ;l ¼ fs;s* ;l
(52)
s;s ;l
s;s ;l
s;s ;l
s;s ;l
Eq.(53) demonstrates the volume of gas, consisting four parts:
Power plant gas supply Gas interaction between areas Reserved gas volume Power plants gas consumption
Regs;l Sgs;l
(54)
The reserve capability of each node is shown by Eqs. (55)e(57).
0 vs;l Vsc
(55)
vs;l vs;l1 Vsr
(56)
vs;l1 vs;l Vsr
(57)
5. Numerical results The Cplex optimizer is implemented to solve the proposed MIP (Mixed Integer Programming) model. The IEEE 14 bus system is used to test the proposed model. Electric and gas networks data and generation units characteristics are demonstrated in Appendix. Hourly time scale is used for power system study but in the gas network study, the time scale is based on 10 min interval. So each day consists of 144 intervals. Also the minimum electrical reserve is neglected for each bus. The electrical load curve is depicted by Fig. 4. Two tests are considered to investigate the effect of gas network quasi dynamic model in UC problem. 5.1. Test1
g
Rel;s
X s*
out fs;s * ;l þ
X s*
Two cases are taken into account in this study: in fs;s * ;l þ vs;l þ vs;l1
XX 1 g 1 g * Pppi;k;s;l þ Rppi;k;s;l * ¼ 0 k E gas i;k;s i
(53)
Case1) UC considering gas network static model. Case2) UC considering gas network quasi dynamic model.
k
Fig. 4. The electrical load curve.
Fig. 6. The output power of gas fueled units located in the 2nd and the 3rd gas areas for both static and quasi dynamic models.
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Fig. 7. The required gas in the 1st gas area for static and quasi dynamic models. Fig. 10. Required gas in the 1st area for both static and quasi dynamic models.
Fig. 8. The gas interaction curves between the 1st and the 2nd gas areas. Fig. 11. Required gas in the 1st area during 24 h.
The output power of gas fueled units located in the 3rd gas area for both cases is illustrated in Fig. 5. Also Fig. 6 shows these outputs for the 2nd and the 3rd gas areas. The output curves point out the similarity between generations in two cases. The gas requirement in the 1st gas area for two cases is demonstrated by Fig. 7. In static model, the pattern of the gas requirement in the 1st area is similar to the output power of gas fueled power plants. On the contrary, in quasi dynamic model these patterns are different. These differences depend on the gas velocity and pipelines lengths between the gas areas. These two important factors influence the interactional behavior between areas. Figs. 8 and 9 show the gas interaction curves between the 1st and the 2nd and the 2nd and the 3rd gas areas.
5.2. Test2 This study focuses on the role of dual fuel units and gas reservation to reduce the gas network dynamic effect. Three cases are considered as follows: Case1) UC considering gas network static model and 800 (KSCF/h) non-power plant gas consumption in time period 20e23 in the 3rd gas area.
Fig. 9. The gas interaction curves between the 2nd and the 3rd gas areas.
Case2) UC considering gas network quasi dynamic model and 800 (KSCF/h) non-power plant gas consumption in time period 20e23 in the 3rd gas area. Case3) Case 2 but with 200 (KSCF/h) gas reservoir in the 3rd area. In the first case, the non power plant gas consumption occurs in time period between 20 and 23, when the electric load is very low. Fig. 10 shows when static model is applied, the gas requirement in the 1st area changes only in this period and its differences in other time periods are negligible. On the contrary to the first case, as a result of gas network dynamic behavior, in the second one, the non-power plant consumption is shifted on the load peak period. Consequently, unit G41 uses its alternative fuel for compensating this shortage in time periods 8e24. The utilization of the second fuel results in lowering gas consumption level. Fig. 11 depicts required gas in the 1st area during 24 h. In the last case, this shortage is resolved by presence of appropriate reservoir in area 3. Therefore G41 can use its primary and cheaper fuel.
6. Conclusion In this paper, a gas network quasi dynamic model was proposed for modeling the impact of gas velocity and distances between gas areas in unit commitment problem. Also different kinds of generation units like simple gas fired units, combined cycle units, and dual fuel units were modeled. The application of proposed MILP model was demonstrated by some case studies. The numerical results showed the impact of gas velocity and distances between areas. It demonstrated that ignoring these factors may result in significant error in results. On the other hand the results depicted the impact of dual fuel units and the usage of alternative fuel in gas shortage periods. Although the alternative fuel has high price and increase the
M. Yazdani Damavandi et al. / Energy 36 (2011) 6243e6250
operation costs of the system, but can be a good replacement for gas fuel in contingency times.
Ifout
Pi;k;s;t
The binary variable (1 if gas flow direction is outlet otherwise zero). Three-state dependent variable (1 if the flow is from s to s* gas areas; 1 if the flow is in the opposite direction and 0 if there is no flow). Generation level of a unit (MW).
PLk;k ;t
Power transfer between electric buses (MW).
Pi;k;s;t
Generation level of a gas fueled unit (MW).
s;s ;l
Ifd
s;s ;l
Appendix A. List of symbols Indices i k l s & s* t u
Index of a generating unit. The electric bus of a generating unit. Gas network time scale (10 min). Gas area of a generating unit. Power system time scale (h). Segment index.
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g
g Ppp i;k;s;l
Output power based on gas network time scale (MW).
Ri;k;s;t
Reserve level of a unit (MW).
Rgi;k;s;t Rgppi;k;s;l
Reserve level of a gas fueled unit (MW).
Parameters max C i;k;s;u Maximum cost of a segment ($).
RLk;k ;t
Transfer capability of the reserve between areas (MW).
min C i;k;s;u
Minimum cost of a segment ($).
Res;l
Required gas volume in a gas area (KSCF/h).
Ei;k;s
Efficiency of a generating unit.
vs;l
Stored gas volume in a gas area (KSCF).
max fs;s
Gas volume transfer capability between gas areas (KSCF/h).
kgas
Gas to electric energy transformation coefficient.
