Solving Natural Gas Loadflow Problems Using Electric Loadflow Techniques

Solving Natural Gas Loadflow Problems Using Electric Loadflow Techniques Qing Li Computer Science Department Oklahoma State University Stillwater, Oklahoma 74078 1053 Email: lqing@okstate.edu Seungwon An School of Electrical and Computer Engineering Oklahoma State University Stillwater, Oklahoma 74078 5032 Email: aseung@okstate.edu Thomas W. Gedra School of Electrical and Computer Engineering Oklahoma State University Stillwater, Oklahoma 74078 5032 Email: gedra@okstate.edu Abstract Methods to solve natural gas loadflow problems using electric loadflow techniques are presented. The motivation is to integrate a natural gas network with an electric power transmission network so that the network analysis for a combined natural gas and electric power distribution network can be performed in a consistent manner. The issues arising from solving gas loadflow problems are discussed for the sake of electrical engineers. The application method and related issues are demonstrated through a case study on a gas network with compressors. I. INTRODUCTION The share of electricity generation from natural gas is projected to increase from 17% in 2001 to 29% in 2025, including generation by electric utilities, IPPs, and CHP generators, according to the annual DOE report released on January 9, 2003 [1]. Natural-gas-fired turbines and combined-cycle plants will likely dominate the market for new generating capacity when more competition is introduced because natural gas generation technologies are generally less capital-intensive and often more efficient than other alternatives. Both electric power and natural gas industries are network industries, in which energy sources are connected to energy users through transmission and distribution networks. As the deregulation of electricity markets continues, the development of institutions such as futures contract markets and electronic auction markets could lead to greater integration of the electricity and natural gas industries and the emergence of competitive energy markets. The interdependence of the electricity and natural gas industries naturally requires integrated optimization of combined natural gas and electric power networks. There are competitive advantages to optimize the combined natural gas and electric power networks than to optimize the two networks separately for system operation, system planning, and economic analysis. There are few studies found focusing on the combination of the two networks, although each of gas [2], [3], [4], [5] and electric networks [6], [7], [8] has been well studied individually. There are both similarities and differences in network analysis for these two networks. But few references in the literature can be found to specifically address these issues. For integrated gas and power network analysis, it is important for a power engineer to realize that the power loadflow techniques, which are already familiar, can be used to solve gas loadflow problems, although some special care should be taken. This work addresses issues arising from using power loadflow techniques to solve natural gas loadflow problems for the sake of electrical engineers who are to do network analysis for a combined natural gas and electric power network. Basic gas flow modeling and gas network component modeling are presented in a way consistent to their counterparts in power networks so that a power engineer can easily apply previous knowledge in power network analysis directly to a gas network or to a combined gas and power network. II. GAS NETWORK MODELING A typical gas network consists of one or more gas sources (gas producers or storages), one or more gas loads (electrical generators, other networks, or storages), pipelines, compressors, and other devices, such as valves and regulators. The compressors are installed in the network to increase the gas pressure so that the gas can flow through the pipeline to the locations it is consumed. Valves and regulators are devices that allow selected sections of the gas networks to be cut off (at break-down, for overhaul etc.), and they also provide control of the gas flow rate, prevent excessive growth of pressure in the network, and prevent the flow of gas in an undesirable direction. Three basic types of components of a gas network are considered for network modeling purposes: 1) pipelines, 2) compressors, valves, regulators and other devices, and 3) interconnection points (nodes or busses). A directed graph is an efficient way to represent a gas network. The pipelines are represented by branches (also called arcs or edges). The compressors, valves, and regulators, which are sometimes called non-pipe elements, are also represented as branches (special branches). The interconnection points of pipelines or non-pipeline elements are represented by nodes (or called vertices).

