Stochastic Security-Constrained Scheduling of ... - IEEE Xplore

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Stochastic Security-Constrained Scheduling of Coordinated Electricity and Natural Gas Infrastructures Ahmed Alabdulwahab, Abdullah Abusorrah, Xiaping Zhang, and Mohammad Shahidehpour, Fellow, IEEE

Abstract—This paper proposes a coordinated stochastic model for studying the interdependence of electricity and natural gas transmission networks (referred to as EGTran). The coordinated model incorporates the stochastic power system conditions into the solution of security-constrained unit commitment problem with natural gas network constraints. The stochastic model considers random outages of generating units and transmission lines, as well as hourly forecast errors of day-ahead electricity load. The Monte Carlo simulation is applied to create multiple scenarios for the simulation of the uncertainties in the EGTran model. The nonlinear natural gas network constraints are converted into linear constraints and incorporated into the stochastic model. Numerical tests are performed in a six-bus system with a seven-node gas transmission network and the IEEE 118–bus power system with a ten-node gas transmission network. Numerical results demonstrate the effectiveness of EGTran to analyze the impact of random contingencies on power system operations with natural gas network constraints. The proposed EGTran model could be utilized by grid operators for the short-term commitment and dispatch of power systems in highly interdependent conditions with relatively large natural gas-fired generating units. Index Terms—Load forecasting, mixed-integer programming, natural gas and electricity infrastructures, optimization, stochastic security-constrained unit commitment (SCUC), uncertainty.

N OMENCLATURE Index: S i sp l br b t

Index of scenarios. Index of generating units. Index of natural gas suppliers. Index of natural gas loads. Index of electricity transmission branches. Index of buses. Index of hours.

Manuscript received March 31, 2014; revised November 4, 2014, January 12, 2015, and March 26, 2015; accepted April 3, 2015. This work was supported in part by the Deanship of Scientific Research at King Abdulaziz University in Saudi Arabia under Grant Gr/34/6. A. Alabdulwahab and A. Abusorrah are with the Department of Electrical Engineering and Computer Engineering and the Renewable Energy Research Group, King Abdulaziz University, Jeddah 21589, Saudi Arabia. X. Zhang is with the Robert W. Galvin Center for Electricity Innovation, Illinois Institute of Technology, Chicago, IL 60616 USA. M. Shahidehpour is with the Robert W. Galvin Center for Electricity Innovation, Illinois Institute of Technology, Chicago, IL 60616 USA, and also with the Renewable Energy Research Group, King Abdulaziz University, Jeddah 22254, Saudi Arabia. Digital Object Identifier 10.1109/JSYST.2015.2423498

η m, n j, k p, q

Index of natural gas supply contracts. Index of nodes in natural gas network. Index of buses in electricity network. Index of triangle.

Variables: Iit Pit SUit , SDit on Xi(t−1) off Xi(t−1) c Fi W F gas Vsp Ll π fmn pfbr Θ qp

Status indicator of generating unit i. Generation dispatch of unit i. Start-up and shutdown cost of unit i. On time of unit i at time t, in hours. Off time of unit i at time t, in hours. Cost function of generating unit i. Cost of natural gas contract. Natural gas consumption of gas-fired unit. Amount of natural gas delivery of supplier sp. Natural gas load of load l. Pressure of node. Natural gas flow from node m to n. Power flow through branch br. Bus voltage angle. Binary indicator variable of the pth triangle.

Constants: PS NS NB NT Xjk pfmax br Pimin, Pimax Riup, Ridn Tion , Tioff PL PLoss ρgas Fηo NGS NGL GU GC(m) GP Lmin , Lmax π min , π max V min, V max

Probability of scenario s. Number of scenarios. Number of load buses. Number of time periods. Reactance between bus j and k. Power flow limits of branch br. Minimum and maximum capacity of unit i. Up/down limits for corrective dispatch of unit i. Minimum on/off time of unit i. Forecast load of power system. Estimated loss in power transmission system. Price of natural gas. Maximum daily natural gas quantity. Number of natural gas suppliers. Number of natural gas loads. Set of natural gas-fired generating units. Set of nodes connected with m. Set of node pair (m, n) for pipeline from node m to node n. Minimum and maximum of natural gas load. Mxinimum and maximum pressure. Minimum and maximum of natural gas injection.

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NP a P , bP , c P A B C D E

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Number of triangles. Constants in the pth triangle. Node-gas provider incidence matrix. Node-gas load incidence matrix. Bus-generator incidence matrix. Bus-electrical load incidence matrix. Bus-branch incidence matrix. I. I NTRODUCTION