Pipe
Ls;s lek;t M NSi;k;s
Distance between gas areas (mile). Electric demand of a bus (MW). Very large positive number. Maximum number of fuel switching for a unit.
max Pi;k;s
Maximum generation of a unit (MW).
min Pi;k;s
Minimum generation of a unit (MW).
P i;k;s;u
Minimum generation of a segment (MW).
max
min P i;k;s;u PLmax k;k Rmin k;t
Cost of turning on of a unit during the period ($).
Sgs;l
Gas volume, deliverable to a gas area (KSCF/h).
TSi;k;s
Time scale of fuel switching.
on Ti;k;s
Minimum up time of a unit (h).
off Ti;k;s Vsc Vsr g Ves;s
Minimum down time of unit (h). Gas reservoir capacity in a gas area (KSCF). Gas volume, transferred to or from a reservoir (KSCF/h). Gas velocity between gas areas (mile/10 min). Line impedance between buses. Voltage angel of bus. Time interval between gas areas. The ratio between gas and electric time scales.
Variables Ci;k;s;t;u Generation cost of a unit in each segment ($). Costi;k;s;t SUi;k;s;t SDi;k;s;t in fs;s ;l
Generation cost of a unit ($). Start up indicator of a generating unit. Shut down indicator of a generating unit. Gas volume transferred between gas areas (KSCF/h).
out fs;s ;l
Gas volume transferred between gas areas (KSCF/h).
Ii;k;s;t;u
Decision variable (1 if unit generation is in segment u, otherwise zero). Decision variable (1 if unit is available, otherwise zero). Decision variable (zero if unit is on gas fuel, 1 if is on the second fuel type). The binary variable (1 if gas flow direction is inlet otherwise zero).
Ifin
s;s ;l
Fig. B.1. IEEE 14 Bus Diagram.
Minimum reserve requirement for a bus (MW). Ramp rate of a unit (MW/h).
Iai;k;s;t Isi;k;s;t
Appendix B. Electric and gas networks data and generation units characteristics
Transmission capability between buses (MW).
c SUi;k;s
qk;t as;s* b
g
Maximum generation of a segment (MW).
Rri;k;s
Xk;k*
Generating unit reserve based on gas network time scale (MW).
Table B.1 Transmission Line Data. Line No.
From
To
X (P.u.)
Flow Limit (MW)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 2 2 2 3 4 4 4 5 6 6 6 7 7 9 9 10 12 13
2 5 3 4 5 4 5 7 9 6 11 12 13 8 9 10 14 11 13 14
0.05917 0.22304 0.19797 0.17632 0.17388 0.17103 0.04211 0. 20912 0.55618 0.25202 0.1989 0.25581 0.13027 0.17615 0.11001 0.0845 0.27038 0.19207 0.19988 0.34802
100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100
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Table B.2 Gas Area Interaction Data. Pipeline No.
From
To
Flow Limit (KSCF/h)
Length (mile)
Gas Average Velocity (mile/h)
1 2
1 2
2 3
3000 2500
70 75
50 45
Table B.3 The Data Of The Generators. Bus No. 1 2
ST GT GT
3 6
1 2
8
a ($/MW.h2)
b ($/MW.h)
c ($/h)
0.004683 0.021575 0.001534 0.001534 0.00473 0.00578 0.01184 0.00611
7.5629 5.92653 7.10787 7.10787 10.7154 13.28 27.28 19.02
140.15 245.894 509.781 509.781 143.0288 176.7752 176.7752 250.124
Table B.4 Generator Operation Data. Unit Bus Gas Pmax Pmin Ramp Min No. No. Node (MW) (MW) Rate Down (MW/h) Time No. (h) G11 G21 G22 G23 G31 G41 G51
1 2 2 2 3 6 8
1 1 1 1 2 3 3
100 70 50 50 120 100 200
20 10 10 10 20 20 50
40 30 30 30 40 40 40
3 3 3 3 1 3 4
Min Efficiency Dual Combined Up (%) Fuel Cycle Time (h) 2 1 3 2 2 2 4
36 40 40 40 40 38 53
0 0 0 0 0 1 0
e ST GT GT e e e
References [1] Li Z. Natural gas for generation: a solution or a problem. IEEE Power and Energy Magazine July 2005;3(4):16e21. [2] Chen H, Baldick R. Optmaizing short term natural gas supply portfolio for electric utility companies. IEEE Transaction on Power Systems February 2007; 22(1):232e9. [3] Pantos M. Market-based congestion management in electric power systems with increased share of natural gas dependent power plants. Elsevier, Energy Journal; May 2011:1e12. [4] Shahidehpour M, Fu Y, Wiedman Y. Impact of natural gas infrastructure on electric power systems. Proceedings of the IEEE May 2005;93(5):1042e56. [5] Koeppel G, Andersson G. Reliability modeling of multi-carrier energy systems. Elsevier, Energy Journal March 2009;34(3):235e44. [6] Unsihuay C, Marangon-Lima JW, de Souza ACZ. Integrated power generation and natural gas expansion planning. IEEE Power Tech conference Lausanne; July 2007:1404e9. [7] Bakken BH, Skjelbred HI, Wolfgang O. eTransport: Investment planning in energy supply systems with multiple energy carriers. Elsevier, Energy Journal September 2007;32(9):1676e89. [8] Mejia JM, Brugman A. Natural gas and electricity market issues in Colombia. IEEE Power Engineering Society General Meeting June 2005;2:1983e5. [9] Arroyo G, Madrigal M. Growing interactions between mexican gas markets and electricity system planning. IEEE Power Engineering Society General Meeting; June 2005:781e3. [10] Yazdani Damavandi M, Seifi H, Pedram MM. Generation units maintenance scheduling considering gas network constraints. IEEE International Conference on Electric Power and Energy Conversion Systems; 2009. [11] Li T, Eremia M, Shahidehpour M. Interdependency of natural gas network and power system security. IEEE Transaction on Power Systems November 2008; 23(4):1817e24. [12] Liu C, Shahidehpour M. Security-constrained unit commitment with natural gas transmission constraints. IEEE Transaction on Power Systems August 2009;24(3):1523e36. [13] An S, Li Q, Gedra TW. Natural gas and electricity optimal power flow. IEEE Transmission and Distribution Conference and Exposition September 2003; 11:138e43. [14] Arnold M, Andersson G. Decomposed electricity and natural gas optimal power flow. 16th Power Systems Computation Conference (PSCC), Glasgow, Scotland; July 2008. 1e7. [15] Shashi Menon E. Gas pipeline hydraulics. Boca Raton: Taylor and Francis Group; 2005.