In a directed graph representing a gas network, each branch has been assigned a flow direction. If the actual direction of gas flow is opposite to the assigned direction, the flow rate is negative; otherwise, the flow rate is positive (or zero). The two main parts of gas network modeling are pipeline gas flow modeling and compressor modeling. Valves and regulators are not modeled in this paper. The major equations for calculating gas flow rate in a pipe, and the equations for calculating compressor power consumption are discussed in the following sections. A. Flow Equations for Gas Flowing in Pipes The flow equations for pipeline gas describe the relation among the gas flow rate, the pressures at the two pipe ends, and related gas properties, pipe characteristics and operating conditions. An important property of the gas is its viscosity, which is the gas resistance to flow. Due to the viscosity, there is shearing stress within a flowing gas and between a flowing gas and its container. The viscosity is defined as the ratio of the shearing stress to the rate of change in flow velocity. The Reynolds number is a dimensionless number which is used to characterize the gas flow conditions. It is dependent on the properties of the gas, the gas velocity and the diameter of the pipe. The Reynolds number is defined as Re = Dvd/µ, D is the inner diameter of the pipe, v is the velocity of the gas, d is the the density of the gas, and µ is the gas viscosity. There are three basic types of flow in pipes: laminar flow, partially turbulent flow, and fully turbulent flow. Laminar flow is observed at very low Reynolds numbers (<2000) and very smooth pipes [2]. The laminar flow breaks down to turbulent flow when the flow rate increases above a certain limit (usually with a so-called critical Reynolds number around 2000), and eddy currents develop in the center of the flow stream. In practice, the flow will usually become turbulent at Reynolds numbers much lower than the critical value because the irregularities in the pipe wall and joints, bends, etc. tend to produce eddy currents in the flow stream even at fairly low flow rates. There exist two types of turbulent flows, depending on whether the eddy currents occur only in the center of the pipe (partial turbulence) or whether the eddies fill the whole pipe (full turbulence) [2]. In partially turbulent flow, a laminar layer adjacent to the pipe wall known as the boundary layer still exists, while the center of the flow stream is turbulent. The frictional resistance to flow is due to viscous effects in the laminar layer and depends on the Reynolds number. As the flow rate increases, the eddy currents expand and the laminar layer shrinks towards the pipe wall. When this layer becomes so small that the roughness of the pipe affects the flow in the turbulent core, the flow is fully turbulent. The frictional resistance now depends on the roughness of the pipe wall. As the flow rate continues to increase, the frictional resistance becomes less and less dependent on the Reynolds number (i.e., flow rate and viscosity) and reaches a limiting value which is dependent on the characteristics of the pipe (i.e., diameter and roughness) only. There are a lot of flow equations used to describe the steadystate flow rate of gas in a pipe at different flow conditions [2], but none are universally accepted. The variations in the flow equations are due to the difficulties in quantifying the effects of gas flow friction. Many factors should be considered when deriving the flow equations, such as pipe shape, size, angle to the horizon, gas properties, temperature, pressure, and so on. There is not a single flow equation (such as Ohm s law, which, luckily, is linear) that applies under all conditions. The (nonlinear) flow equations commonly used for gas flows are dependent on the region of flow (laminar, fully turbulent, or mixed) in a particular segment of the network. Assuming an isothermal steady gas flow in a long horizontal pipeline, with negligible kinetic energy change, constant friction coefficient, and constant gas compressibility over the length of the pipe, the general steady-state flow rate can be derived from the energy balance. The general equation for calculating flow rate (in standard ft 3 /hr, or SCF/hr) for pipe k from node i to node j can be expressed by the following formula [2]: f k = f kij = S ij 6.4774 T ( ) 0 π 2 i πj 2 D 5 k S ij, (1) π 0 F k GL k T ka Z a f kij = pipeline flowrate, SCF/hr S ij = +1 if π i π j 0, = 1 if π i π j < 0, F k = pipeline friction factor, D k = internal diameter of pipe between nodes, inch G = gas specific gravity (air=1.0, gas=0.6), L k = pipeline length between nodes, miles π i = pressure at node i, psia π j = pressure at node j, psia π 0 = standard pressure, psia T 0 = standard temperature, R T ka = average gas temperature, R Z a = average gas compressibility factor. The flow rate equations used in gas industry are modifications of the general flow rate equation (1) with empirical expressions for the friction factor, F k. The friction factor F k in equation (1) depends on the flow region (laminar flow, mixed or transition flow, or fully turbulent flow)[2]. As an approximation, for a fully turbulent flow (Reynolds number 4000) in a high-pressure network, Weymouth ([5], [9]) suggested that the friction factor F k varies as a function