T

HE annual natural gas consumption for power generation in the USA increased more than 30% from 2007 to 2012 [1]. Natural-gas fired units account for 30% of the total U.S. power generation in 2012 as natural gas prices continued to drop. The other impetus of this trend is the tougher emission regulation that has accelerated the scheduled retirement of coalfired plants. The natural gas generation would produce no SO2 emissions and substantially less NOx than coal and oil plants. The natural gas-fired units, which offer high efficiency and reduced construction time, could also support the variability of renewable energy units in a highly uncertain environment [2]. Natural gas plants provide an important linkage between natural gas and electric power transmission systems. In electricity markets, security-constrained unit commitment (SCUC) is applied to minimize the generation cost and determine the unit dispatch for satisfying power network security constraints. The rapidly increasing natural gas-fired unit capacity highlights the interdependence of gas and electricity [3]. Unlike other generating plants, natural gas plants operate on a just-in-time-inventory principle with natural gas purchases and the scheduling of deliveries and transportation on the natural gas pipeline system. There are three typical natural gas transportation service contracts made to natural gas markets, local distribution companies, or end users. These contracts include firm, no-notice, and interruptible transportation. Power generation customers usually choose to rely on interruptible natural gas transportation service, which is priced solely on a volumetric basis without fixed monthly reservation fee. The natural gas delivery for power generation with interruptible services has a lower priority than other nonpower enduse loads with firm services. The availability of interruptible capacity could decline as nonpower gas consumption grows, particularly in severe weather situations when the demand for electricity and natural gas would peak at the same time. This situation would result in fuel shortages, higher marginal costs for natural gas-fired generating units, and eventually higher market prices for electricity. Meanwhile, volatile natural gas prices would affect the hourly commitment and dispatch of generating units and the operation cost of supplying the hourly load. In addition, an interruption or pressure loss in the natural gas pipeline system may lead to a loss of multiple natural gas power plants, which could directly affect the electric power supply availability and power system security. Such conditions necessitate the coordinated modeling of interdependent natural gas and electricity networks. However, the preliminary studies on the economics and the security of coordinated natural gas and electricity systems were conducted by grid operators [4].

The interdependence of electricity and natural gas systems was addressed before for various scheduling horizons. A shortterm interdependence of the two infrastructures was analyzed in [5]–[7]. The impact of natural gas network operation on the short-term security of power system is assessed in [5] with a simplified natural gas network. The natural gas constraints were modeled by daily and hourly consumption limits on pipelines, sub-areas, plants, and generating units. A more accurate natural gas flow model was considered in [6] and [7]. A two-phase nonlinear optimization model was proposed in [6] to model the coordinated operation of natural gas and power systems for reliability studies. Benders decomposition was applied to a coordinated model of electricity and natural gas in [7], in which the nonlinear natural gas allocation problem was solved by the Newton–Raphson substitution method. References [8]–[10] discussed the interdependence concerns in a midterm scheduling problem. The impact of natural gas contracts on a GENCO’s risk-constrained hydrothermal scheduling was studied in [8]. The seasonal operation planning of a hydrothermal system with energy-constrained natural gas plants was discussed in [9]. The coordinated scheduling of interdependent hydrothermal power system and natural gas transmission system was considered in [10] by applying the augmented Lagrangian relaxation method. A long-term integrated planning study conducted in [11] considered the issues pertaining to investment decisions. However, uncertainties were not considered in the aforementioned studies on the coordinated natural gas and electricity systems. The authors in [12] and [13] incorporated stochastic factors into the coordination problem. Reference [12] proposed an optimization model of coordinated midterm hydro and natural gas systems, which considered water inflow uncertainties, load forecast errors, and random outages of system components. Reference [13] developed a two-state optimization model to analyze the impact of infrastructure outages with predefined contingency cases and corresponding probabilities. This paper offers the stochastic EGTran model for analyzing the coordination of electricity and natural gas transmission systems by grid operators. Compared with the previous studies, the main contributions of this work are summarized as follows. 1) A scenario-based stochastic unit commitment model is proposed to minimize the total expected cost of power production while satisfying power system security and natural gas network constraints. 2) The Monte Carlo simulation is applied to generate scenarios in which each scenario represents a possible system status with component outages and load forecasting inaccuracies. 3) A linearization strategy is developed by applying a piecewise linear approximation in a 3-D Euclidean space to convert the nonlinear natural gas flow equations into a set of linear constraints. In the proposed work of EGTran, system uncertainties are represented by hourly load forecast errors as random variables with a truncated normal distribution, and forced outages of generating units and transmission lines as independent Markov processes. Scenario reduction is adopted as a tradeoff between

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ALABDULWAHAB et al.: STOCHASTIC SCHEDULING OF ELECTRICITY AND NATURAL GAS INFRASTRUCTURES

the solution speed and the modeling accuracy [14]. The coordinated model of EGTran is formulated as a mixed-integer linear programming (MILP) problem, which would minimize the total expected cost of power production in all scenarios. The rest of this paper is organized as follows: The stochastic simulation is presented in Section II. The coordinated model is formulated in Section III. Section IV presents and discusses in detail the coordination of a six-bus power system with the natural gas system. In addition, the IEEE 118-bus power system is considered in this section for the coordination studies with the natural gas system. The conclusion drawn from the study is provided in Section V.