[16] Mokhatab S, Poe WA, Speight JG. Handbook of natural gas transmission and processing. Burlington: Elsevier; 2006. [17] Tabors RD. Coal to natural gas seasonal fuel switching: an option for acid rain control. IEEE Transaction on Power Systems May 1989;4(2):457e62. [18] Lu B, Shahidehpour M. Unit commitment with flexible generating units. IEEE Transaction on Power Systems May 2005;20(2):1022e34. [19] Zhai A, Snyder W, Waight J, Farah J, Gonzalez A, Vallejo P. Fuel constrained unit commitment with fuel mixing and allocation. 22nd IEEE Power Engineering Society International Conference on Power Industry Computer Applications, PICA 2001:11e6. [20] Yan Y, Wen F, Wang J. Large-scale long-term generation scheduling with fuel limits and emission limits. IEEE Power and Energy Society General Meeting e Conversion and Delivery of Electrical Energy in the 21st Century; July 2008:1e5. [21] Lu B, Shahidehpour M. Short-term scheduling of combined cycle units. IEEE Transaction on Power Systems August 2004;19(3):1616e25. [22] Liu C, Shahidehpour M, Li Z, Fotuhi- Firuzabad M. Component and mode models for the short-term scheduling of combined-cycle units. IEEE Transaction on Power Systems May 2009;24(2):976e90.
Maziar Yazdani Damavandi was born in Iran, in 1985. He received his B.Sc. degree in Electrical Engineering in 2007 from Noshirvani Industrial University, Babol, Iran and M.Sc. degree in Electrical Engineering in 2010 from Tarbiat Modares University (TMU), Tehran Iran. Currently, he is an expert of the Iran Power System Engineering Research Center (IPSERC). His research interests are in power system planning, combined heat & power systems (CHP), Gas Network Modeling and Common Gas & Electric Retailers.
Iman Kiaei was born in Iran in 1986. He received the B.Sc. degree with honour in Electrical Engineering from Ferdowsi University of Mashhad, Mashhad, Iran in 2008 and the M.Sc. degree with honour in Electrical Engineering from Tarbiat Modares University, Tehran, Iran, in 2011, respectively. Currently, he is an expert of the Iran Power System Engineering Research Center (IPSERC). His main research interests are power system planning, coordination of energy infrastructures with unit commitment, power system restructuring, electricity market and power system reliability.
Mohammad Kazem Sheikh-EI-Eslami received his B.Sc. degree in Electrical Engineering from Tehran University, Tehran, Iran, in 1992, M.Sc. and Ph.D. degrees from Tarbiat Modares University, Tehran, Iran, in 2001 and 2005, respectively. Currently, he is an Assistant Professor in the Department of the Electrical Engineering, Tarbiat Modares University, Tehran, Iran. His research interests include power market simulation and generation expansion planning.
Hossein Seifi was born in Shiraz, Iran, in 1957. He received the B.Sc. degree from Shiraz University, Shiraz, Iran, in 1980 and the M.Sc. and Ph.D. degrees from the University of Manchester Institute of Science and Technology, Manchester, U.K., in 1987 and 1989, respectively. He then joined Tarbiat Modares University, Tehran, Iran, where he is currently a full Professor, and at the same time is the Head of the Iran Power System Engineering Research Center (IPSERC). His research interests include power system planning and operational issues.
Contents lists available at ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
New approach to gas network modeling in unit commitment Maziar Yazdani Damavandi, Iman Kiaei, Mohamad Kazem Sheikh-El-Eslami*, Hossein Seifi Faculty of Electrical and Computer Engineering, Tarbiat Modares University (TMU), Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 December 2010 Received in revised form 2 June 2011 Accepted 21 July 2011 Available online 24 August 2011
Expansion of gas fired units installation in power network results in more interdependency between gas and electric infrastructures. Thus new research has paid attention to the impact of gas network in power system studies. One such issue is the impact of gas network in a unit commitment problem. Many linear and nonlinear models are proposed to demonstrate this impact. However, the dynamics of these two infrastructures are different. In contrary to an electric network, the dynamic of a gas network lasts over a long period. In large scale networks, the impact of gas volume variations may remain for some hours or even for a whole day. Earlier research considered a gas network static model by which its dynamic behavior was not observed. In this paper, the quasi dynamic model is proposed for a gas network by which gas velocity and distances between gas areas may be considered. This model is assessed for a unit commitment problem on a test network and its capabilities are appreciated. 2011 Elsevier Ltd. All rights reserved.