of the diameter D k only: F k = 0.128. (2) D 1 3 k Since there exist extra drag losses other than the viscous losses, the pipeline efficiency factor should be included in the flow equation (1) to predict the actual flow rate in practice. These additional losses can be due to weld beads, fittings, bends, and dirt and rust scaling on the internal pipe wall. The efficiency factor normally varies between 0.8 and 1 for most gas pipes [2]. Therefore, for a fully turbulent flow in high pressure networks, the flow equation (1) can be approximated by the following equation: f k = f kij = S ij M k S ij ( π 2 i π 2 j ), SCF/hr (3) M k = ɛ 36.124 T 0 D 8/3 k, π 0 GLk T ka Z a ɛ = pipeline efficiency. Equation (3), known as modified Weymouth equation, is most satisfactory for large diameter ( 10 inches) lines with high pressures [5]. As indicated in equation (3), the gas flow can be determined once π i and π j are known for given conditions. Note that different flow regions have different friction factor expressions, and therefore have different flow equations. For example, in laminar flow region, the friction factor is a function of the Renolds number Re only: F k = 64 (4) Re For partially turbulent flow, various expressions for the friction factor can be found in many standard references ([2], [10]). B. Compressor Equations Compressors are installed in the gas network to transport gas and compensate for the loss of energy due to frictional resistance, which results in a loss of pressure at the downstream. Compressors and other non-pipe elements are special branches, that is, the pressures at the two ends of the branch and the flow rate between the two ends of the branch are related by special equations rather than the pipeline gas flow equation. Compressors consume a large amount of energy to operate. The most economic source of energy to drive the major compressors in a large gas network is the natural gas that goes through the compressors, although steam or electricity could also be used as the energy source. Centrifugal compressors are versatile, compact, and generally used in the range of 1,000 to 100,000 inlet ft 3 /min for process and pipeline compression applications [11]. In a centrifugal compressor, work is done on the gas by an impeller. Gas is discharged at a high velocity into a diffuser. The velocity of gas is reduced and its kinetic energy is converted to static pressure. In general, the nature of the compressor work function is very complex. The key equation used in network analysis for a compressor is the horsepower consumption, which is a function of the amount of gas that flows through the compressor and the pressure ratio between the suction and the discharge. After empirical modifications to account for deviation from ideal gas behavior, the actual adiabatic compressor horsepower equation [5] at T 0 = 60 F (= 520 R) and π 0 = 14.65 psia becomes [ (πj ) ] Zki( α 1 α ) H k = H kij = B k f k 1, HP (5) B k = 3554.58 T ( ) ki α, η k α 1 f k = flow rate through compressor, SCF/hr π i = compressor suction pressure, psia π j = compressor discharge pressure, psia π i Z ki = gas compressibility factor at compressor inlet, T ki = compressor suction temperature, R α = specific heat ratio (c p /c V ), η k = compressor efficiency. C. Conservation of Flow and Incidence Matrix The mass-flow balance equation at each node can be written in a matrix form as A ik = U ik = (A + U)f + w T τ = 0, (6) +1, if branch k enters node i, 1, if branch k leaves node i, 0, if branch k is not connected to node i. +1, if the kth unit has its outlet at node i, 1, if the kth unit has its inlet at node i, 0, otherwise. { +1, if the kth turbine gets gas from node i, T ik = 0, otherwise. f = a vector of mass flow rates through branches, w = a vector of gas injections at each node, τ = a vector of gas consumption rates for each turbine = 0, if a compressor is not driven by a gas turbine. The matrix A, known as the branch-nodal incidence matrix [2], represents the interconnection of pipelines and nodes. In addition, we define the unit-nodal incidence matrix U, which describes the connection of units (compressors) and nodes. The vector of gas injections w is obtained by w = w S w L, (7)