A set of possible scenarios is generated in EGTran, which is based on the Monte Carlo simulation method for modeling system uncertainties. Each scenario represents a possible system state, including the availability of system components and a possible system load demand. The load forecasting errors as well as random outages of generating units and transmission lines are considered in various scenarios. A sequential Monte Carlo simulation is applied to model the random outages of system components for the scheduling periods. The simulation is based on sampling the probability distribution of the component state duration. We assume that the probability distributions associated with mean time to failure and mean time to repair are available, and the coordinate system is initially at the normal operating state. We use a two-state continuous-time Markov chain model in EGTran to represent available and unavailable states of a system component. In a two-state component representation, the distribution functions representing underlying operating and repair state durations are generally assumed to be exponential. The ith component’s repair and failure rates in the period is represented using μi and λi , respectively. The steady-state availability of the ith generating unit is μi /(λi + μi ), and its unavailability is λi /(λi + μi ). In the two-state continuous-time Markov model, the associated conditional probabilities for the ith component are defined as μi λi + ∗ e−(λi +μi )∗(t−t0) λi + μi λi + μi λi ∗ 1−e−(λi +μi )∗(t−t0) p(ϕt = 0|ϕt0 = 1) = λi + μi μi p(ϕt = 1|ϕt0 = 0) = ∗ 1−e−(λi +μi )∗(t−t0) λi + μi λi μi p(ϕt = 0|ϕt0 = 0) = + ∗ e−(λi+μi )∗(t−t0) . λi +μi λi +μi p(ϕt = 1|ϕt0 = 1) =

The day-ahead load forecast error can be represented by a hyperbolic distribution, a normal distribution or a truncated normal distribution function. The truncated normal distribution is more practical since it would eliminate extremities of random load values for unrealistically significant forecast errors [15]. The probability distribution function of the truncated normal distribution is given as ⎧ ⎪ −∞ ≤ ε < εmin or εmax < ε ≤ +∞ ⎪ ⎨0, PDF N (ε) , εmin ≤ ε ≤ εmax (5) PDFTND (ε) = εmax ⎪ PDFN (ε)dε ⎪ ⎩ εmin

1 ε−ε0 2 1 e− 2 ( σ ) , PDFN (ε) = √ 2 2πσ

II. S TOCHASTIC M ODELING

(1) (2) (3) (4)

For each component, we generate a successive series of states in EGTran by drawing samples of time to failure λi or time to repair μi with exponential distribution functions. The simulation is applied to represent generator and transmission line outages. A power system state represents a combination of individual component states. We apply {Vit , i = 1, . . . , NG, t = 1 . . . NT} to each scenario for representing component availabilities, in which Vit = 1 indicates the ith component is available at time t and Vit = 0 indicates otherwise.

3

−∞ ≤ ε ≤ +∞.

(6)

The truncated normal distribution for the load forecast error in EGTran is represented by zero mean and a standard deviation that is 5% of the hourly load forecast value. When we create multiple scenarios, the scheduling period will be divided into several time intervals, in which each interval τ spans several hours. For each time interval, several scenarios (e.g., u) are generated based on historical data that are different from the corresponding forecasts. The scenario tree has uτ scenarios each with a possibility of 1/uτ . A low-discrepancy Monte Carlo simulation method, i.e., the Latin hypercube sampling (LHS) technique, is employed to decrease the variance of simple Monte Carlo simulation. LHS stratifies the input probability distributions, and the effect is that each simulation sample is constrained to match the input distribution very closely. Thus, we can use a relatively smaller number of samples to reach the same convergence. The computation burden rapidly ascends with the number of scenarios for solving scenario-based large-scale scheduling problems. Therefore, a scenario reduction method is adopted as a tradeoff between the computation efficiency and the modeling accuracy [16]. The scenario reduction technique would control the goodness-of-fit of approximation by measuring a distance of probability distributions as a probability metric. In this paper, we use SCENRED for the reduction of scenarios and modeling the random data processes, which is a tool provided by General Algebraic Modeling System (GAMS). The scenario reduction algorithm provided by SCENRED determines a scenario subset (of prescribed cardinality or accuracy) and assign optimal probabilities to the preserved scenarios. The outcome would present a smaller number of scenarios with a reasonable approximation of the original system. After the scenario reduction, S scenarios are retained, and a weight Ps is assigned to each scenario that reflects the possibility of its occurrence. The sum of probabilities for all scenarios is equal to 1, that is, PS = 1. III. S TOCHASTIC F ORMULATION OF C OORDINATED S YSTEMS The stochastic EGTRan model would determine the hourly SCUC solution with natural gas transmission constraints for minimizing the expected operation cost while satisfying the



coordinated constraints. The framework of the coordination in the EGTran model is described as follows: Min: Expected system operation cost s.t. 1) power balance constraints; 2) generator unit constraints (including unit capacity limits, ramp rate limits, min on/off time, etc.); 3) power transmission constraints; 4) natural gas system constraints.

The objective of the EGTran model (7) is to minimize the expected cost of supplying the hourly load while satisfying various system constraints over the scheduling horizon, i.e.,

min

Ps

t

s=1

Wηt +

η

i/ ∈GU

t

×

[Fic (Pits )

·

s Iit

+ SUit + SDit ] .