Keywords: Combined cycle units Dual fuel units Gas network modeling Infrastructures planning Unit commitment
1. Introduction With the occurrence of energy crises in recent years and the importance of environmental issues, gas consumption has increased as a cheap and clean energy carrier. As a result of the advancements in gas turbine technology and utilization of combined cycle power plants, the installed capacity of these types of units is increased. In recent years, 98% of the constructed power plants in USA use gas fuel for generating the electrical power [1]. With ever increasing dependence on gas and power system infrastructures, the role of gas network in various power system studies has increased. Ref. [2] maximizes the profit of generating units, by simultaneously considering the gas and electricity contracts. Ref. [3] addresses market-based congestion management in electric power systems, taking into account the constraints of the electric power system and the natural gas system. The effects of gas network contingencies on power system security are discussed in Ref. [4]. Ref. [5] proposes a reliability model for multi-carrier energy systems with focus on the role of gas fuel as the most important energy carrier. The combined gas and electric infrastructure expansion planning is observed in Ref. [6]. In Ref. [7], a novel optimization model ‘eTransport’ is presented that takes into account both the topology
* Corresponding author: IPSERC e Tarbiat Modares University, Tehran, Iran. Tel./fax: þ98 21 88220121. E-mail addresses: [email protected] (M. Yazdani Damavandi), eiman. [email protected] (I. Kiaei), [email protected] (M. K. Sheikh-El-Eslami), seifi_ [email protected] (H. Seifi). 0360-5442/$ e see front matter 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2011.07.034
of multiple energy infrastructures such as gas network and the technical and economic properties of different investment alternatives. Furthermore, in Refs. [8,9], gas and electric markets are investigated in some countries. The effect of gas network on generating units maintenance scheduling is observed in Ref. [10]. Some research concentrates on the effect of gas network on unit commitment planning. The purpose of this planning is making an effort to specify unit participations in order to meet system constraints and to achieve higher social welfare. In Ref. [11], a linear model for gas network is proposed, including a set of nodes with predetermined inter-node transfer capability. The role of gas pressure is not considered and the gas transfer between nodes is assumed as SCF (Standard Cubic Foot). This model is extended in Ref. [12], and a nonlinear model for gas network based on the pressure difference between nodes is presented. This nonlinear model was already proposed in Refs. [3,13,14], for combined gas and electric network OPF (optimal power flow). The deficiency of these models is disregarding the impact of different dynamics of gas and electric networks. The power system dynamic is very fast and the variation in load levels is compensated in a short period so that the network reaches fast to the steady state condition. On the other hand, the slow velocity of natural gas results in remains its network dynamics even for some hours. The gas network infrastructure may affect both planning and operation of a power system. In terms of the planning phase, the generation units should be so allocated that their fuel requirements may be met through the gas network. The planning phase is not considered in this paper.
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On the other hand, a unit commitment problem is an operation issue in a power system with its time scale of several hours to a week or so. Its aim is finding out the units combinations to meet the hourly load variations in such a way that both technical and non-technical constraints are met. A generation unit operation is, typically, dependent on fuel (here, natural gas) availability. The fuel availability itself, depends on the gas network infrastructure. The gas network static model is suitable for the planning like gas and electric networks common OPF in one snapshot or limited and long term snapshots planning like gas and electric networks expansion. But the Refs. [15,16] determine these nonlinear equations are only valid in steady state condition and in unit commitment planning with hourly time snapshots the gas network is in transient condition. This paper proposes a MILP quasi dynamic model for gas network considering the gas velocity and distances between nodes. On the contrary of nonlinear models, the linear models are simple and the linear solvers are very efficient and fast. If a static model is used for this network, the unit commitment problem may result in improper results; implying either higher or lower cost than the actual that may be achieved in practice. That is why a dynamic gas network modeling is a must in unit commitment problem. This issue is addressed in this paper. The paper is organized as follows. In Section 2, the unit commitment problem is discussed. In Section 3, the dual-fuel and combined cycles units are modeled. The proposed gas network model is illustrated in Section 4. Numerical results are investigated in Section 5. Concluding remarks are provided in Section 6.
2. Unit commitment model The unit commitment problem may be formulated as an optimization problem. The objective function and various constraints of this problem are discussed as follows.
Minimization of system operation costs is the most important objective of unit commitment problem. The major portion of the system operation costs is power plants costs which consists of the units generation and start up costs. Therefore the proposed objective function consists of two terms. The first term is the generation costs which is derived from units cost functions. Due to the important role of dual fuel units in a power network, these units are modeled in this paper. Therefore two cost function curves are namely, one for natural gas and the other for alternative fuel considered. The second term is the start up cost, calculated from multiply of single cold startup cost to number of startup.
t
s
k
i
Costi;k;s;t þ
XXXX s
k
i
c SUi;k;s *SUi;k;s;t
(1)
t
The generation cost curves of the units are assumed to be as piecewise linear functions. Eqs. (2) and (3) show the operation cost of each unit in terms of the unit output power. Eqs. (4)e(6) model the linear segments limitations of output power and operation cost for each segment. Eq. (7) demonstrates the total cost of each generation unit for a specified time scale. The relationship between units availability and linearized segment indicators is represented by Eq. (8). max min C i;k;s;u C i;k;s;u min * P P i;k;s;t i;k;s;u max min P i;k;s;u P i;k;s;u min þ C i;k;s;u þ 1 Ii;k;s;t;u *M
Ci;k;s;t;u
(3)
Ii;k;s;t;u *ðMÞ Ci;k;s;t;u Ii;k;s;t;u *ðMÞ
(4)
max Pi;k;s;t Ii;k;s;t;u *P i;k;s;u þ 1 Ii;k;s;t;u *M
(5)
min Pi;k;s;t Ii;k;s;t;u *P i;k;s;u þ 1 Ii;k;s;t;u *ðMÞ
(6)
Costi;k;s;t ¼
X
Ci;k;s;t;u
(7)
u
X
Ii;k;s;t;u ¼ Ia
(8)
u
The dual-fuel unit generation cost consists of two terms. The first term shows the generation cost of gas fuel (modeled by Eqs. (10) and (11)) and the second term shows the generation cost based on alternative fuel (modeled by Eqs. (12) and (13)). Eqs. (14) and (15) limit operation cost of units considering the value of the dual fuel indicator.