w S = a vector of gas supplies, w L = a vector of gas demands. Thus, a negative gas injection means that gas is taken out of the network. The matrix T and the vector τ represent gas is withdrawn to power a gas turbine to operate the compressor. So if a gas compressor, say k, between nodes i and j, is driven by a gas-fired turbine, and the gas is tapped from the suction pipeline i, we have the following representation: T ik = +1, T jk = 0, and τ k = amount tapped. Conversely, if the gas were tapped at the compressor outlet, we would have T ik = 0, T jk = +1, and τ k = amount tapped. Analytically, τ k can be approximated as τ k = α T k + β T k H kij + γ T k H 2 kij, (8) H k = H kij is the horsepower required for the gas compressor k in equation (5). III. NATURAL GAS LOADFLOW Whenever significant changes in gas demands or supplies are expected to occur, the gas loadflow analysis is needed to see whether the network has enough capability to carry the amount of gas to the generator while satisfying various network constraints, such as pressure limits at each node and compressor operation limits. The electric loadflow is well established ([6], [7]), and the gas loadflow problem is introduced in Osiadacz s book [2]. In this section, we will formulate and solve the gas loadflow problem in a similar way for the electric loadflow problem, for the sake of electrical engineers. We assume that the compressors in the network are driven by gas turbines which tap gas from the gas network, and we include the gas consumption rates τ k at compressor stations. A. Loadflow Problem Statement The problem of simulation of gas network with N N nodes in steady state, known as loadflow, is usually that of computing the values of node pressures and flow rates in the individual pipes for known values of N S source pressures (N S 1) and of gas injections in all other nodes. The gas loadflow problem is stated below: Given a natural gas system described by a branch-nodal incidence matrix A and a unit-nodal incidence matrix U, and given a set of gas injections except at the N S known-pressure sources (injections at these nodes initially unknown), and each unit s operating condition (such as the compression ratio, the flow rate through the compressor, or the suction or discharge pressure), determine all other pressures, and calculate the flow rates of all branches and the gas consumptions at compressor stations. Simply speaking, one of two quantities, nodal pressure π i and gas injection w i at each node, and one compressor operating condition are specified, and other values are to be determined. Specified quantities are chosen based on the following definitions: Nodes: Known-Injection Node: For a node i of this type, we assume that we know the gas injection w i, and the pressure π i is to be determined. Generally, source and load nodes, and junctions with no gas injections belong to this node. Electrically, this is analogous to a load bus. In fact, solving the loadflow problem with only this type of node is not in general possible. The first reason is that, in the flow equation (1), the pressures never appear by themselves, but instead appear only as a squared-pressure difference of the form πi 2 π2 j. Therefore, there are only N N 1 pressures which affect the loadflow. We therefore pick N S 1 nodes to provide reference pressures. These nodes are generally the external gas sources supplying our system. Another reason with solving a loadflow for a network containing only known-injection nodes is that this would imply that we know the gas injections at every single node. In fact, we cannot mathematically specify all N N gas injections, as it may not be possible to find a solution to the loadflow equations. Specifying the injections at all nodes requires specifying the gas supplies to gas turbines driving gas compressors, which we cannot know until the loadflow is solved. Instead, we must pick at least one node, allowing the set of gas injection(s) to be whatever is required to solve the loadflow equations. Thus, we have to specify another node type: Known-Pressure Node: Each is typically one of the source nodes, and the pressures of such nodes serve as references for all other pressures. We assume that we know {π i, i = 1,..., N S }, but we do not know the corresponding gas injections. Electrically this is analogous to a (possibly distributed) slack bus. In addition to nodes, the other main components are branches, which connect the nodes. Branches: Pipelines: Pipeline flow modeling has already been discussed above. Other than the physical characteristics of the pipeline, the only variables that the flow f k = f kij on pipeline k depends on are the pressures π i and π j at the ends of the pipeline. Compressors: The other key component we will model in a gas network is a compressor (also called a unit). The connection between the unit s inlet and outlet nodes is not defined by the branch-nodal incidence matrix A, but by the unit-nodal incidence matrix U. The compression ratio between the compressor inlet and outlet, and the flow rate through the compressor are governed by the horsepower equation (5), not by the flow equation (1).

