A Joint Energy and Transmission Rights Auction on a Network with Nonlinear Constraints: Design, Pricing and Revenue Adequacy

Abstract

The forward and real-time (spot) auction markets operated by independent system operators (ISOs) allow for trade in multiple wholesale electricity products, differentiated by time and location on the transmission network. This chapter presents a general auction model that implements key features of the ISO markets, including definition of several market products, the rules for joint auctioning of the products in a sequence of forward and spot markets, the rules for financial settlement of those products, and the requirements to ensure revenue adequacy of the auctioneer. The model formulation is focused on a joint energy and transmission rights auction (JETRA; henceforth, the ‘auction model’ or ‘auction’), along with a non-linear representation of the transmission network constraints. However, the formulation can be extended, in some cases with modification, to other market products. Our earlier paper (O’Neill et al. 2002) explored properties of this auction with linear transmission constraints.

Appendix: Proof of Revenue Adequacy for the Auction Sequence

This appendix provides a set of sufficient conditions and a proof of revenue adequacy of the auction sequence. This proof extends the revenue adequacy proofs for transmission in Harvey et al. (1997) and O’Neill et al. (2002), both of which considered the case of linear transmission constraints, to an auction with both flowgate or flow-based and point-to-point rights together with nonlinear transmission constraints that define a convex feasible region. To simplify the presentation, the auction model is mapped into a more compact and general non-linear program (NLP) representing an auction in the following way:

As before, the rights bid for and awarded in the s-th auction in a sequence of auctions determine the distribution of revenues from the subsequent auction, s − 1. Meanwhile, the prices obtained in the s-th auction determine how the rights awarded in the previous auction s + 1 are financially settled, as well as how much winning bidders in auction s pay for the rights they win.

Define g ^s as the vector of quantities awarded to P- and G-type bids (encompassing t ^P, g ⁺ and g ⁻ in the JETRA-NL model) with upper bound G ^s in the s-th auction in the sequence. Define a general benefit function c(g ^s) (based on the bids by those seeking rights) for the bid award level, g ^s. The vector y ^s represents net injections in the s-th auction associated with rights g ^s. K ^s(y) represents the flows induced by y ^s as a result of the applicable load flow equations. Define t ^s as the vector of F transmission rights (t ^F in the JETRA-NL model) with upper bound T ^s in the s-th auction. F ^s is the vector of bounds in auction s for transmission elements and network flow constraints. Define π as the vector of dual values for the nodal energy balance constraint, which can be interpreted as the shadow or clearing prices for energy. Finally, define μ as the vector of dual values associated with transmission constraints, which can be interpreted as the shadow prices for transmission rights.

Using the resulting model NLP, the sth auction in the auction sequence s + 1, s, s − 1, …, 0, termed NLP^s, is:

$$ \mathrm{ NL}{{\mathrm{ P}}^s}:\max{b^s}t+{c^s}(g) $$

$$ Ag{-} y = 0\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (\pi ) $$

$$ {B^s}t+{K^s}(y)\le {F^s}\quad \quad \quad \quad \quad \quad \quad (\mu ) $$

$$ t\ \le\ {T^s}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ \ \ (\psi ) $$

$$ g\ \le\ {G^s}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ \ (\rho ) $$

Note that all constraint and objective function parameters can depend on s.

The optimal solution to NLP^s is defined as {y ^s, t ^s, g ^s} and the corresponding optimal dual variables are {π ^s, μ ^s, ψ ^s, ρ ^s}. To demonstrate revenue adequacy of the auction sequence, prices and payments must be defined for the bids for g and t that are accepted. Duals π ^s are the market prices for g ^s, and μ ^s are the market prices for F ^s, and are treated as row vectors in the below. The rights held as a result of the s + 1^st auction in the sequence are g ^s + 1 and t ^s + 1. Financial settlements (payments by the auctioneer) in NLP^s for rights to its revenues, analogous to those defined above for the full auction model, are π^s Ag ^s + 1 and μ ^s B ^s + 1 t ^s + 1 for the two types of rights awarded in the previous auction NLP^s + 1, where the superscript T is the transpose operator Meanwhile, the winning bidders for the two types of rights awarded by NLP^s pay π^{s T} Ag ^s and μ ^sT B ^s t ^s, respectively.

The following theorem concerns the revenue adequacy of this sequence of auctions, and is a generalization of our earlier results for the linear JETRA (O’Neill et al. 2002):

Theorem 1

If B ^s(g) is concave, K ^s(y) is convex, K ^s(y) ≤ K ^s+1(y) for all y, and F ^s+1  ≤ F ^s, then each auction in the sequence of auctions {S − 1, …, s, …, 1, 0}, is revenue adequate; that is:

$${\pi^s}^{\mathrm{ T}} (A^s g^s-A^{s+1} g^{s+1}){+ }{\mu^s}^{\mathrm{ T}}({B^s}{t^s}-B^{s+1} t^{s+1}) \ge 0.$$

Proof

By convexity of K ^s,

$$K^s (y^{s+1}) \ge K^s (y^s)+\nabla K^s (y^s) (y^{s+1}-y^s).$$

Rearranging, we obtain,

$$\nabla K^s (y^s) y^s \ge \nabla K^s (y^s) y^{s+1} + K^s (y^s)-K^s (y^{s+1}).$$

Premultiplying by the row vector of transmission capacity shadow prices μ ^s ≥ 0,

$$\mu^s \nabla K^s (y^s) y^s \ge \mu^s \nabla K^s (y^s) y^{s+1} + \mu^s K^s (y^s) - \mu^s K^s (y^{s+1})$$

(4.13)

From the KKTs to NLP^s,

$$\mu^s (B^s t^s + K^s (y^s)) = \mu^s F^s$$

(4.14)

Since K ^s(y) ≤ K ^s+1(y) and F ^s ≥ F ^s + 1 and because (B ^s+1 t ^s+1, y ^s+1) is a feasible solution to NLP^s+1,

$$B^{s+1} t^{s+1} + K^s (y^{s+1}) \le F^s$$

(4.15)

Multiplying both sides by μ ^s ≥ 0,

$$\mu^s (B^{s+1} t^{s+1} + K^s (y^{s+1})) \le \mu^s F^s$$

(4.16)

Combining (4.14) and (4.16) and multiplying both sides by −1 (which requires reversing the inequality),

$$-\mu^s (B^{s+1} t^{s+1} + K^s (y^{s+1})) \ge -\mu^s(B^s t^s + K^s (y^s))$$

(4.17)

Adding (4.13) and (4.17), eliminating terms that cancel, and finally rearranging,

$$\mu^s \nabla K^s (y^s) y^s-\mu^s (B^{s+1} t^{s+1}) \ge \mu^s \nabla K^s (y^s) y^{s+1} - \mu^s (B^s t^s)$$

Substituting π ^s = μ ^s∇K ^s(y ^s) from the KKT condition for y ^s for problem NLP^s and rearranging,

$$\pi^s (y^s - y^{s+1}) + \mu^s (B^s t^s - B^{s+1} t^{s+1}) \ge 0$$

Finally, in NLP^s, A ^s g ^s = y ^s while in NLP^s+1, A ^s+1 g ^s+1 = y ^s+1; substitution of these constraints establishes the desired result:

$$\pi^s(A^s g^s-A^{s+1} g^{s+1}){+ }\mu^s (B^s t^s-B^{s+1} t^{s+1}) \ge 0.$$

Note that this result does not explicitly depend on the form of the objective function NLP^s. The objective can be linear or nonlinear, as long as it is concave so that the KKT conditions describe an optimal solution, then the use of the KKT conditions in the above proof remains valid.