Locational Marginal Prices Obey DC Circuit Laws

Abstract

Electricity markets often utilize the DC approximation of the AC power flow equations to facilitate solving an otherwise complex nonconvex optimization problem. These DC power flow equations have analogies to DC circuit laws such as Kirchhoff's Laws, resulting in an intuitive understanding of power flows under this model. Variables derived from the Lagrangian dual of the DC optimal power flow problem, such as locational marginal prices (LMPs) and congestion cost, are less intuitive without an understanding of optimization theory. Even with this understanding, LMP behavior, such as the conditions in which negative prices occur or the impact of individual congested lines on network-wide LMPs, remain somewhat mysterious. In this paper, we show that prices also obey DC circuit laws, which can help facilitate an intuitive understanding of their behavior and relationships throughout a network without explicitly understanding duality. In particular, prices can be modeled as voltages, and their differences can be modeled as flows, allowing for a physical interpretation of prices. This analogy also lends itself to the use of well-understood DC circuit concepts such as superposition and Kirchhoff's Laws, which can further facilitate a clearer understanding of price behavior.

Figures (5)

• Figure 1: Left: A 3-bus network with 3 marginal generators and one load. LMPs are shown in blue at each bus, MW flows are shown between buses in green, and congestion (and its associated cost $\mu$) is shown with a dashed line. Right: The DC circuit equivalent: line susceptances become resistors, congestion becomes a voltage (or current) source, and ground is set at the node of the lowest cost marginal generator. The voltages in the circuit are analogous to LMPs in the DC OPF problem.
• Figure 2: 7-bus network with mesh topology. Dashed lines indicate congestion. Blue text indicates LMPs, and green text indicates LMP differences between buses multiplied by the susceptance of the line between those buses. Three generators are marginal (not at a lower or upper generation limit).
• Figure 3: 7-bus network from Fig. \ref{['fig:7bus']} and its DC circuit analogy (top: with current sources representing congestion; bottom: with voltage sources).
• Figure 4: 7-bus network with nodal voltage contributions from each source (congested line). The resulting nodal voltages (LMPs) are the sum of the contribution of the individual sources. Note that the sum may be off by a few decimal places when compared to the original voltages shown in Fig. \ref{['fig:7bus_total']} due to rounding.
• Figure 5: 3-bus network with negative prices due to one congested line (left), and corresponding DC circuit equivalent (right). Note that because ground (the cheapest marginal generator) is not located at the node with the lowest potential, negative voltages (LMPs) necessarily result.