Compressor data (other than the physical characteristics of the compressor) can be specified in several ways[2] for compressor k: relative boost R k = R kij = π j /π i, absolute boost π j π i, mass-flow rate f k = f kij. The inlet pressure π i or the outlet pressure π j could also be specified. B. Loadflow Problem Formulation We assume that the pressures at the N S known-pressure nodes are known, and that the injections at the knowninjection nodes are specified. Also, some operating parameter for each compressor, say (for purposes of illustration), the relative boost R k = R kij, is specified, and let s say there are N P branches in the system, of which N C are compressors. We can state the loadflow problem this way: Given : π 1 π NS w NS +1 w NN R 1 R NC Find : w 1 w NS π NS +1 π NN f 1 f NC f NC +1 f NP It appears that there are N N + N C quantities that are given, while N N + N P must be found. It is clear that N C < N P. This inequality is strict, unless we have no pipelines at all, only N P compressors connected to each other without any intervening pipelines a silly situation. So since N N +N C < N N + N P, the system appears to be underdetermined. Note, however, that from the flow equation (3), the flow f k depends only on the pressures π i and π j of the nodes it connects. Likewise, the horsepower H k required by the compressor depends only on the flow f k and the ratio R k = π j /π i, and thus also only depends on the pressures π i and π j. The tap-off loss τ k depends only on the horsepower H k and thus on the nodal pressures. So, if we knew {π i, i = 1,..., N N }, we would know all other quantities we have discussed. But we are only given the pressures for {i = 1,..., N S }. Let s use π to indicate the part of pressures π we know, and π for the unknown part. Likewise for w and w. The objective, then, is to calculate π, giving us the entire vector π, from which all other quantities can be calculated. We can use the mass-balance equation (6) to write w = T τ(π, π) (Ã + Ũ)f(π, π), Ã, Ũ, and T are obtained from A, U, and T as in equation (6) after we have removed the rows corresponding to the N S nodes (the known-pressure nodes with unknown injections), that is, we have dropped the first N S equations of (6) corresponding to w. This gives us N N N S equations (for the elements of w) in N N N S unknowns (the elements of π). At this point, standard Newton-Raphson or other iterative methods can be employed to drive the mismatch w (the difference between the values of w computed from above and the true values of w given as part of the data) to zero by correcting our current guess for the correct value of π. After π are solved, then all pressures π are known, and thus all other unknown variables can be determined. Fig. 1. A Gas Network of 14 Nodes, 12 Pipeline Branches and 4 Compressors IV. NATURAL GAS LOADFLOW CASE STUDY Figure 1 shows a gas network, which consists of 14 nodes (including one source node and five load nodes), 12 pipeline branches, and four compressors. The four compressors are driven by four gas-fired turbines, and the gas is tapped from the suction pipeline. Table I lists the pipeline data including pipe lengths, diameters, and efficiencies. For the gas loadflow problem, the following conditions are given: π 1 = 1000 psia, source node pressure, w L2 = 30 MMCFD, load flow rate at node 2, w L3 = 90 MMCFD, load flow rate at node 3, w L12 = 110 MMCFD, load flow rate at node 12, w L13 = 40 MMCFD, load flow rate at node 13, w L14 = 90 MMCFD, load flow rate at node 14, π 5 π 4 = 1.6, compression ratio for compressor 1, π 7 π 6 = 1.8, compression ratio for compressor 2, π 9 = 1000 psia, discharge pressure for compressor 3, π 10 = 1100 psia, discharge pressure for compressor 4, T k = 60 o F(= 520 o R), compressor suction temperature, Z k = 0.9987, gas compressibility factor at compressor inlet, α = 1.3049, gas specific heat ratio. The four compressors are driven by four gas turbines.

