Financial Transmission Rights: Point-to Point Formulations

Abstract

Transmission rights stand at the center of market design in a restructured electricity industry. Beginning with the intuition that electricity markets require some rights to use the transmission system, simple models of transmission rights soon founder after confronting the limited capacity and complex interactions of a transmission grid. The industry searched for many years without success looking for a workable system of physical rights that would support decentralized decisions controlling use of the grid.

Appendix: Generic Transmission Line Representation

The generic transmission line analysis employs complex variables. To avoid confusion here, the indexes for the two terminals of the line are k and m. For a development of the model transmission line and transformer model, see Grainger and Stevenson (1994). By choice of parameters, this generic transmission line representation allows for a $\varPi$-equivalent representation of a line with no transformer, an ideal transformer, or a combination of both.

Here we follow Weber (1997)’s notation and conventions. This is useful in that Weber also provides an extensive detail on the characterization of the Jacobian of the power flow equations to provide further insight into the implications of the AC power flow model, including calculation of the derivatives with respect to the transformer parameters. As shown in Fig. 1.2 let V _k represent the complex voltage with magnitude $\left| {{V_k}} \right|$ and angle ${\theta_k}$. The data include the line resistance (r), reactance (x). The transformer includes turns ratio (t _km ) and angle change (${\alpha_{km }}$). The line charging capacitance is the complex Y _cap .

Fig. 1.2

Transmission line and transformer

The line admittance (y) is the inverse of the line impedance (z) formed from the resistance and reactance.

$$ y=\frac{1}{z}=\frac{1}{r+jx }=\frac{1}{r+jx}\frac{{{{{\left( {r+jx} \right)}}^{*}}}}{{{{{\left( {r+jx} \right)}}^{*}}}}=\frac{1}{r+jx}\frac{r-jx }{r-jx }=\frac{r-jx }{{{r^2}+{x^2}}}=g+jb. $$

With P as the real power and Q as the reactive power, the general rules for complex power (S) have:

$$ S=P+jQ=V{I^{*}}=zI{I^{*}}=z{{\left| I \right|}^2}={{\left( {P-jQ} \right)}^{*}}={{\left( {{V^{*}}I} \right)}^{*}}. $$

The line capacitance is represented here as:

$$ \frac{{{Y_{cap }}}}{2}=0+j{B_{cap }}. $$

Following Weber, for the generic representation in Fig. 1.2, complex current (${I_k}$) from k towards m satisfies:

$$ {I_k}={V_k}\left( {y+\frac{{{Y_{cap }}}}{2}} \right)-{V_m}\frac{{{e^{{-j{\alpha_{km }}}}}}}{{{t_{km }}}}y. $$

Therefore, the complex power flow from k to m is:

$$ \begin{array}{lll} {S_k}={V_k}I_k^{*}={V_k}V_k^{*}{{\left({y+\frac{{{Y_{cap }}}}{2}} \right)}^{*}}-{V_k}V_m^{*}\frac{{{e^{{j{\alpha_{km }}}}}}}{{{t_{km}}}}{y^{*}} \\ ={{\left| {{V_k}} \right|}^2}{{\left({y+\frac{{{Y_{cap }}}}{2}} \right)}^{*}}-\frac{{\left| {{V_k}} \right|\left| {{V_m}} \right|{e^{{j\left({{\theta_k}-{\theta_m}+{\alpha_{km }}} \right)}}}}}{{{t_{km}}}}{y^{*}}, \\ ={{\left| {{V_k}} \right|}^2}\left( {g-j\left({b+{B_{cap }}} \right)} \right)-\frac{{\left| {{V_k}} \right|\left|{{V_m}} \right|}}{{{t_{km }}}}\left( {\cos \left({{\theta_k}-{\theta_m}+{\alpha_{km }}} \right)} +j\sin \left({{\theta_k}-{\theta_m}+{\alpha_{km }}} \right) \right)\left( {g-jb} \right), \\ ={{\left| {{V_k}} \right|}^2}g-\frac{{\left| {{V_k}} \right|\left| {{V_m}} \right|}}{{{t_{km }}}}\left( {g\cos \left({{\theta_k}-{\theta_m}+{\alpha_{km }}} \right)+b\sin \left({{\theta_k}-{\theta_m}+{\alpha_{km }}} \right)} \right) \\ +j\bigg(-\frac{{\left| {{V_k}} \right|\left| {{V_m}} \right|}}{{{t_{km }}}}\bigg( {g\sin \left({{\theta_k}-{\theta_m}+{\alpha_{km }}} \right)-b\cos \left({{\theta_k}-{\theta_m}+{\alpha_{km }}} \right)} \bigg)-{{{\left| {{V_k}} \right|}}^2}\left( {b+{B_{cap }}} \right) \bigg).\end{array}$$

