Consider a duopoly in which two generators with capacities K1 and K2 com-
pete to serve an inelastic demand d. It is assumed that capacity is perfectly divisible and K1 = K2 = K. Generator i could be either off-line, denoted by
ui = 0, or online, as specified by ui = 1. In the instance of the latter, the
ith generator can produce ki MWh during a given hour, where ki ∈ [0, K].
Recall that the generator can provide positive electricity output only if it is online. Suppose generator i’s cost of generating ki MWh is given by an affine
function si+ ciki, where si and ci are positive parameters that denote the
start-up cost and the marginal cost, respectively. Without loss of generality, assume that s1 ≤ s2.
5.2.1 Firm Strategies
In the duopolistic regime, the action or strategy of each generator is given by a single marginal cost offer pi ∈ [0, pmax] for its entire capacity, which
does not necessarily reflect its true cost, i∈ {1, 2}. Recall that pmax denotes
the price cap which is assumed to be greater than either c1 or c2. Assume
that both firms are rational profit-maximizing firms and have access to their payoffs as well as that of their rivals.
5.2.2 Allocation Mechanism
Consider an electricity market organized as a sealed-bid auction. The auc- tion allocation and payment rules specify which bids are accepted, fully or partially, and at what price the transaction is settled. This allocation is usu- ally determined by solving the unit commitment problem and the economic dispatch problem. Furthermore, assume that the auctioneer determines the
allocation to minimize the total “as-bid” cost based on not only the marginal cost offers, but also the start-up costs.
Definition 46 (Allocation Model). The auctioneer determines allocation by solving the mixed-integer linear program:
min u1,k1,u2,k2 u1(s1+ p1k1) + u2(s2+ p2k2) subject to u1k1+ u2k2 = d, ui ∈ {0, 1}, i= 1, 2, ki ∈ [0, K], i= 1, 2. (5.1)
Such a model is prototypical of currently operational day-ahead markets in North America. In practice, other factors, such as a more complicated cost structure and physical/operational constraints, may result in discontinuities and lumpiness in cost. These impacts are similar to those arising from the presence of start-up costs in the model in question.
It is possible that problem (5.1) yields multiple solutions. A tie-breaking rule that favors the allocations with lower actual cost is adopted; if a tie still persists, player 1 is dispatched first. Such rules differ from the commonly used pro-rata tie-breaking rules employed elsewhere, because with positive start-up cost, a convex combination of multiple solutions, in general, is not a solution. Such rules are adopted solely for identifying the existence of pure-strategy equilibria. When studying the mixed-strategy equilibria, the probability of a tie is zero and hence the tie-breaking rule has no impacts on market outcomes. In terms of prices, if a pricing rule leads to multiple solutions, the minimal one is selected.
Suppose the solution to (5.1) is denoted by{u∗
1, k1∗, u∗2, k∗2}, based on which
a payment scheme is considered and comprises of a pricing rule and an uplift rule.
5.2.3 Marginal-Cost Pricing with Make-Whole Uplift
Consider a payment scheme based on marginal-cost pricing with a make- whole uplift. This scheme is currently adopted in most day-ahead markets in North America.
Definition 47(Marginal-Cost Price). The marginal-cost price is determined by the optimal dual variable ρmc associated with the supply-demand balance constraint in the following linear programming problem,
min k1,k2 u∗1(s1+ p1k1) + u∗2(s2+ p2k2) subject to u∗1k1 + u∗2k2 = d, (ρmc) ki ∈ [0, K], i= 1, 2. (5.2)
In the linear program (5.2), u∗
1s1 and u∗2s2 are constants, and have no im-
pact on the problem. Therefore, marginal-cost pricing may be insufficient in reimbursing the start-up cost. In such cases, a “make-whole” uplift payment is introduced to compensate the generators.
Definition 48 (Make-Whole Uplift). The make-whole uplift payment wi to
generator i is defined as wi, where
wi , max{0, u∗i(si+ pik∗i − ρmck ∗
i)}. (5.3)
In effect, this uplift ensures the successful firms to get at least their total offered prices. Given the allocation and payment rules, the player’s problem under marginal-cost pricing and make-whole uplift is defined as follows. Definition 49 (Player’s Problem under Marginal-Cost Pricing with Make- Whole Uplift). Gmci (p−i) : max pi∈[0,pmax] ρmcu∗iki∗+ wi− u∗i(si+ cik∗i) subject to {u∗1, k1∗, u∗2, k∗2} solves (5.1), ρmc solves (5.2).