(7)

The first term in the objective function is the natural gas contract cost for gas-fired generating units. The second term is the generation cost of other thermal units, which includes fuel cost and start-up and shutdown cost. Ps is assigned to each scenario that reflects the possibility of its occurrence. B. Electric Power System Constraints In the stochastic model, reserve constraints are relaxed because of the specific consideration of each potential contingency in prevailing scenarios. The objective is subject to the following constraints with the details provided in [17]: Power balance constraints

s s Pits · Iit + Pits · Iit = PL + PLoss . (8) i/ ∈GU

i∈GU

Generating unit constraints: Generating units have physical constraints for capacity limits (9), ramp rate limits (10), and minimum on/off time (11), i.e., s s ≤ Pits ≤ Pimax Iit (9) Pimin Iit

s s s s s 1−Ii(t−1) Riup +Iit 1−Ii(t−1) Pimin ≤ 1−Iit Pits −Pi(t−1)

s s s s s Pi(t−1) −Pits ≤ 1−Ii(t−1) (1−Iit ) Ridn +Ii(t−1) (1−Iit ) Pimin

s,on Xi(t−1) − Tion

E · pf = C · Pit − D · PL pfbr = (θj − θk )/xjk (j, k ∈ br) |pfbr | ≤ pfmax br θref = 0.

(12) (13) (14) (15)

C. Modeling of the Natural Gas System

A. Objective Function

NS

flow limits extending from bus j to bus k. Constraint (15) corresponds to the reference bus. The aforementioned equations are given as follows:

s s Ii(t−1) ≥0 − Iit s,off s s Xi(t−1) ≥ 0. − Tioff Iit − Ii(t−1)

(10)

(11)

A brief review of linearization steps for nonlinear constraints is presented in Appendix A Power transmission constraints: Constraint (12) is the bus power balance equation. Constraints (13) and (14) are branch

There are certain similarities between electricity and natural gas systems. Both are designed to supply the end-use loads through the electrical or the gas transmission system. Delivering the natural gas from gas suppliers to retail customers is represented by an enormous section of the gas industry, which includes gas wells and storage facilities, pipelines, compressors, and valves. The natural gas network and its coupling constraints with the electricity system presented in EGTran are discussed as follows. 1) Natural gas fuel constraints for gas units A utility who owns the natural gas-fired units may sign firm or interruptible contracts with natural gas suppliers [18]. Firm natural gas contracts would be consumed regardless of the fuel cost. We consider flexible natural gas contracts for natural gas-fired unit in this model. Accordingly, the fuel cost in (16) depends on natural gas consumption, and natural gas-fired units are considered as hourly natural gas loads (17) in the natural gas network model. Equation (18) shows that the supply of natural gas to generating units is limited by the maximum daily quantity in a natural gas day, i.e., gas Wηt = ρgas η · Fηt

(16)

gas Llt = Fηt

gas Fηt ≤ t

(17) Fηo .

(18)

2) Natural gas network constraints Natural gas transmission system is composed of natural gas wells, storage facilities, transmission pipelines (high pressure) and distribution pipelines (low pressure), and natural gas customers [19]. As one of the most complex nonlinear systems, the natural gas transmission system can be represented by its steady-state and dynamic characteristics. The transient state of natural gas flow through a pipeline is described as a 1-D dynamics along the axis of the natural gas pipeline, which requires distributed parameters and time-varying state variables [20]. These dynamic characteristics would introduce additional challenges to both the modeling and the computation complexity of the stochastic scheduling the coordinated problem, which we will present at a later work. We present in this paper a steady-state natural gas flow model composed of a group of nonlinear equations. Mathematically, the steady-state natural gas problem will determine the state


variables corresponding to nodal pressures and flow rates in individual pipelines, which are based on the known injection of natural gas supply and load values. Supply and Load: Gas suppliers are gas wells and storage facilities that are modeled as positive gas injections at related nodes. The lower and upper limits of natural gas suppliers in each period are defined in (19). Gas consumers are classified into industrial and residential loads in which residential loads have higher priorities. Natural gas loads are represented as negative gas injections at related nodes with lower and upper limits given in (20). Flow Conservation: The flow conservation equation (21) ensures the natural gas balance in the natural gas system. For each node, the steady-state natural gas flow injected is equal to the flow extracted from the system. Pipeline: Natural gas is carried from suppliers to customers through pipelines. There are two types of pipelines: passive (regular) and active. Active pipelines are regular pipelines with compressors, which enlarge the pressure difference between two end nodes and thus increase the transmission capacity. Natural gas flow driven by the pressure difference of two nodes is dependent on factors such as the length and diameter of pipelines, operating temperatures and pressures, type of natural gas, altitude change over the transmission path, and the roughness of pipelines. We only consider regular pipelines in EGTran as we optimize the flows in the natural gas pipeline system. The gas flow between nodes m and n through a regular pipeline is a quadratic function of the pressure at end nodes (22), where Cmn is the pipeline constant that depends on the temperature, length, diameter, friction, and gas compositions. In (22), the distribution of gas flow through a network of pipelines is formulated as nonlinear relationship between the gas flow and node pressure. Similar to bus voltage limits in power transmission lines, the natural gas network would maintain the nodal pressure within an appropriate region (24) that is guaranteed to customers. The aforementioned equations are given as follows: min max ≤ vsp ≤ vsp vsp

(19)

≤ Ll ≤

(20)

Lmin l NGS

sp=1

A vsp −

Lmax l NGL

B Ll −

l=1

fmn = 0

(21)

n∈GC(m)

2 − π2 | fmn = sgn(πm , πn ) · Cmn |πm n 1 πm ≥ πn sgn(πm , πn ) = −1 πm < πn π min ≤ π ≤ π max .