Ci;k;s;t;u ¼ Cgi;k;s;t;u þ Cmi;k;s;t;u
(9)
max min C gi;k;s;u C gi;k;s;u min min þ C gi;k;s;u Cgi;k;s;t;u max * P P i;k;s;t i;k;s;u min P i;k;s;u P i;k;s;u þ 1 Ii;k;s;t;u *M
(10)
max min C gi;k;s;u C gi;k;s;u min * Pi;k;s;t P i;k;s;u max min P i;k;s;u P i;k;s;u þ 1 Ii;k;s;t;u *ðMÞ
(11)
max min C mi;k;s;u C mi;k;s;u min min þ C mi;k;s;u * P P Cmi;k;s;t;u max i;k;s;t i;k;s;u min P i;k;s;u P i;k;s;u þ 1 Ii;k;s;t;u *M
(12)
max min C mi;k;s;u C mi;k;s;u min min þ C mi;k;s;u * P P Cmi;k;s;t;u max i;k;s;t i;k;s;u min P i;k;s;u P i;k;s;u þ 1 Ii;k;s;t;u *ðMÞ
(13)
Cgi;k;s;t;u
2.1. Objective function
XXXX
max min C i;k;s;u C i;k;s;u min min * Pi;k;s;t P i;k;s;u þ C i;k;s;u max min P i;k;s;u P i;k;s;u þ 1 Ii;k;s;t;u *ðMÞ
Ci;k;s;t;u
min
þ C gi;k;s;u
1 Isi;k;s;t *ðMÞ Cgi;k;s;t;u 1 Isi;k;s;t *ðMÞ
(14)
Isi;k;s;t *ðMÞ Cmi;k;s;t;u Isi;k;s;t *ðMÞ
(15)
The number of startup and shut down of the units is defined by Eqs. (16) and (17).
SUi;k;s;t SDi;k;s;t ¼ Iai;k;s;t Iai;k;s;t1
(16)
SUi;k;s;t þ SDi;k;s;t 1
(17)
(2)
M. Yazdani Damavandi et al. / Energy 36 (2011) 6243e6250
2.2. Constraints The constraints to be observed can be categorized as: a) System constraints b) Dual-fuel and combined cycle units constraints c) Gas network constraints
2.2.1. System constraints 2.2.1.1. Line flow. DC load flow equations are defined as follows:
XX s
Pi;k;s;t lek;t þ
X
PLk;k* ;t ¼ 0
(18)
PLk;k* ;t ¼
1 Xk;k*
(19)
(20)
2.2.1.2. System adequacy. To achieve system reserve adequacy, a minimum reserve for each bus is considered. Eqs. (21)e(24) are implemented to prevent the reserve capture in electric buses. Eq. (21) shows that, the reserve in each bus should be greater than a specific amount with regard to transfer between that bus and the others. The reserve level and its transfer capability are described by Eq. (22).
XX
Ri;k;s;t þ
X
RLk;k* ;t Rmin k;t
(21)
k*
i
RLk;k* ;t PLmax k;k* PLk;k* ;t
(22)
RLk;k* ;t ¼ RLk* ;k;t
(23)
Ri;k;s;t 0
(24)
2.2.1.3. System security. The N1 criteria for transmission lines, is considered as the system security constraint. 2.2.1.4. Power plants generations. The generation and reserve of each power plant are restricted by Eqs. (25) and (26):
(25)
min Pi;k;s;t Iai;k;s;t *Pi;k;s :
(26)
Also, Eqs. (27) and (28) are applied to define the resultant reserve and output power from gas fueled units.
g Pi;k;s;t
Ri;k;s;t Rri;k;s
(31)
2.2.1.6. Min up time and min down time. Eqs. (32) and (33) describe the minimum up time and minimum down time of the units, respectively. off
tþTi;k;s X off off Ti;k;s * Iai;k;s;t Iai;k;s;tþ1 þ Iai;k;s;h Ti;k;s
þ
Rgi;k;s;t
max Iai;k;s;t Isi;k;s;t *Pi;k;s
Iai;k;s;t Isi;k;s;t
(32)
h ¼ tþ2
on Ti;k;s *
tþTi;k;s X on Iai;k;s;tþ1 Iai;k;s;t þ 1 Iai;k;s;h Ti;k;s
(33)
h ¼ tþ2
3. Dual-fuel & combined cycle units As a result of the growth in the price of natural gas, the installation of high efficiency units such as dual-fuel and combined cycle units has increased. Some research models these units in unit commitment problem. 3.1. Dual-fuel units Ref. [17] is the first paper which used seasonal switching of dualfuel units for minimizing greenhouse gases. The dual fuel and fuel blending units are modeled in the unit commitment problem in [11,18,19]. g If such a unit is on the gas fuel, the gas resultant power (Pi;k;s;t ) will be equal to (Pi;k;s;t ). Otherwise, if it is on the alternative fuel, the gas consumption will be zero while the electric power is still produced. Eqs. (34)e(36) are the constraints to be applied to these types of the units:
if
i˛Dual fuel units
if
g Isi;k;s;t ¼ 00Pi;k;s;t ¼ Pi;k;s;t
(34)
if
g ¼ 0 Isi;k;s;t ¼ 10Pi;k;s;t
(35)
h ¼X tþTSi;k;s
jIsi;k;s;h Isi;k;s;h1 j NSi;k;s :
(36)
h ¼ tþ1
max Pi;k;s;t þ Ri;k;s;t Iai;k;s;t *Pi;k;s
g Pi;k;s;t
(30)
on
* qk;t qk* ;t
max PLmax k;k* PLk;k* ;t PLk;k* :
s
min Pi;k;s;t1 Pi;k;s;t Rri;k;s *Iai;k;s;t1 þ Pi;k;s * 1 Iai;k;s;t1
k*
i
6245
min : *Pi;k;s
Isi;k;s;t Isi;k;s;t1 Zi;k;s;t
(37)
Isi;k;s;t1 Isi;k;s;t Zi;k;s;t
(38)
(27) h ¼X tþTSi;k;s
(28)
2.2.1.5. Ramp rates. The output power variation of each generation units is limited by its ramp rates as follows:
min Pi;k;s;t Pi;k;s;t1 Rri;k;s *Iai;k;s;t þ Pi;k;s * 1 Iai;k;s;t1
Eq. (36) is linearized to Eqs. (37)e(39) as follows:
(29)
Zi;k;s;t NSi;k;s :
(39)
h ¼ tþ1
3.2. Combined cycle units Combined cycle power plants consist of a set of gas turbines and a set of steam units. The output power of steam unit depends on the
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M. Yazdani Damavandi et al. / Energy 36 (2011) 6243e6250
output power of the gas turbine units. Refs. [20e22] model combined cycle units, considering their roles in the unit commitment problem. These units are modeled based on CCM (Combined Cycle Modes) in Ref. [21]. In this model, each mode is considered as virtual generation units which can generate only one virtual unit in every moment. Also, transition between modes is limited to specific states. Fig. 1 demonstrates this state transition diagram. Inaccuracy and complexity of CCM model lead to propose CCC (Combined Cycle Component) model in Ref. [17]. The steam and gas turbine units are separately modeled there. The steam production of gas turbine units, which determines the limitation of steam unit maximum power, dependents on their output powers (Fig. 2). The principle of energy conservation is applied in this paper. The proposed model is a linear simplification of CCC model. The output power limitation of steam unit is determined based on units efficiency and output power of gas turbine units; as shown by Eqs. (40) and (41).