The complex current (${I_m}$) from m towards k is

$$ {I_m}=-{V_k}\frac{{{e^{{j{\alpha_{km }}}}}}}{{{t_{km }}}}y+{V_m}\left( {\frac{1}{{t_{km}^2}}y+\frac{{{Y_{cap }}}}{2}} \right). $$

Hence,

$$ \begin{array}{lll} & {S_m}=-{V_m}V_k^{*}\frac{{{e^{{-j{\alpha_{km }}}}}}}{{{t_{km }}}}{y^{*}}+{V_m}V_m^{*}{{\left( {\frac{1}{{t_{km}^2}}y+\frac{{{Y_{cap }}}}{2}} \right)}^{*}}, \\& \ \ \ \ \ \ \ \ \ \ \ \ =-\frac{{\left| {{V_m}} \right|\left| {{V_k}} \right|{e^{{j\left( {{\theta_m}-{\theta_k}-{\alpha_{km }}} \right)}}}}}{{{t_{km }}}}\left( {g-jb} \right)+{{\left| {{V_m}} \right|}^2}\left( {\frac{g}{{t_{km}^2}}-j\left( {\frac{b}{{t_{km}^2}}+{B_{cap }}} \right)} \right), \\& \ \ \ \ \ \ \ \ \ \ \ ={{\left| {{V_m}} \right|}^2}\frac{g}{{t_{km}^2}}-\frac{{\left| {{V_m}} \right|\left| {{V_k}} \right|}}{{{t_{km }}}}\left( {g\cos \left( {{\theta_m}-{\theta_k}-{\alpha_{km }}} \right)+b\sin \left( {{\theta_m}-{\theta_k}-{\alpha_{km }}} \right)} \right) \\& \ \ \ \ \ \ \ \ \ \ \ \ \ \ +j\left( {-\frac{{\left| {{V_m}} \right|\left| {{V_k}} \right|}}{{{t_{km }}}}\left( {g\sin \left( {{\theta_m}-{\theta_k}-{\alpha_{km }}} \right)-b\cos \left( {{\theta_m}-{\theta_k}-{\alpha_{km }}} \right)} \right)-{{{\left| {{V_m}} \right|}}^2}\left( {\frac{b}{{t_{km}^2}}+{B_{cap }}} \right)} \right).\end{array} $$

If the system is normal and the angle change is fixed, then the angle change can be included in the line admittance. Similarly for normal systems, if the transformer tap setting is fixed, the turns ratio can be included in the per unit normalization of the voltages, which would produce appropriately modified values of y but with the elimination of the separate transformer parameters (t, $\alpha$).⁴² Ignoring the line capacitance, this simplified representation would be

$$ \begin{array}{lll} {S_k} = {{\left| {{V_k}} \right|}^2}\hat{g}-\left| {{V_k}} \right|\left| {{V_m}} \right|\left( {\hat{g}\cos \left( {{\theta_k}-{\theta_m}} \right)+\hat{b}\sin \left( {{\theta_k}-{\theta_m}} \right)} \right) \hfill \\ \ \ \ + j\left( {-\left| {{V_k}} \right|\left| {{V_m}} \right|\left( {\hat{g}\sin \left( {{\theta_k}-{\theta_m}} \right)-\hat{b}\cos \left( {{\theta_k}-{\theta_m}} \right)} \right)-{{{\left| {{V_k}} \right|}}^2}\hat{b}} \right). \hfill\end{array} $$

and

$$ \begin{array}{lll} {S_m} = {{\left| {{V_m}} \right|}^2}\hat{g}-\left| {{V_m}} \right|\left| {{V_k}} \right|\left( {\hat{g}\cos \left( {{\theta_m}-{\theta_k}} \right)+\hat{b}\sin \left( {{\theta_m}-{\theta_k}} \right)} \right) \hfill \\ \ \ \ + j\left( {-\left| {{V_m}} \right|\left| {{V_k}} \right|\left( {\hat{g}\sin \left( {{\theta_m}-{\theta_k}} \right)-\hat{b}\cos \left( {{\theta_m}-{\theta_k}} \right)} \right)-{{{\left| {{V_m}} \right|}}^2}\hat{b}} \right). \hfill\end{array} $$

This is a familiar simplification often seen in the electrical engineering literature. However, if the system is not normal, tap ratios are variable, or phase angle adjustments are variable, it will be necessary to use the more general representation as shown above.

The notation translation to the discussion in the main text has:

$$ {G_k}=g,\;{\Omega_k}=-b,\;{\delta_i}={\theta_k},\;{Z_{ij }}={S_k},\;{\alpha_k}={\alpha_{km }},\;{t_k}={t_{km }}. $$