(22) (23) (24)

D. Solution Method for the Stochastic EGTran A fast forward substitution method that is based on the Newton–Raphson method [12] is used to solve the coordination problem in EGTran. The approximate solution would require a large number of iterations, which is also sensitive to the initial natural gas operating point. If the initial point is not close

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enough to the global optimal point, its final solution may result in a local optimal solution. Meanwhile, the nonlinear pipeline flow rate constraints would have a significant impact on the computation of the stochastic EGTran problem. In this paper, we convert (22) and (23) into a set of linear constraints by using piecewise linear approximation in 3-D Euclidean space, which is briefly discussed below. Given a node pair (m, n), the feasible region of node pressure is expressed as: min max ≤ πm ≤ πm , πnmin ≤ πn ≤ πnmax , m, n ∈ GP}. {(πm , πn )|πm (25)

The piecewise linear approximation would decompose the feasible region (25) into multiple convex polytopes (triangles). A plane in 3-D Euclidean space is then generated in each polytope to represent the relationship between gas flow and node pressure. The gas flow when πm ≥ πn is expressed as fmn = Cmn ·

2 −π 2 = πm n

NP

p (ap · πm +bp · πnp +cp · q p ) (26)

p p where ap , bp , and cp are constants in the p-th triangle; πm p p and πn are node pressures in the p-th triangle; and q is binary indicator variable. The following linear constraints are introduced based on the preceding notations: NP

qp = 1

p=1 min πm πnmin NP

p=1 NP

p max · q p ≤ πm ≤ πm · qp p p max · q ≤ πn ≤ πn · q p

(27) (28) (29)

p πm = πm

(30)

πnp = πn .

(31)

p=1

Equations (7)–(31) represent the coordinated electricity and natural gas infrastructure problem in EGTran, which are converted into an MILP unit commitment problem and a natural gas network subproblem, as shown in Fig. 1. Once the hourly unit commitment problem is solved, the natural gas consumption of gas-fired units is determined. Then the subproblem will check the natural gas network feasibility in all scenarios. If any violation is encountered, the natural gas network constraints are generated (Benders cuts) and fed back to the master problem and renew the solution [7]. IV. N UMERICAL S IMULATIONS We use two case studies consisting of a six-bus power system with a six-node gas system and the IEEE 118-bus system with a ten-node natural gas system to test the proposed EGTran model. A. Six-Bus System The six-bus system, as depicted in Fig. 2, has three gasfired units, seven transmission lines, and three loads. The characteristics of generators, buses, transmission lines, and the hourly load distribution are given in Tables VII–X, respectively (see Appendix B). The unit start-up and shutdown costs are



TABLE I H OURLY S CHEDULE OF C ASE 1 IN A S IX -B US S YSTEM

TABLE II H OURLY S CHEDULE OF C ASE 2 IN 6 B US S YSTEM

Fig. 1. EGTran for the stochastic coordination of electricity and natural gas transmission networks.

Fig. 2. Six-bus power system.

Fig. 3. Six-node natural gas system.

assumed negligible and equal to zero in this case. We assume that all the three gas-fired units have the same fuel price. The six-node natural gas system is shown in Fig. 3, which has five pipelines, two natural gas suppliers, and five natural gas loads. Natural gas loads 1, 5, and 3 are determined by the hourly

generation dispatch of gas-fired units 1, 2, and 3. Gas loads 2 and 4 represent all the other kinds of gas loads. The natural gas transmission parameters are listed in Tables XI–XIII. Three cases are presented to discuss the efficiency of the proposed stochastic model and illustrate the impact of natural gas transmission constraints. Cases 1 and 2 are deterministic, and Case 3 applies a stochastic model for representing system uncertainties. Case 1: This is the deterministic base case without considering the natural gas transmission constraints. Case 2: Natural gas transmission constraints are taken into account based on the linear model discussed in Section III. Case 3: Stochastic conditions are introduced into SCUC with natural gas transmission constraints. These cases are discussed as follows. Case 1: Natural gas transmission constraints are not considered in this case. We calculate the hourly schedule in 24 h considering the dc power transmission constraints. The hourly commitment schedule is shown in Table I, in which hour 0 represents the initial status. The least expensive unit (Unit 1) is committed during the whole schedule horizon, whereas the expensive units are only committed for the periods with higher load demand. Unit 2 is committed between hours 12 and 21, whereas Unit 3 is committed between hours 10 and 22. The daily operation cost in this case is $81 741. The peak load appears at hour 17 when the fuel consumptions of three units are 2951, 1339, and 493 kcf/h, respectively. Case 2: The natural gas transmission constraints are incorporated in this case. The hourly commitment schedule is shown in Table II. In this case, the total generation dispatch of the least expensive unit (Unit 1) is decreased from 4771.8 MWh in Case 1 to 4396.6 MWh in Case 2. The hourly generation dispatch of the cheapest unit (Unit 1) in Cases 1 and 2 is shown in Fig. 4. The output of the cheapest unit (Unit 1) is curtailed between hours 8 and 24 in Case 2 since natural gas transmission constraints have limited the supply of fuel. At peak load hour 17, the fuel consumptions by the three units are 2764, 1786, and 493 kcf/h, respectively. In this case, the expensive unit (Unit 2) is dispatched more because Unit 1 is located at the same node as that of the residential gas load 2, which has a higher supply priority, and


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TABLE III H OURLY S CHEDULE OF C ASE 3 IN A S IX -B US S YSTEM

TABLE IV C OMPARISON OF R ESULTS FOR D IFFERENT S CENARIOS

Fig. 4. Hourly dispatch of Unit 1 in Cases 1 and 2.