5.2.4 Convex Hull Pricing with Opportunity-Cost Uplift
Another optional payment scheme that has been recently suggested, adopts the convex hull pricing with opportunity-cost uplift. Instead of fixing the discrete decisions, the convex hull prices are derived from the closest convex approximation, or the convex hull, of the allocation problem.
is defined as the subgradient of the convex hull of the value function or
ρch ∈ ∂vch(d), (5.4)
where vch(d) is defined by
vch(d) , min{µ|(d, µ) ∈ conv(epi(v(d)))} , (5.5) value function v(d) is given by
v(d) , min
ui,pi,i=1,2
u1(s1 + p1k1) + u2(s2+ p2k2)
subject to u1k1 + u2k2 = d,
ui ∈ {0, 1}, i = 1, 2.
and epi(f ) denotes the epigraph of a function f and conv(A) denotes the convex hull of set A.
The opportunity-cost uplift payment is designed explicitly to emulate the auction surplus realized by price-takers in free markets. Due to the dis- crete nature of (5.1), in general there is no allocation that can maximize all players’ auction surpluses under any linear price scheme. Consequently, the opportunity-cost uplift payment is added as a side payment such that the surplus of each player is the same as that when individually maximized under linear prices ρch.
Definition 51 (Opportunity-Cost Uplift). Opportunity-cost uplift payment oi to generator i is defined as
oi , max
ui∈{0,1},ki∈[0,Ki]
ui(ρchk
i− si− piki) − u∗i ρchk∗i − si− pik∗i . (5.6)
With the opportunity-cost uplift payment, the surplus of generator i is added to
max
ui∈{0,1},ki∈[0,Ki]
ui(ρchki− si− piki) , (5.7)
which is the maximal surplus of generator i under price ρch. Likewise, the
the player’s problem under convex hull pricing with opportunity-cost uplift rules is defined.
nity-Cost Uplift). Gchi (p−i) : max pi∈[0,pmax] ρchu∗iki∗+ wi− u∗i(si + ciki∗) subject to {u∗1, k1∗, u∗2, k∗2} solves (5.1), ρch solves (5.4). (5.8)
5.2.5 Nash Equilibrium Concepts
Definition 53 (Pure-Strategy Nash Equilibrium). Suppose players 1 and 2 with symmetric capacity K are characterized by start-up costs s1 and s2,
and marginal cost c1 and c2, respectively. If the system level demand is d,
then:
(i) Under marginal-cost pricing with make-whole uplift, the pure-strategy Nash equilibrium is given by the tuple of offers {p∗i}2i=1, where p∗i solves Gmci (p∗
−i) for i = 1, 2 where p−i ,{pj}j6=i.
(ii) Under convex hull pricing with opportunity-cost uplift, the pure-strategy Nash equilibrium is given by the tuple of offers {p∗i}2i=1, where p∗i solves Gchi (p∗
−i) for i = 1, 2 where p−i ,{pj}j6=i.
As pure-strategy Nash equilibria may not always exist, the mixed exten- sion of the original game and the associated mixed-strategy equilibrium are studied. In this case, the player’s strategy is a probability measure on the Borel subsets of the pure strategy action set, [0, pmax]. Since the pure strategy
action set is just a closed range of the real line, the player’s mixed strategy can be captured by a cumulative distribution function (CDF) of its offer price, denoted by Fi(pi).
For this game, the conditions of the existence theorem 5 in [90] do not hold. First, depending the tie-breaking rules, the total payoff of the game may not be upper semi-continuous;2 Second, no matter what tie-breaking rules are
used, at least one of the players’ payoff is not weakly lower semi-continuous at zero.
Consider the expected payoff function of player i adopts strategy fi(pi) as
2For example, the total payoff under pro-rata tie-breaking rules is neither upper semi-
follows: ¯ πi(Fi(p), Fj(p)) , Z pmax 0 ¯ πi(pi, Fj(p))dFi(p), where ¯πi(pi, Fj(p)) , Z pmax 0 πi(pi, pj)dFj(p).
where πi(pi, pj) is the payoff of player i in the pure strategy game. By doing
so, the player’s problem is extended to the mixed-strategy regime, in which the player’s objective is to maximize its expected payoff:
¯
Gi(F−i) : max Fi(p)
[ ¯πi(Fi(p), F−i(p))].
Definition 54 (Mixed-Strategy Nash Equilibrium). Suppose players 1 and 2 with symmetric capacity K are characterized by start-up costs s1 and s2,
and marginal cost c1 and c2, respectively. If the system level demand is d,
then:
(i) Under marginal-cost pricing with make-whole uplift, the mixed-strategy Nash equilibrium is given by the tuple of generalized probability den- sities of offers {Fi∗}2i=1, where Fi∗ solves ¯Gmci (F∗
−i) for i = 1, 2 where
F−i ,{Fj}j6=i.
(ii) Under convex hull pricing with opportunity-cost uplift, the mixed- strategy Nash equilibrium is given by the tuple of generalized probability densities of offers {Fi∗}2i=1, where Fi∗ solves ¯Gchi (F−i∗ ) for i = 1, 2 where
F−i ,{Fj}j6=i.