Pi;k;s;t
Pi0 ;k;s;t * 1 Ei0 ;k;s Ei0 ;k;s
Pi;k;s;t þ Ri;k;s;t
*Ei;k;s
Ei00 ;k;s
(40)
Pi0 ;k;s;t þ Ri0 ;k;s;t * 1 Ei0 ;k;s
þ
þ
! Pi00 ;k;s;t * 1 Ei00 ;k;s
Ei0 ;k;s
! Pi00 ;k;s;t þ Ri00 ;k;s;t * 1 Ei00 ;k;s
*Ei;k;s
Ei00 ;k;s
(41)
4. Gas network quasi dynamic model Two major models have been so far proposed for a gas network; a linear model which is independent from the gas nodes pressure and a nonlinear model which depends to it. The constraints in nonlinear models are classified as follows: The gas flow equations: B The sum of inlet and outlet gas flow to each should balance. B The inlet gas to a sending end of a pipeline equals to the gas outlet from its receiving end in each time snapshot of the problem. A gas pipeline flow equation is proportional to the pressure difference between its two sides. The pressure limits for each gas node should be observed. In all of these constraints, the independent variables are gas pipelines flows, calculated based on gas consumption and supply. On the other hand, the gas nodes pressures are the dependent variables derived from gas pipelines flows. In a linear model, the
Fig. 2. The relationship between gas turbines and steam unit of a combined cycle power plants.
calculations of gas network pressures are disregarded. The linear models are simple which can be solved fast and efficiently. Both linear and nonlinear models are of static type in which the gas velocity is ignored. In practice, due to this velocity, a delay, depending on the distance between gas nodes, would appear in dynamic conditions, which should be properly modeled and taken into account. In the proposed model, gas network is considered as set of areas, capable of supplying gas for non-power plant consumption, power plant consumption, reservation, and interaction with other areas. Difference between the gas input volume and non-power plant consumption demonstrates the amount of gas supply volume in each area. Also it is assumed that the gas reservoir in each area consists of the estimated gas volume within its connected pipelines, because in reality if an interruption occurs in the supply side, available gas in pipelines could supply gas for some hours. Interactions between areas are independent from gas pressure and only considered as SCF (Standard Cubic Feet). Gas network dynamics and the continuous impact of gas volume variation ahead of time imply that a proper time scale should be used to treat the trend of these variations. Unlike the time scale of a power system dynamics, the time scale of a gas network study is considered as a fraction of hour. This ratio is described by b parameter. Since gas fired power plants are the connection point of the gas network and the power system infrastructures, time scale adjustment in reserve and output power of power plants are implemented by Eqs. (42)e(44).
if
ðt 1Þ*b l t*b0
(42)
0GT , 0ST g
g
(43)
g
g
(44)
Pi;k;s;t ¼ Pppi;k;s;l 1GT , 0ST
2GT , 0ST
Ri;k;s;t ¼ Rppi;k;s;l
Quasi dynamic model considers gas flow velocity and distance between areas, with regard to gas flow direction. Thus Eqs. (45) and (46) model gas flow direction in pipelines. In addition, time delay between gas areas is shown by Eqs. (47) and (48). 1GT , 1ST
2GT , 1ST
Fig. 1. Combined cycle units operating modes.
Ifd
s;s* ;l
¼ Ifin * Ifout* s;s ;l
s;s ;l
(45)
M. Yazdani Damavandi et al. / Energy 36 (2011) 6243e6250
6247
Fig. 3. The relationship between a gas pipeline variables.
Ifin * þ Ifout* 1 s;s ;l
(46)
s;s ;l
Fig. 5. The output power of gas fueled units located in the 3rd gas area for both static and quasi dynamic models.
Pipe
as;s* y
Ls;s*
(47)
g
Ves;s*
h ¼ l þ Ifd
s:s* ;l
*as;s*
(48)
Eqs. (49)e(52) describe the gas flow interactions between areas. Transfer limitation between gas areas is demonstrated by Eqs. (49) and (50). Also the relationship between adjacent areas involving time delay is denoted by Eqs. (51) and (52). The Fig. 3 depicts the gas pipeline parameters and indicators. in in max 0 fs;s *fs;s * ;l If * *
(49)
out out max 0 fs;s *fs;s * ;l If * *
(50)
Ifout* ¼ Ifin *
(51)
out in fs;s * ;l ¼ fs;s* ;l
(52)
s;s ;l
s;s ;l
s;s ;l
s;s ;l
Eq.(53) demonstrates the volume of gas, consisting four parts:
Power plant gas supply Gas interaction between areas Reserved gas volume Power plants gas consumption
Regs;l Sgs;l
(54)
The reserve capability of each node is shown by Eqs. (55)e(57).