Fig. 5. System hourly LMP in Cases 1 and 2.

the natural gas pressure of node 2 has reached its maximum in this case. Thus, the fuel supply to the cheap unit (Unit 1) cannot be guaranteed when natural gas transmission constraints are considered in this case. Meanwhile, natural gas transmission constraints enforce the expensive unit (Unit 2) to be committed between hours 10 and 22 and Unit 3 between hours 8 and 24, which is longer than that in Case 1. In addition, the generation dispatch of Units 2 and 3 is increased by 273.4 and 101.9 MWh, respectively. Therefore, the daily operation cost is increased to $88 297, as compared to $81 741 without the inclusion of natural gas transmission constraints. In this case, the natural gas transmission constraints also changed the system locational marginal prices (LMPs). For comparison, the average LMP profile of the six buses is shown in Fig. 5. In Case 1, LMPs only spike in peak load periods between hours 15 and 19, whereas in Case 2, LMPs spikes are extended longer between hours 10 and 22. This corresponds to the dispatch of expensive units when considering gas transmission constraints. Case 3: In this case, uncertainties are introduced in the EGTran model to compare the deterministic and the stochastic solutions with natural gas transmission constraints. The load forecast error follow a truncated normal distribution with a mean value that is equal to the load forecast and a standard deviation of 5% of the mean value. Power system component outages are not considered in this small example. The computation time for the scenario-based problem depends on the number of scenarios. We generate 10 000 scenarios using the Monte Carlo simulation, and the scenario reduction method would reduce the total number to 10 as a tradeoff between the calculation speed and the solution accuracy. The hourly commitment schedule is shown in Table III. Table IV presents the operation costs of various scenarios with probabilities. The operation cost of the deterministic solution

Fig. 6. Ten-node natural gas system.

in Case 2 is $88 297, whereas in the stochastic solution, the operation cost in each scenario varies as load forecast errors are considered. In this case, the operation costs of S3, S4, and S10 are higher than those of the deterministic case. This situation is caused by the prevailing load forecast error. Each scenario represents a combination of random loads over the 24-h scheduling period. After the scenario reduction, the random loads in S3, S4, and S10 summed over the entire scheduling are 5083.4, 5087.8, and 5141.6 MW, respectively, which is higher than the load forecast of 5059.3 MW. The computation time in Case 2 with the inclusion of natural gas constraints is 252 s, which is increased to 2078 s in Case 3 when load forecast errors are considered in the ten retained scenarios. The computation time is closely related to the scale of additional variables, which depends on the number of scenarios. B. Modified IEEE 118-Bus System The modified IEEE-118 bus system has 54 thermal generators, including eight gas units, 186 branches, and 91 load buses. The total capacity of gas units is 725 MW, which is 10% of total generation capacity. The peak load of 5890 MW occurs at hour 17. The natural gas transmission is composed of ten nodes and 10 pipelines, as shown in Fig. 6. L1–L8 are gas loads consumed by Units 1–8, respectively. L9–L12 are fixed residential gas loads. The test data for the 118-bus power system and the ten-node natural gas transmission system are given at motor.ece.iit.edu/data/Gas transsmion_118_10.xls. Two cases are presented here to compare deterministic and



TABLE V C OMPARISON OF R ESULTS FOR C ASE 1

Fig. 7. Total generation of gas-fired units in Case 1. TABLE VI C OMPARISON OF R ESULTS FOR C ASE 2

stochastic solutions. Random outages of generating unit and transmission lines and load forecast errors are considered in the stochastic cases. Case 1: Deterministic solution of EGTran. Case 2: Stochastic solution of EGTran. Case 1: Table V presents the results for the deterministic case in which the daily production cost is increased to $1 799 280 when the natural gas transmission constraints are considered. Fig. 7 shows that the total generation dispatch in the eight natural gas-fired units is curtailed by 2894 MW during the scheduling horizon. The generation dispatch of G3 and G4 was curtailed by 903 and 1636 MW, respectively, because the two generating units are located at the end of the natural gas network with a fuel supply that is restricted by the nodal pressure along the path. The pressure at nodes 2, 4, and 8 is often capped, which lowers the gas flows to loads. The other factor is that these gas-fired units are located at the same node as that of high-priority residential loads L10 and L11. Thus, the natural gas available to the gas-fired unit could be tightly limited at times. The computation time was increased from 434 to 638 s when natural gas transmission constraints were introduced with additional variables in EGTran. Case 2: Table VI provides the stochastic solution with probabilities. The low-discrepancy Monte Carlo simulation method is used to generate 1000 scenarios, each representing possible component outages and load forecasting errors. Four scenarios are retained by the scenario reduction method since the com-

