Electricity Price of Hybrid Power System and Decision Making of Renewable Energy Investment Capacity

Appendix 1

The welfare function of three order Hesse matrix about price, price, daytime and nighttime for the installed capacity of renewable energy:

Three order Hesse matrix:

The three order determinant for Hesse matrices:

1. The first order principal:

   −2γ−c2β+γ−δ2<0.

2. The second order primary principal:

   −2γ−c2β+γ−δ22δ−c2β−δ+γ22δ−c2β−δ+γ2−2γ−c2β−δ+γ2=2γ+c2β+γ−δ22−2δ−c2β−δ+γ22=2γ+c2β+γ−δ2+2δ−c2β−δ+γ22γ+c2β+γ−δ2−2δ+c2β−δ+γ2=4γ+δγ−δ+c2β+γ−δ2>0.

3. The third order primary principal:

   −2γ−c2β+γ−δ22δ−c2β−δ+γ2−c2qd+qnβ+γ−δ2δ−c2β−δ+γ2−2γ−c2β−δ+γ2−c2qd+qnβ+γ−δ−c2qd+qnβ+γ−δ−c2qd+qnβ+γ−δ−c2qd+qn2=4c2δ+γδ−γqd+qn2=4c2δ2−γ2qd+qn2=−4c2γ2−δ2qd+qn2<0.

   So the welfare function is three order Hesse matrix about daytime electricity, nighttime electricity, renewable energy installed capacity. The matrix is negative definite, so the power system’s welfare function is a concave function of price of daytime and nighttime and renewable energy installed capacity.

Appendix 2

The objective function of power system is concave about daytime electricity, nighttime electricity and renewable energy installed capacity. The three order Hesse matrix is negative definite matrix, so there exists optimal prices of white day and night and renewable energy installed capacity which makes the objective function of power system to reach the maximum.

The optimal combination of daytime electricity price, nighttime price and renewable energy installed capacity:

Appendix 3

According to the optimal daytime price and the optimal nighttime price, the difference between the optimal daytime price and the nighttime price can be expressed as:

According to the optimal daytime price and optimal nighttime price, the average electricity price of day price and night price can be expressed as:

Appendix 4

Calculate the first order derivative of installed capacity of renewable energy with respect to renewable energy unit installed capacity of renewable energy cost and carbon emission reduction units, traditional energy and traditional energy supply unit generation cost price sensitivity coefficient, we can obtain: 1) The derivative with respect to renewable energy installed capacity per unit cost is negative. That is, the higher the cost of unit installed capacity, installed capacity of renewable energy is smaller; 2) The derivative with respect to renewable energy carbon emission reduction are positive. That is, renewable energy carbon emission reduction units is bigger, the installed capacity of renewable energy increased; 3) The derivative with respect to the traditional energy generating unit cost is positive. That is, the traditional energy unit generation cost is higher, the installed capacity of renewable energy is higher; 4) The relationship between the energy installed capacity and the sensitivity coefficient of the traditional energy supply price varies according to the sensitivity coefficient of the traditional energy price. It may be made as follows: θ = (4α(γ–δ)+(q_d+q_n)(a_d+a_n+2(δ–γ)(2L+v)))/(4((q_d+q_n) (L+v)–α)).

1. When β < θ

   ∂kr∂β=4αβ+γ−δ+qd+qnad+an−4βL+v+2δ−γ2L+v2δ−γqd+qn2=4βqd+qnL+v−α−4αγ−δ−qd+qnad+an+2δ−γ2L+v2γ−δqd+qn2<0.

   The relationship between the installed capacity of renewable energy on the price sensitivity coefficient of traditional energy supply, when the traditional energy supply price sensitive coefficient: β < θ, the installed capacity of renewable energy on the price sensitive coefficient of transmission energy supply is negatively related, namely the price of conventional energy supply sensitive coefficient is, the smaller the installed capacity of renewable energy.

2. When β > θ

   ∂kr∂β=4αβ+γ−δ+qd+qnad+an−4βL+v+2δ−γ2L+v2δ−γqd+qn2=4βqd+qnL+v−α−4αγ−δ−qd+qnad+an+2δ−γ2L+v2γ−δqd+qn2>0.

   The relationship between the installed capacity of renewable energy on the price sensitivity coefficient of traditional energy supply, when the traditional energy supply price sensitive coefficient: β > θ, the installed capacity of renewable energy on the price sensitivity coefficient is positively related to the transmission of energy supply, namely the price of traditional energy supply sensitive coefficient is, the greater the installed capacity of renewable energy.