0 vs;l Vsc
(55)
vs;l vs;l1 Vsr
(56)
vs;l1 vs;l Vsr
(57)
5. Numerical results The Cplex optimizer is implemented to solve the proposed MIP (Mixed Integer Programming) model. The IEEE 14 bus system is used to test the proposed model. Electric and gas networks data and generation units characteristics are demonstrated in Appendix. Hourly time scale is used for power system study but in the gas network study, the time scale is based on 10 min interval. So each day consists of 144 intervals. Also the minimum electrical reserve is neglected for each bus. The electrical load curve is depicted by Fig. 4. Two tests are considered to investigate the effect of gas network quasi dynamic model in UC problem. 5.1. Test1
g
Rel;s
X s*
out fs;s * ;l þ
X s*
Two cases are taken into account in this study: in fs;s * ;l þ vs;l þ vs;l1
XX 1 g 1 g * Pppi;k;s;l þ Rppi;k;s;l * ¼ 0 k E gas i;k;s i
(53)
Case1) UC considering gas network static model. Case2) UC considering gas network quasi dynamic model.
k
Fig. 4. The electrical load curve.
Fig. 6. The output power of gas fueled units located in the 2nd and the 3rd gas areas for both static and quasi dynamic models.
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Fig. 7. The required gas in the 1st gas area for static and quasi dynamic models. Fig. 10. Required gas in the 1st area for both static and quasi dynamic models.
Fig. 8. The gas interaction curves between the 1st and the 2nd gas areas. Fig. 11. Required gas in the 1st area during 24 h.
The output power of gas fueled units located in the 3rd gas area for both cases is illustrated in Fig. 5. Also Fig. 6 shows these outputs for the 2nd and the 3rd gas areas. The output curves point out the similarity between generations in two cases. The gas requirement in the 1st gas area for two cases is demonstrated by Fig. 7. In static model, the pattern of the gas requirement in the 1st area is similar to the output power of gas fueled power plants. On the contrary, in quasi dynamic model these patterns are different. These differences depend on the gas velocity and pipelines lengths between the gas areas. These two important factors influence the interactional behavior between areas. Figs. 8 and 9 show the gas interaction curves between the 1st and the 2nd and the 2nd and the 3rd gas areas.
5.2. Test2 This study focuses on the role of dual fuel units and gas reservation to reduce the gas network dynamic effect. Three cases are considered as follows: Case1) UC considering gas network static model and 800 (KSCF/h) non-power plant gas consumption in time period 20e23 in the 3rd gas area.
Fig. 9. The gas interaction curves between the 2nd and the 3rd gas areas.
Case2) UC considering gas network quasi dynamic model and 800 (KSCF/h) non-power plant gas consumption in time period 20e23 in the 3rd gas area. Case3) Case 2 but with 200 (KSCF/h) gas reservoir in the 3rd area. In the first case, the non power plant gas consumption occurs in time period between 20 and 23, when the electric load is very low. Fig. 10 shows when static model is applied, the gas requirement in the 1st area changes only in this period and its differences in other time periods are negligible. On the contrary to the first case, as a result of gas network dynamic behavior, in the second one, the non-power plant consumption is shifted on the load peak period. Consequently, unit G41 uses its alternative fuel for compensating this shortage in time periods 8e24. The utilization of the second fuel results in lowering gas consumption level. Fig. 11 depicts required gas in the 1st area during 24 h. In the last case, this shortage is resolved by presence of appropriate reservoir in area 3. Therefore G41 can use its primary and cheaper fuel.
6. Conclusion In this paper, a gas network quasi dynamic model was proposed for modeling the impact of gas velocity and distances between gas areas in unit commitment problem. Also different kinds of generation units like simple gas fired units, combined cycle units, and dual fuel units were modeled. The application of proposed MILP model was demonstrated by some case studies. The numerical results showed the impact of gas velocity and distances between areas. It demonstrated that ignoring these factors may result in significant error in results. On the other hand the results depicted the impact of dual fuel units and the usage of alternative fuel in gas shortage periods. Although the alternative fuel has high price and increase the
M. Yazdani Damavandi et al. / Energy 36 (2011) 6243e6250
operation costs of the system, but can be a good replacement for gas fuel in contingency times.
Ifout
Pi;k;s;t
The binary variable (1 if gas flow direction is outlet otherwise zero). Three-state dependent variable (1 if the flow is from s to s* gas areas; 1 if the flow is in the opposite direction and 0 if there is no flow). Generation level of a unit (MW).
PLk;k ;t
Power transfer between electric buses (MW).
Pi;k;s;t
Generation level of a gas fueled unit (MW).
s;s ;l
Ifd
s;s ;l
Appendix A. List of symbols Indices i k l s & s* t u
Index of a generating unit. The electric bus of a generating unit. Gas network time scale (10 min). Gas area of a generating unit. Power system time scale (h). Segment index.
6249
g
g Ppp i;k;s;l
Output power based on gas network time scale (MW).
Ri;k;s;t
Reserve level of a unit (MW).
Rgi;k;s;t Rgppi;k;s;l
Reserve level of a gas fueled unit (MW).
Parameters max C i;k;s;u Maximum cost of a segment ($).
RLk;k ;t
Transfer capability of the reserve between areas (MW).
min C i;k;s;u
Minimum cost of a segment ($).
Res;l
Required gas volume in a gas area (KSCF/h).
Ei;k;s
Efficiency of a generating unit.
vs;l
Stored gas volume in a gas area (KSCF).
max fs;s
Gas volume transfer capability between gas areas (KSCF/h).
kgas
Gas to electric energy transformation coefficient.
Pipe
Ls;s lek;t M NSi;k;s
Distance between gas areas (mile). Electric demand of a bus (MW). Very large positive number. Maximum number of fuel switching for a unit.
max Pi;k;s
Maximum generation of a unit (MW).
min Pi;k;s
Minimum generation of a unit (MW).