putation burden rapidly increases with the number of scenarios for solving scenario-based large-scale EGTran. Random outages for Unit 28 at hours 12 and 13 occur in Scenario 1, which is close to the peak load periods. Unit 28 is one of the cheapest units, which is fully committed in the deterministic solution. In the stochastic solution, Unit 28 is only committed between hours 13 and 24, which does affect the operation cost of EGTran. The other factor leading to the higher operation cost is the inclusion of variable loads in the scenarios, which follow a truncated normal distribution with a mean value equal to the hourly load forecast and a standard deviation of 5% of the mean value. The operation costs of Scenarios 1 and 3 are lower than those of Scenarios 2 and 4, when random loads (based on load forecast errors) in Scenarios 1 and 3 are lower than the forecast. In the deterministic solution of EGTran, the daily operation cost is 1 799 280, which does not change much when natural gas transmission constraints are also introduced. This is mainly because gas-fired generation only accounted for a small portion (10%) of total generation capacity in this system. However, the production cost is increased to 1 825 453 when random load errors are considered in the stochastic example. The load forecast errors have an obvious impact, as compared to that of the natural gas network constraints, on the power system operation cost in this specific example. The computation time for the deterministic problem is several hundred seconds. However, the computation burden rapidly ascends with the system size because of the number of variables introduced by the stochastic factors. The computation times for four scenarios without and with natural gas network constraints are 65 221 and 85 878 s, respectively, in the stochastic model of EGTran. The solution accuracy using the Monte Carlo simulation method could improve if we consider more scenarios. However, that would increase the computation burden considerably for solving the EGTran problem. Thus, an appropriate number of scenarios represent a tradeoff between the computation efficiency and the modeling accuracy. Meanwhile, the linearization of pipeline flow rate constraints would also introduce a number of ancillary variables to the problem, which increases computation burden even further. In practical cases, we would consider a higher number of Monte Carlo scenarios by simplifying the natural gas network constraints or enforcing the natural gas constraints simply on gas-fired units located in natural gas flow constrained regions. We may also resort to parallel processing and other numerical methods that can speed up the EGTran solution process. Compared with contingency analysis, the stochastic technique can easily incorporate load forecasting errors, renewable energy variability, and random outages of system components. V. C ONCLUSION This paper has proposed a steady-state EGTran model for the solution of the stochastic hourly SCUC with natural gas constraints. EGTran applies the MIP approach for solving the coordinated model and considers the random outages of generating units and transmission lines, as well as load forecast errors in the coordination studies of electricity and natural gas. We should point out that the alternative modeling of a transient


TABLE VII G ENERATORS ’ DATA FOR A S IX -B US S YSTEM

natural gas flow in stochastic systems would involve partial differential equations with a large number of distributed parameters and time-varying state variables, which would introduce more challenges to both the modeling and the computation of a stochastic scheduling problem in EGTran. A steady-state natural gas model adopted in this paper is widely used in the natural gas industry for modeling the natural gas network with an acceptable level of accuracy. In this paper, we converted the nonlinear equations in EGTran into a group of linear constraints for the natural gas network for representing generating unit fuel contracts, gas suppliers and loads, and pipelines and natural gas flows. The natural gas-fired generating units linked the two interdependent systems of electricity and natural gas. The examples demonstrated that the coordinated stochastic model is an effective way of representing uncertainties in the hourly solution of the interdependent electricity and natural gas systems.

TABLE VIII H EAT C URVE OF G ENERATORS

TABLE IX PARAMETERS OF P OWER T RANSMISSION B RANCH

TABLE X E LECTRICITY L OAD DATA OF 24 h

A PPENDIX A L INEARIZATION M ETHOD By introducing integer variables, start-up indicator yit and shutdown indicator zit , the ramp rate limits (10) will be revised to s Pits − Pi(t−1) ≤ [1 − yit ]Riup + yit Pimin s Pi(t−1) − Pits ≤ [1 − zit ]Ridn + zit Pimin .

TABLE XI PARAMETERS OF G AS P IPELINE AND G AS L OAD

The minimum on/off time constraint (11) will be revised to on UTi = max{0, min[NT , (Tion − Xi(t=0) )Ii(t=0) ]} UT

i (1 − Iit ) = 0 ∀ t = 1, . . . , UTi t=1 t+Tion −1

Iiτ ≥ Tion ∗ yit

∀ t = UTi + 1, . . . , NT − Tion + 1

τ =t

NT

(Iiτ − yit ) ≥ 0 ∀ t = NT − + 2, . . . , NT

off 1 − Ii(t=0) DTi = max 0, min NT , Tioff − Xi(t=0) Tion

TABLE XII PARAMETERS OF N ODES IN A G AS T RANSMISSION S YSTEM

τ =t

DT

i

Iit = t=1 t+Tioff −1

0

∀ t = 1, . . . , DTi

(1−Iiτ ) ≥ Tioff ∗ zit ∀ t = DTi+1, . . . , NT −Tioff +1

τ =t

NT

(1 − Iiτ − zit ) ≥ 0

∀ t = NT − Tioff + 2, . . . , NT .

τ =t

A PPENDIX II Table VII–XIII are given here.

