*4.1. Model Settings*

We formulated the following equations as an original model. Suppliers can produce electricity at *t* = 2, and they can trade them at *t* = 1. The trading good for *t* = 1 is not electricity; rather, it is the right to sell electricity at *t* = 2. Therefore, at date 1, the suppliers who expect higher prices at date 2 have an incentive to buy them, and those who expect lower prices at date 2 have an incentive to sell them.

There are two types of electricity suppliers: renewable suppliers and normal suppliers.

Renewable suppliers can produce electricity by utilizing renewable resources, and their marginal costs thus tend to be zero. They are small firms and have no budgets for speculation.

Normal suppliers are conventional suppliers. They produce electricity at marginal costs *a* > 0. They have budgets *I* > 0 for speculative trading.

At *t* = 1, the electricity supplied at *t* = 2 is traded in the market. *p*1 is *t* = 1 price, and *p*2 is *t* = 2 price.

At *t* = 1, all firms can watch *p*1 and determine the trading volume. All suppliers can sell electricity, but only normal suppliers can buy electricity in the market (renewable suppliers have no budgets for this activity).

All suppliers have beliefs about renewable supply at *t* = 2, *R*2. If some firms expect high volumes of renewable supply at *t* = 2, they must also expect low electricity prices at *t* = 2, *Ei*[*p*2].

They determine their trading strategy based on their beliefs about *Ei*[*p*2].

Profit maximization at *t* = 1 is as follows:

$$\begin{aligned} \text{Max}\_{\mathbf{x}\_1, \mathbf{x}^b} \ (p\_1 - a)\mathbf{x}\_1 + E\_i[\mathbf{v}(\mathbf{x}\_1, \mathbf{x}^b)] \\ \text{s.t.} \ x\_1 \le X, \ p\_1 \mathbf{x}^b \le I, \ \mathbf{x}\_1, \mathbf{x}\_b \ge 0 \end{aligned} \tag{1}$$

*x* is the selling volume, *xb* is the purchase volume, and *<sup>v</sup>*(*<sup>x</sup>*1, *x<sup>b</sup>*) is the value function for *t* = 2.

$$\begin{aligned} \mathbf{v}(\mathbf{x}\_1, \mathbf{x}^b) = \text{Max}\_{\mathbf{x}\_2} \ (p\_2 - a)\mathbf{x}\_2 + p\_2 \mathbf{x}^b\\ \text{s.t.} \ \mathbf{x}\_2 \le \mathbf{X} - \mathbf{x}\_1, \ \mathbf{x}\_2 \ge \mathbf{0} \end{aligned} \tag{2}$$

They determine the appropriate date for selling their electricity by comparing the prices and marginal costs. For example, if *p*1 is higher than their expected forward price *Ei*[*p*2] and higher than marginal cost *a*, they sell the electricity.

Renewable suppliers are authorized to sell *R*1 units of electricity. *R*1 is interpreted as the minimum supply value for renewable energy sources (*R*1 can be zero). The main problem of renewable suppliers can be expressed as follows:

$$\begin{aligned} \text{Max}\_{x\_1} \ p\_1 \mathbf{x}\_1 + E\_i[v(\mathbf{x}\_1)] \\ \text{s.t.} \ x\_1 \le R\_{1\prime} \ x\_1 \ge 0 \end{aligned} \tag{3}$$

*x* is the selling volume, and *<sup>v</sup>*(*<sup>x</sup>*1) is the value function for *t* = 2.

At *t* = 2, renewable suppliers can sell an additional *R*2 units of electricity. *R*2 is a random variable, and it is realized at date *t* = 2.

$$\begin{aligned} v(\mathbf{x}\_1) &= \text{Max}\_{\mathbf{x}\_2} \ p\_2 \mathbf{x}\_2 \\ &\text{s.t.} \ x\_2 \le R\_1 + R\_2 - \mathbf{x}\_1, \ x\_2 \ge 0 \end{aligned} \tag{4}$$

Because their marginal costs amounts to 0, their strategy is simpler than that of the normal suppliers. If *p*1 > *Ei*[*p*2], they usually sell electricity at *t* = 1; otherwise, they tend to wait until *t* = 2.

The main way to solve the problem can be expressed as follows:

Normal suppliers' strategy at *t* = 1 can be expressed as follows:

$$\mathbf{x}\_1 = \mathbf{X}\_\prime \; \mathbf{x}^b = \mathbf{0} \qquad \text{if } p\_1 > E\_i[p\_2] \text{ and } p\_1 \ge a \tag{5}$$

$$\mathbf{x}\_1 = \mathbf{0}, \mathbf{x}^b = \mathbf{0} \qquad \text{if } p\_1 > E\_i[p\_2] \text{ and } p\_1 < a \tag{6}$$

$$\text{fix}\_1 = 0, \mathbf{x}^b = \frac{1}{p\_1} \qquad \text{if } p\_1 \le E\_i[p\_2] \tag{7}$$

Renewable suppliers' strategy at *t* = 1 is:

$$\mathbf{x}\_1 = R\_1 \qquad \text{ if } p\_1 > E\_i[p\_2] \tag{8}$$

$$\mathbf{x}\_1 = \mathbf{0} \qquad\qquad\text{if } p\_1 \le E\_i[p\_2] \tag{9}$$

Normal suppliers' strategy at *t* = 2 can be expressed as follows:

$$\mathbf{x}\_2 = X - \mathbf{x}\_1 \qquad \text{ if } p\_2 \ge a \tag{10}$$

$$\mathbf{x}\_2 = \mathbf{0} \qquad\qquad\text{if } p\_2 < a \tag{11}$$

Renewable suppliers' strategy at *t* = 2 can be expressed as follows:

$$\mathbf{x}\_2 = R\_1 + R\_2 - \mathbf{x}\_1 \tag{12}$$

The consumers' demand is *d*1(*p*1) at *t* = 1. This is the industrial firm's demand. The consumers' demand at *t* = 2 is *d*2(*p*2). We assumed that little elasticity would exist. That is, *<sup>d</sup>*1(*p*1) < 0, and *d*2(*p*2) < 0. We assumed *limp*1→∞*d*1(*p*1) = 0, and *limp*2→∞*d*2(*p*2) = 0.