P i;k;s;u
Minimum generation of a segment (MW).
max
min P i;k;s;u PLmax k;k Rmin k;t
Cost of turning on of a unit during the period ($).
Sgs;l
Gas volume, deliverable to a gas area (KSCF/h).
TSi;k;s
Time scale of fuel switching.
on Ti;k;s
Minimum up time of a unit (h).
off Ti;k;s Vsc Vsr g Ves;s
Minimum down time of unit (h). Gas reservoir capacity in a gas area (KSCF). Gas volume, transferred to or from a reservoir (KSCF/h). Gas velocity between gas areas (mile/10 min). Line impedance between buses. Voltage angel of bus. Time interval between gas areas. The ratio between gas and electric time scales.
Variables Ci;k;s;t;u Generation cost of a unit in each segment ($). Costi;k;s;t SUi;k;s;t SDi;k;s;t in fs;s ;l
Generation cost of a unit ($). Start up indicator of a generating unit. Shut down indicator of a generating unit. Gas volume transferred between gas areas (KSCF/h).
out fs;s ;l
Gas volume transferred between gas areas (KSCF/h).
Ii;k;s;t;u
Decision variable (1 if unit generation is in segment u, otherwise zero). Decision variable (1 if unit is available, otherwise zero). Decision variable (zero if unit is on gas fuel, 1 if is on the second fuel type). The binary variable (1 if gas flow direction is inlet otherwise zero).
Ifin
s;s ;l
Fig. B.1. IEEE 14 Bus Diagram.
Minimum reserve requirement for a bus (MW). Ramp rate of a unit (MW/h).
Iai;k;s;t Isi;k;s;t
Appendix B. Electric and gas networks data and generation units characteristics
Transmission capability between buses (MW).
c SUi;k;s
qk;t as;s* b
g
Maximum generation of a segment (MW).
Rri;k;s
Xk;k*
Generating unit reserve based on gas network time scale (MW).
Table B.1 Transmission Line Data. Line No.
From
To
X (P.u.)
Flow Limit (MW)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 2 2 2 3 4 4 4 5 6 6 6 7 7 9 9 10 12 13
2 5 3 4 5 4 5 7 9 6 11 12 13 8 9 10 14 11 13 14
0.05917 0.22304 0.19797 0.17632 0.17388 0.17103 0.04211 0. 20912 0.55618 0.25202 0.1989 0.25581 0.13027 0.17615 0.11001 0.0845 0.27038 0.19207 0.19988 0.34802
100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100
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Table B.2 Gas Area Interaction Data. Pipeline No.
From
To
Flow Limit (KSCF/h)
Length (mile)
Gas Average Velocity (mile/h)
1 2
1 2
2 3
3000 2500
70 75
50 45
Table B.3 The Data Of The Generators. Bus No. 1 2
ST GT GT
3 6
1 2
8
a ($/MW.h2)
b ($/MW.h)
c ($/h)
0.004683 0.021575 0.001534 0.001534 0.00473 0.00578 0.01184 0.00611
7.5629 5.92653 7.10787 7.10787 10.7154 13.28 27.28 19.02
140.15 245.894 509.781 509.781 143.0288 176.7752 176.7752 250.124
Table B.4 Generator Operation Data. Unit Bus Gas Pmax Pmin Ramp Min No. No. Node (MW) (MW) Rate Down (MW/h) Time No. (h) G11 G21 G22 G23 G31 G41 G51
1 2 2 2 3 6 8
1 1 1 1 2 3 3
100 70 50 50 120 100 200
20 10 10 10 20 20 50
40 30 30 30 40 40 40
3 3 3 3 1 3 4
Min Efficiency Dual Combined Up (%) Fuel Cycle Time (h) 2 1 3 2 2 2 4
36 40 40 40 40 38 53
0 0 0 0 0 1 0
e ST GT GT e e e
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Maziar Yazdani Damavandi was born in Iran, in 1985. He received his B.Sc. degree in Electrical Engineering in 2007 from Noshirvani Industrial University, Babol, Iran and M.Sc. degree in Electrical Engineering in 2010 from Tarbiat Modares University (TMU), Tehran Iran. Currently, he is an expert of the Iran Power System Engineering Research Center (IPSERC). His research interests are in power system planning, combined heat & power systems (CHP), Gas Network Modeling and Common Gas & Electric Retailers.
Iman Kiaei was born in Iran in 1986. He received the B.Sc. degree with honour in Electrical Engineering from Ferdowsi University of Mashhad, Mashhad, Iran in 2008 and the M.Sc. degree with honour in Electrical Engineering from Tarbiat Modares University, Tehran, Iran, in 2011, respectively. Currently, he is an expert of the Iran Power System Engineering Research Center (IPSERC). His main research interests are power system planning, coordination of energy infrastructures with unit commitment, power system restructuring, electricity market and power system reliability.
Mohammad Kazem Sheikh-EI-Eslami received his B.Sc. degree in Electrical Engineering from Tehran University, Tehran, Iran, in 1992, M.Sc. and Ph.D. degrees from Tarbiat Modares University, Tehran, Iran, in 2001 and 2005, respectively. Currently, he is an Assistant Professor in the Department of the Electrical Engineering, Tarbiat Modares University, Tehran, Iran. His research interests include power market simulation and generation expansion planning.
Hossein Seifi was born in Shiraz, Iran, in 1957. He received the B.Sc. degree from Shiraz University, Shiraz, Iran, in 1980 and the M.Sc. and Ph.D. degrees from the University of Manchester Institute of Science and Technology, Manchester, U.K., in 1987 and 1989, respectively. He then joined Tarbiat Modares University, Tehran, Iran, where he is currently a full Professor, and at the same time is the Head of the Iran Power System Engineering Research Center (IPSERC). His research interests include power system planning and operational issues.