TABLE XIII PARAMETERS OF A NATURAL G AS S UPPLIER

9



ACKNOWLEDGMENT The authors would like to thank the Deanship of Scientific Research for the technical and financial support. R EFERENCES [1] Annual energy review, EIA, Washington, DC, USA, 2011. [Online]. Available: http://www.eia.doe.gov/emeu/aer/ [2] M. Shahidehpour, Y. Fu, and T. Wiedman, “Impact of natural gas infrastructure on electric power systems,” in Proc. IEEE, vol. 93, no. 5, pp. 1042–1056, May 2005. [3] R. D. Tabors, S. Englander, and R. Stoddard, “Who’s on first? The coordination of gas and power scheduling,” Elect. J., vol. 25, no. 5, pp. 8–15, Jun. 2012. [4] ISO New England, Natural gas and power generation in New England. [Online]. Available: http://www.iso-ne.com/ [5] T. Li, M. Eremia, and M. Shahidehpour, “Interdependency of natural gas network and power system security,” IEEE Trans. Power Syst., vol. 23, no. 4, pp.1817–1824, Nov. 2008 [6] J. Munoz, N. Jimenez-Redondo, J. Perez-Ruiz, and J. Barquin, “Natural gas network modeling for power systems reliability studies,” in Proc. IEEE Bologna Power Tech Conf., Jun. 2003, vol. 4, pp. 1–8. [7] C. Liu, M. Shahidehpour, Y. Fu, and Z. Li, “Security-constrained unit commitment with natural gas transmission constraints,” IEEE Trans. Power Syst., vol. 24, no. 10, pp. 1523–1535, Aug. 2009. [8] C. Sahin, Z. Li, and M. Shahidehpour, “Impact of natural gas on riskconstrained midterm hydrothermal scheduling,” IEEE Trans. Power Syst., vol. 26, no. 12, pp. 520–528, May 2011. [9] G. Skugge, J. A. Bubenko, and D. Sjelvgren, “Optimal seasonal scheduling of natural gas units in hydro-thermal power system,” IEEE Trans. Power Syst., vol. 9, no. 2, pp. 848–854, May 1994. [10] C. Liu, M. Shahidehpour, and J. Wang, “Application of augmented Lagrangian relaxation to coordinated scheduling of interdependent hydrothermal power and natural gas system,” IET Gener., Transmiss. Distrib., vol. 4, no. 12, pp. 1314–1325, Aug. 2010. [11] S. Hecq, Y. Bouffioulx, P. Doulliez, and P. Saintes, “The integrated planning of the natural gas and electricity systems under market conditions,” in Proc. IEEE Porto Power Tech Conf., Porto, Porfugal, 2001, pp. 1–5. [12] L. Wu and M. Shahidehpour, “Optimal coordination of stochastic hydro and natural gas supplies in midterm operation of power systems,” IET Gener., Transmiss. Distrib., vol. 5, no. 5, pp. 577–587, May 2011. [13] C. M. Correa Posada and P. Sanchez Martin, “Stochastic contingency analysis for the unit commitment with natural gas constraints,” in Proc. IEEE POWERTECH, Jun. 2013, pp. 1–6. [14] L. Wu, M. Shahidehpour, and Z. Li, “Comparison of scenario-based and interval optimization approaches to stochastic SCUC,” IEEE Trans. Power Syst., vol. 27, no. 2, pp. 913–921, May 2012. [15] “Integration of renewable resources: Technical appendices for California ISO renewable integration studies version,” California ISO, Folsom, CA, USA, Oct. 2010. [Online]. Available: http://www.caiso.com/282d/ 282d85c9391b0.pdf

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Ahmed Alabdulwahab received the Ph.D. degree in electrical engineering from the University of Saskatchewan, Saskatoon, SK, Canada. He is currently an Associate Professor with the Department of Electrical Engineering and Computer Engineering and affiliated with the Renewable Energy Research Group, King Abdulaziz University, Jeddah, Saudi Arabia. He has been a Consultant to the Electricity and Co-Generation Regulatory Authority in Saudi Arabia and a Visiting Scientist at Kinectrics and the University of Manchester, Manchester, U.K. His field of interest includes power system planning and reliability evaluations.

Abdullah Abusorrah received the Ph.D. degree in electrical engineering from the University of Nottingham, Nottingham, U.K. He is currently an Associate Professor with the Department of Electrical Engineering and Computer Engineering, King Abdulaziz University, Jeddah, Saudi Arabia. He is the Coordinator of the Renewable Energy Research Group and a member of the Energy Center Foundation Committee at King Abdulaziz University. His field of interest includes power quality and system analyses.

Xiaping Zhang received the B.S. and M.S. degrees in electrical engineering from Shanghai Jiaotong University, Shanghai, China, in 2006 and 2009, respectively. She is currently working toward the Ph.D. degree at the Illinois Institute of Technology, Chicago, IL, USA.

Mohammad Shahidehpour (F’01) received the Honorary Doctorate from the Polytechnic University of Bucharest, Bucharest, Romania. He is the Bodine Chair Professor and Director of the Robert W. Galvin Center for Electricity Innovation, Illinois Institute of Technology, Chicago, IL, USA. He is also a Research Professor with the Renewable Energy Research Group, King Abdulaziz University, Jeddah, Saudi Arabia.