Jo1n>Mo-j
If both the intercept and slope of the bidding curve are chosen to be independent strategy variables by the supplier, the maximum profit of supplier i can be achieved when the following differential equation is satisfied Jo1n>Mo-j
Jo1n>Mo-j
Eqs. (8) and (9) show that the intercept and slope of the bidding curve are not independent. The 2N variables, xi and bi, need to be calculated by means of N equations, so there exists an infinite number of equilibrium states. That means only one of the parameters between the intercept and slope of the bidding curve is independent in order to achieve a definite equilibrium. 6z3 Yq{1
If the slope of the bidding curve is chosen to be the strategy 6z3 Yq{1
variable by supplier i, Eqs. (6)–(8) yield the following optimal reaction function of supplier i 6z3 Yq{1
is the point that the bidding curve passes through. 6z3 Yq{1
If the intercept of the bidding curve is chosen to be the strategy variable by supplier i, Eqs. (6), (7) and (9) yield the following optimal reaction function of supplier i 6z3 Yq{1
6z3 Yq{1
The equilibrium bidding strategy can be obtained by solving Eqs. (10) or (11). 6z3 Yq{1
When the slope of the bidding curve is chosen to be the strategy variable by supplier i, if the bidding curve passes through the point 6z3 Yq{1
, which means the supplier I is bidding its marginal cost at generation of forward contracts, the bidding curve is 6z3 Yq{1
6z3 Yq{1
Then, the generation of supplier i is 6z3 Yq{1
6z3 Yq{1
Correspondingly, the system marginal price is
& t b
& t b
According to differential equations akin to Eq. (8), the following optimal reaction function of supplier i exists,
& t b
& t b
Eq. (15) shows that the strategy variable is independent of the forward contracts if the bidding curve passes through point .
& t b
3 The equilibrium of supply function model with generation constraints
& t b
Generation constraints have not been considered in the optimal reaction function of the models mentioned above. When generation constraints are considered, if the generation of supplier i calculated from Eqs. (10), (11) or (15) is above the upper limit of generation of supplier i, his generation will be set to the limited value. If the market price calculated from the optimal reaction function of the suppliers is low and the calculated generation is below the minimum generation of supplier i, it is possible that the profit of supplier i is negative. As a result, supplier i will buy electricity from the spot market to meet the forward contracts and maximize its own profit. The necessary condition of supplier to generate electricity is
& t b
& t b
where is the market price without suppliers If inequality (16) is met, let = , otherwise, let = 0.
& t b
If there exist suppliers whose generation constraints are active, then suppliers whose generation constraints are not active will be faced with the following residual demand function:
& t b
& t b
where M–L and N–M are the numbers of suppliers whose generation is over maximum and under minimum generation, respectively. and are the maximum and minimum generations of supplier i, respectively. Js qze'BGY
The generations of suppliers whose generation constraints are not active are calculated by using Eqs. (17),(10), (11) or (15) again. If there still exist suppliers whose generation exceeds their generation limit, the residual demand function will be calculated again. The calculation processes are repeated until there no longer exist suppliers whose generation exceeds their generation limit, and then, the threshold value of the bidding strategy of supplier i whose generation constraints are active is Js qze'BGY
Js qze'BGY
where is the maximum and minimum generation of supplier i. Js qze'BGY
The threshold value of the bidding strategy for supplier indicates that the supplier will choose any strategy that is less than the threshold value if the maximum generation is active, while the supplier will choose the threshold value if the minimum generation is active. Js qze'BGY
4. Numerical examples Js qze'BGY
An example with six generators is employed to calculate the market equilibrium state of the different bidding strategy models. Firstly, with or without forward contracts, the variety of equilibrium states is analyzed when either a different intercept is chosen or the bidding curve passes through point if the slope of the bidding curve Js qze'BGY
is used as the strategy variable. The market equilibrium states in the various bidding strategy models of the suppliers who chose the slope or the intercept of the bidding curve as the strategy variable are compared when the slope is selected as strategy variable by some suppliers while Js qze'BGY
other suppliers choose the intercept as strategy variable. Js qze'BGY
In the paper, the inverse demand curve is assumed as Accordingly, the demand curve is The cost coefficients of the suppliers are listed in Table 1.
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