Branch Node Node Length Diameter Efficiency mile inch 1 1 2 80.5 19.6 0.90 2 1 3 88.3 19.6 0.90 3 2 3 55.9 19.6 0.90 4 2 4 61.1 19.6 0.90 5 3 6 67.9 19.6 0.90 6 5 8 93.5 19.6 0.90 7 7 10 79.7 16.7 0.90 8 9 12 73.5 16.7 0.85 9 11 13 87.9 16.7 0.85 10 12 13 86.6 16.7 0.90 11 12 14 79.7 16.7 0.90 12 13 14 83.5 16.7 0.85 TABLE I GAS NETWORK PIPELINE DATA The compressor efficiencies and the gas turbine fuel rate coefficients in equation (8) are listed in Table II. Compressor Efficiency Turbine Fuel Rate Coefficients α β γ 1 0.83 0 0.2 10 3 0 2 0.84 0 0.2 10 3 0 3 0.83 0 0.2 10 3 0 4 0.84 0 0.2 10 3 0 TABLE II COMPRESSOR AND GAS TURBINE DATA Table III shows the the gas loadflow results, including the pressures at each node, the flow rates at each pipeline branch and each compressor branch, and the gas consumption rates for each gas turbine. V. DISCUSSION The presented method is also called the Newton-nodal method. In the Newton-nodal method, an initial approximation is made to the nodal pressures. This approximation is then successively corrected until the final solution is reached. The major equation used to solve the loadflow problem is Kirchhoff s first law: the sum of flows into and leaving each node is zero, as shown in equation (6). The main advantages of the nodal method are the straightforward formulation and high degree of sparsity of the Jacobian matrix. The main disadvantage of the nodal method is the poor convergence characteristics for gas loadflow problems. The nodal equations contain square-root or close to square-root type terms as in equation (1). These terms are computationally inefficient and the method is very sensitive to starting values. If the initial estimates for the iterative process are far away from the solution, the computational process may diverge. Therefore, a user of the nodal method should have enough knowledge and experience to give a reasonable initial guess of the pressures at all nodes for the nodal method to converge. There are other methods to solve gas loadflow problems. One commonly used method in gas network is the Newtonloop method. In Newton-loop method, the major equation is Node Pressure Branch Node Node Flowrate psia MMCSF 1 1000 1 1 2 184.5 2 683 2 1 3 178.5 3 672 3 2 3 36.5 4 551 4 2 4 118.0 5 882 5 3 6 125.0 6 499 6 5 8 117.4 7 898 7 7 10 124.2 8 729 8 9 12 117.0 9 1000 9 11 13 123.0 10 498 10 12 13-33.6 11 1100 11 12 14 40.6 12 698 12 13 14 49.4 13 729 Compressor Flowrate Gas Consump 14 654 MMCSF MMCSF 1 4 5 117.4 0.623 2 6 7 124.2 0.825 3 8 9 117.0 0.410 4 10 11 123.0 1.129 TABLE III GAS LOADFLOW RESULTS Kirchhoff s second law: the sum of the pressure drops (or more precisely, the squared pressure drops) around any loop is zero. The loop method requires to define a set of loops through a spanning tree (dendrite) of the gas network. For each loop, a loop flow is defined as the flow correction to be added to the approximated values of all branch flows in the loop to yield the true values. From an initial guess of all branch flows, the standard Newton-Raphson or other iterative methods can be used to drive the pressure drops around all loops to zero by iteratively correcting the loop flows. The advantage of the loop method over the nodal method is its good convergence characteristics. The loop equations are quadratic or nearly quadratic because the (squared) pressure drops are quadratic function of flow rates as shown in the flow equation (1). This gives rise to fast convergence and less sensitivity to the initial conditions. The major disadvantage of the loop method is that the loops in the network have to be defined through a minimum cut set, which is not unique. In order to give a Jacobian matrix with a high degree of sparsity, we should choose the loops that are least interconnected. The formulation of the loadflow problems using the loop method is more complex than the nodal method. The equations in the loop-methods become very complicated and error-prone when the gas turbines tap gas from the pipeline to drive the compressors. For both the Newton-nodal method and the Newton-loop method, using the numerical methods to calculate the Jacobian matrix can simplify the equation formulations and codewriting. It is because numerical calculation eliminates the necessity of the analytical derivation of the Jacobian matrix, which are usually very complex and error-prone due to the complexities of the flow equation (1) and the compressorrelated equations (5 and 8). The overhead of the numerical calculation of the Jacobian matrix is that more computation time are needed for large networks.

