Implementation of a Scheduling and Pricing Model for Natural Gas

Abstract

Since 1999, the Australian state of Victoria has operated a natural gas spot market to both determine daily prices for natural gas and develop an optimal schedule for the market based on an LP (Linear Programming) approximation to the underlying inter-temporal nonlinear aspects of the gas flow optimization problem. This market employs a dispatch optimization model and a related market clearing model. Here we present the model employed for both the operational scheduling and price determination. The basic dispatch optimization formulation covers the key physical relationships between pressure, flow, storage, with flow controlled by valves, and assisted by compressors, where flow and storage are measured with respect to energy rather than in terms of mass. But we also discuss a range of sophisticated mathematical techniques which have had to be employed to create a practical dispatch tool, including iterating between piecewise and successive linearization; iterating between barrier and simplex algorithms to manage numerical accuracy and solution speed issues, and special methods developed to deal with scheduling flexibility. The market clearing model is a variation on the dispatch optimization model which replaces the gas network with an infinite storage tank with unlimited transport capacity. We address the performance of the model including accuracy and run time.

The authors wish to thank the Australian Energy Market Operator (AEMO) for providing information used in this chapter and for review of the content.

Appendix: Detailed Pressure Flow Equations for Flow Rates and Linepack

In this Appendix we present the key equations for deriving natural gas flows and pressures in the Victorian pipe network. The derivation is based on six initial equations described in Eqs. 16, 17, 18, 19, 20, 21) below.

$$ {P_l} = {\rho_l}{\hbox{RTz}} $$

(16)

The ideal gas law equation 1 describes pressure, P _l at a point l along a pipeline of length 0 < l < L. R is the ideal gas constant in units of kPa*m³/(°K × kg), T is the temperature of the pipeline in °K, and z is the supercompressibility of gas, while ρ _l is the density in units of kg/m³.

$$ {q_l} = {\rho_l}{v_l}{\hbox{A}} $$

(17)

The gas flow rate q _l at a point l, in kg/s, is described in (17). Where A is the pipe cross-sectional area in units of m², ν _l is the velocity of gas in m/s in the pipe at point l, at the point l measured at sea level.

$$ \frac{{dP}}{{dl}} \times 1000 = - \frac{f}{\hbox{2D}}{\rho_l}v_l^2 $$

(18)

The Fanning equation 18 describes the rate of change of pressure at position l along a pipeline. D is the pipe diameter in units of m while f is the Fanning friction factor.

$$ Re = \frac{{{\rho_l}{v_l}{\text{D}}}}{{{\mu }}} $$

(19)

The Reynolds Number, Re, is defined by (19). In this equation μ is the viscosity of gas measured in kg/ms.

$$ f = \frac{{0.316}}{{{\eta }}}{\left( {\frac{1}{{Re}} + {{\left( {\frac{{fturb}}{{0.316}}} \right)}^4}} \right)^{{0.25}}} $$

(20)

The Fanning friction factor formulation in (20) utilizes the Blasius formulation [18], where η is the pipeline efficiency (a fraction). It describes friction as gas flows along a pipe. This makes use of a turbulent friction factor, fturb.

$$ fturb = 0.0053 + 0.1662{\left( {\frac{\text{e}}{\hbox{1000D}}} \right)^{{0.35}}} $$

(21)

Smooth pipelines have no turbulence while rough pipelines have more turbulence. Turbulence becomes more important as the Reynolds Number increases. Given e, a measure of the roughness of the pipeline in mm, then the relative roughness of a pipeline can be defined with respect to its diameter as (e/D). Using empirical data the form of the turbulent friction factor shown in (21) was developed by gas system engineers working on the development of the MCE. Now, using Eq. 17 to substitute for ν _l in (18) we get:

$$ \frac{{dP}}{{dx}} = - \left[ {\frac{\text{f}}{\hbox{D}} \times \frac{{q_l^2}}{{{{\hbox{A}}^2}{\rho_l} \times 2}}} \right] \div 1000 $$

(22)

Equation 22 can be further refined by using Eq. 16:

$$ \frac{{dP}}{{dx}} = - 0.5{{\hbox{G}}_{\rm{f}}}\frac{1}{{{P_l}}}q_l^2 $$

(23)

Where:

$$ {{\hbox{G}}_{\rm{f}}}{ = }\frac{\text{fRTz}}{{{\hbox{1,000D}}{{\hbox{A}}^{{2}}}}} $$

(24)

By integrating (23) with respect to l and observing that for l = 0, P _l = P _o, the origin pressure of the pipeline gives:

$$ {P_l} = {\left( {P_0^2 - {{\hbox{G}}_{\rm{f}}}q_l^2 \times l} \right)^{{0.5}}} $$

(25)

Using (25) to define the value of P _l where l = L, i.e. the destination pressure, and assuming a constant flow (q _l = q), friction factor, and supercompressibility, then the steady state flow rate can be derived as:

$$ q = {\left( {\frac{{P_0^2 - P_L^2}}{{{{\hbox{G}}_{\rm{f}}}{\hbox{L}}}}} \right)^{{0.5}}} $$

(26)

Here Eq. 26 is closely related to the Weymouth panhandle equation referred to by Zheng et al. [8] and Midthun et al. [13]. It is used later to define friction factors as a function of flow rate. However, we can derive another flow rate equation by assuming no friction arises from turbulence (fturb = 0), and substituting the Reynolds Number from (20) into (19) and the resultant equation for f into (22) to give:

$$ {P_l} \times \frac{{d{P_l}}}{{dl}} = - 0.5{\hbox{G}} \times q_l^{{1.75}} $$

(27)