Special care should be taken while solving gas loadflow problems by an electrical engineer. The flow equations are highly nonlinear, and dependent on the flow regions. Many other important factors, such as changes along the pipeline in pipeline diameter or elevation, or in gas temperature, are not presented here. The compressor power requirements and the gas turbine fuel consumption equations are also nonlinear and dependent on the operating conditions. It is important to use the flow equations and compressor-related equations that are calibrated by field tests. Also note that the compressors in the gas network have operating constraints on inlet and outlet pressures, pressure ratio, and prime mover power input. There also exist regulators and valves in a gas network. Modeling a regulator is similar to that of a compressor. Usually, a regulator has its outlet pressure controlled to a set pressure. Also, the gas should not flow backward through the regulators (similarly to the compressors). For loadflow purposes, valves are not explicitly modelled in this work, and instead are assumed to be either fully open or fully closed. [2] Andrzej J. Osiadacz. Simulation and Analysis of Gas Network. Gulf Publishing Company, February 1987. [3] Roland W. Jeppson. Analysis of Flow in Pipe Networks. Ann Arbor Science, 1976. [4] Peter J. Wong and Robert E. Larson. Optimization of natural-gas pipeline systems via dynamic programming. IEEE Transactions on Automatic Control, AC-13(5):475 481, October 1968. [5] Festus Oladele Olorunntwo. Natural Gas Transmission System Optimization. PhD thesis, The University of Texas at Austin, May 1981. [6] J. Duncan Glover and Mulukutla Sarma. Power System Analysis & Design. PWS Publishing Company, second edition, 1994. [7] John J. Grainger and Jr. William D. Stevenson. Power System Analysis. McGraw-Hill, Inc., 1994. [8] Allen J. Wood and Bruce F. Wollenberg. Power Generation, Operation, and Control. John Wilen & Sons. Inc., 1996. [9] T. R Weymouth. Problems in natural gas engineering. ASME Transactions, 34:185 234, 1942. [10] Robert H. Perry and Don W. Green. Perry s chemical engineers handbook. McGraw Hill, 1997. [11] E. W. McAllister. Pipe Line Rule of Thumb Handbook. Gulf Publishing Company, second edition, 1988. VI. CONCLUSIONS The Newton-nodal method is presented to solve the gas loadflow problems for gas networks with compressors, which are driven by gas turbines. The problem and solution of the gas loadflow are formulated in a similar way to the electric loadflow methods to facilitate the network analysis for a combined natural gas and electric power network. It is found that the commonly used Newton-nodal method in the electric loadflow can be used to solve gas loadflow problems, but special cautions should be taken to note the differences in the gas network compared with a power network: (1) different equations should be used for different flow regions of pipeline gas, (2) the flow equations and compressor-related equations are highly nonlinear, (3) the Newton-nodal method has a simple equation formulation, but its convergence is sensitive to the initial values, and (4) the Newton-loop method is not sensitive to the initial conditions, and easy to converge, but it is more complex to implement. It requires definition of loops by the choice of a minimal cut set, which is not unique (and therefore is arbitrary), and becomes very complicated when the fuel used to drive the compressors comes from the pipeline itself. Gas modelling, while challenging in and of itself, is more interesting when combined with electric loadflow, and more interesting still when combined in an integrated gas and electric optimal power flow (GEOPF). Research into such a GEOPF is ongoing. ACKNOWLEDGMENT The authors would like to thank OG&E for partial financial support for this work. REFERENCES [1] U.S. Department of Energy. Annual energy outlook 2003 with projections to 2025. Technical report, DOE/EIA-0383(2003), U.S. Department of Energy, January 2003.