Where:

$$ {{\rm G = }}\frac{{{0}{.316}{{\mu }}^{{{0}{.25}}}}{\text{RTz}}}{{{{1,000 \eta Di}}{{{a}}^{{{1}{.25}}}}{{{\rm A}}^{{{1}{.75}}}}}}$$

(28)

Integrating (27) with the assumption that the flow is constant along the pipe segment and the requirement that for l = 0, P _l = P _o (the origin pressure) we get:

$$ {P_l} = {\left( {P_0^2 - {\hbox{G}}{q^{{1.75}}} \times l} \right)^{{0.5}}} $$

(29)

Using (29) to define the value of P _l where l = L, i.e. destination pressure, then for a non-constant friction factor and constant supercompressibility the steady state flow rate can be derived as:

$$ {q_l} = {\left( {\frac{{P_0^2 - P_L^2}}{{{\hbox{G}}L}}} \right)^{{({{1} \left/ {{1.75}} \right.})}}} $$

(30)

Equations 6 and 30 describe flow rates under different assumptions about friction factors. These are used later to define more general flow rate equations. However, before exploring that, it is necessary to consider the linepack equations. The linepack, I, in a pipe segment can be calculated by integrating the volume of gas in each slice of pipeline along the length of the pipeline:

$$ I = \int\limits_0^{\rm{L}} {{\rho_l}{\hbox{A}}dl = } \int\limits_0^{\rm{L}} {\frac{{{P_l}{\text{A}}}}{\hbox{RTz}}dl} $$

(31)

Assuming a constant friction factor this can be rewritten as:

$$ I = \frac{\text{A}}{\hbox{RTz}} \times \int\limits_0^{\rm{L}} {{P_l}dl} $$

(32)

We also have²²:

$$ dF(l) = {P_l}dl $$

(33)

Substituting this into (32), we derive:

$$ I = \frac{\text{A}}{\hbox{RTz}} \times \left\{ {F({\hbox{L}}) - F(0)} \right\} $$

(34)

Substituting the expression for P _l from (25), which was based on a constant fraction factor, into (33), then integrating over l, we get, for a non-zero flow rate:

$$ dF(l) = {P_l}dl = {\left( {P_0^2 - {{\hbox{G}}_{\rm{f}}}q_1^2 \times l} \right)^{{0.5}}}dl $$

(35)

Given the rule (f(x))ⁿ differentiated by x gives n × df/dx × f(x)^n-1 then we can integrate dF(l) by the reverse transformation to give:

$$ F(l) = \frac{{{{\left( {P_0^2 - {{\text{G}}_{\rm{f}}}{q^2} \times l} \right)}^{{1.5}}}}}{{ - 1.5{{\hbox{G}}_{\rm{f}}}{q^2}}} $$

(36)

Evaluation of (36) for F(0) and for F(L), and substituting these into (34) gives:

$$ I = \frac{\text{A}}{\hbox{RTz}}\frac{{\left( {P_0^3 - P_{\rm{L}}^3} \right)}}{{1.5{{\hbox{G}}_{\rm{f}}}{q^2}}} $$

(37)

Further, substituting for q from (26) gives linepack for non-zero flow of:

$$ I = \frac{\text{AL}}{{{1}{\hbox{.5RTz}}}}\frac{{\left( {P_0^3 - P_{\rm{L}}^3} \right)}}{{\left( {P_{{0}}^2 - P_{\rm{L}}^2} \right)}} $$

(38)

Where the flow rate is zero, then P _l = P ₀ for all l, and (16) implies

$$ I = {P_0}\frac{\text{AL}}{{\hbox{RTz}}} = \frac{\text{AL}}{{\hbox{RT}}}\frac{{{P_0}}}{\hbox{z}} $$

(39)

Equations 38 and 39 can be represented in terms of an average pressure on the pipeline P _a:

$$ I = \frac{\text{AL}}{{\hbox{RT}}}\frac{{{P_a}}}{\hbox{z}} $$

(40)

Hence

$$ \begin{array}{lllll} {{P_a} = \frac{2}{3}\frac{{\left( {P_{{0}}^3 - P_{\rm{L}}^3} \right)}}{{\left( {P_0^2 - P_{\rm{L}}^2} \right)}}} & {:{\hbox{if}}} & {{P_0}\, >\, {P_{\rm{L}}}} \end{array} $$

(41)

Or:

$$ \begin{array}{llllll} {{P_a} = {P_0}} & {:{\hbox{if}}} & {{P_{{0}}} = {P_{\rm{L}}}} \\\end{array} $$

(42)

Given user defined values of typical low and high pressures in the system, P _low and P _high, and corresponding supercompressibility values z _low and z _high it is possible to define an average supercompressability z _a as:

$$ {z_a}({P_a}) = {z_{{low}}} + \frac{{({P_a} - {P_{{low}}})}}{{({P_{{high}}} - {P_{{low}}})}} \times ({z_{{high}}} - {z_{{low}}}) $$

(43)

To take advantage of the linear relationship between linepack and pressure in (40), the MCE formulation uses this equation to compute linepack for all cases, including the case when the pipeline inlet and outlet pressure values are different. The term (P ₀ /z) is replaced by the average value of the supercompressibility-adjusted-pressures at the inlet and outlet nodes. Further adjustments are made to these equations to allow for altitude. The model also combines the results of (26) and (30) to determine flows which address the impact of a varying friction factor, non-zero fturb, and varying supercompressibility along the pipeline, for given pressures at the origin and destination of the pipe. All of these values are refined using an iterative approach that converges quickly and tests have demonstrated that a further iteration past the current stopping point would typically impact final flows by less than 0.2%.