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

Jiaping XIE; Weisi ZHANG; Yu XIA; Ling LIANG; Lingcheng KONG

doi:10.21078/JSSI-2018-193-21

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Journal of Systems Science and Information ›› 2018, Vol. 6 ›› Issue (3) : 193-213. DOI: 10.21078/JSSI-2018-193-21

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

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Abstract

In the existing electricity market, the traditional power suppliers and renewable energy generators coexist in the power supply side. In the power supply side, renewable energy generators generate power by wind and other natural conditions, leading renewable energy output a certain randomness. However, the low marginal generating cost and the reduction of carbon emissions, and thus brings a certain advantage for renewable energy compared to alternative energy. Electricity, as a special commodity, stable and adequate power supply is a necessary guarantee for economic and social development. Power shortage situation is not allowed in the power system, and the extra power needs to be handled for the purpose of safety. In this paper, the hybrid power generated by renewable energy generators and traditional energy generators is used as power supply, and then the electricity market sells hybrid power to electricity consumers, the hybrid power system determines the optimal daytime price, nighttime price, and the optimal installed capacity of the renewable energy suppliers. We find that the installed capacity of renewable energy increases first and then decreases with the increase of the price sensitivity coefficient of traditional energy supply. Electricity demand is negatively related to electricity price in the current period, and is positively related to price in the other period. The average price of day and night is only related to the total potential demand of day and night and the total generation probability of renewable energy. The price difference between daytime and nighttime is positively related to potential electricity demand, and negatively related to the sensitivity coefficient of electricity price.

Key words

hybrid power generation / carbon emission reduction / power handling cost / electricity price / installed capacity

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Jiaping XIE , Weisi ZHANG , Yu XIA , Ling LIANG , Lingcheng KONG. Electricity Price of Hybrid Power System and Decision Making of Renewable Energy Investment Capacity. Journal of Systems Science and Information, 2018, 6(3): 193-213 https://doi.org/10.21078/JSSI-2018-193-21

1 Introduction

Since there is no carbon dioxide emission from renewable sources such as wind and solar, and does not require the traditional energy consumption for power generation, more and more people are interested in renewable energy power. The governments of each country are also vigorously promoting renewable energy generation, and are actively promoting the rapid and healthy development of renewable energy industry. However, the renewable energy generation is still in the primary stage, so there are still some problems to be solved. In China, abandoned wind power phenomena and the unreasonable prices problem are restricting the further development of renewable energy sources. The Chinese government aims to increase the proportion of renewable energy to primary energy consumption to 15% by 2020. To meet this objective, the government will vigorously support the development of wind power and solar energy in the future, despite the current weak growth in electricity demand. Unfortunately, in recent years, China's new energy consumptive problem has not been effectively resolved. In 2016, national "abandoned water, wind and light" is nearly 110 billion kW

\cdot

h, causing great waste in clean energy investment. The transmission and consumption problem of large-scale development of clean energy, will exists for a certain period of time in the future, which needs to attach great importance to efforts to solve. In order to achieve the goal of China's carbon emission commitments, clean energy will still need to maintain rapid growth over the next long period of time.

In real life, under normal circumstances, traditional energy generators and renewable energy generators coexist together as the power supply side for hybrid electric power consumer market. At the same time, compared to the traditional energy which uses coal and other resources as raw material for power generation and is characterized with stability, renewable energy which depends on the natural conditions, has the characteristics of intermittent. It is known that power, as a special commodity, needs to have a stable supply of strong power, to meet different market demand. The power shortage situation is usually not allowed. Therefore, for electric power, the supply of electricity should be greater than the demand of electricity, so as to ensure the smooth progress of production and life. For renewable energy power generation, which generates power from wind and solar and has no other energy consumption, the marginal cost of power generation is relatively low. According to the marginal cost of the lowest power priority access principle and the benefits in carbon emissions reduction, the priority on renewable energy in power network is high. However, due to the intermittent characteristics of renewable energy power generation, it cannot guarantee the power supply stableness and needs to cope with the traditional energy power supply, so as to ensure the power supply is greater than the demand for electricity. Because the power supply is greater than the demand for electricity, extra cost is needed for the power system disposal. Renewable energy installed capacity investment requires a certain investment costs, while traditional power generation also requires a certain power generation costs. The power system determines the price of electricity during the daytime, the price of electricity at nighttime, and the installed capacity of renewable energy sources, depending on the daytime and nighttime.

The main contribution of this paper is mainly in the following three aspects: First, different from the traditional power supply mode, we consider a hybrid power supply including renewable energy and traditional energy. We try to find out the overall optimal daytime price, nighttime price and renewable power installed capacity. Secondly, this paper studies the effect of renewable energy power on carbon emission reduction, which can provide theoretical and practical guidance for the replacement of renewable energy power supply. To achieve the goal of carbon emission reduction, it is important to improve the green degree of energy. Finally, as electricity is a special commodity, there is no shortage of electricity in general. However, the excess power needs to be handled safely, so the supply of electricity is not as good as possible. Traditional energy generators and renewable energy generators are no longer just competitive, but a closer competition.

The remainder of this paper is arranged as follows: In the second part, this paper expounds the previous research. We review the previous studies for hybrid power supply, grid electricity price decision, renewable energy capacity investment and other aspects of the literature; In the third part we introduce the model, including model parameters and decision variables, and introduce the related functions involved in this paper; In the fourth part, the objective function of power system is introduced, the daytime price and nighttime price of power system and the capacity investment scale of renewable energy are determined; In the fifth part, we analyze influencing factors during the daytime and nighttime electricity price, and the relationship between the daytime price and the nighttime price, to explore the influential factors of renewable energy installed capacity; In the sixth part, a numerical example of the above model is carried out to analyze the influence of each parameter on the decision variables and to validate the model conclusion. In the seventh part, we summarize this paper and provide directions for the future research. We also put forward relevant management strategies and suggestions.

2 Literature Review

Renewable can effectively alleviate the energy crisis and they do not have carbon emissions compared to conventional sources. However, because of renewable energy is highly dependent on natural conditions, so that renewable energy is intermittent. In order to balance the power supply and demand, the general power grid will make use of hybrid power supply to ensure the smooth and orderly power. Kalinci, et al.^[1] studied the two scenarios, which are only wind turbine and wind turbine/PV hybrid systems. Using the wind turbine/PV array system, instead of wind turbine only, decreases the net present cost (NPC) from ＄14,624,343 to ＄11,960,698. Because the marginal generation costs of renewable energy sources can be neglected, according to the so-called merit order dispatch rule, different types of power plants are brought online in the ascending order of their variable operating costs. Thus, Kök, et al.^[2] assumed the renewable energy source is dispatched into the grid first to satisfy the demand, followed by the newly invested conventional source, and then finally by the existing fleet. Al-Gwaiz, et al.^[3] considered a system consisting of inflexible generators, flexible generators, and variable (or intermittent) generators. The demand and the variable generators' output are uncertain. Wu and Kapuscinski^[4] considered a power system featuring a large share of intermittent renewables and inflexible thermal generators, efficiency gains could be achieved by curtailing the production of renewables. However, as renewables feature very low variable production costs, over-curtailment can be costly. Henriot^[5] showed that while curtailing renewables when their variability is high and the system flexibility is low can reduce generation costs, not all participants benefit from a decentralized decision.

In the power system, electricity price is an important tool to regulate the power supply and demand. When the electricity price is too low, the supply of electricity will be reduced and the demand for electricity will rise, probably leading a supply shortage. When the price of electricity is too high, the supply of electricity will rise, the demand for electricity will decline, there may be oversupply situation. Therefore, the low or excessive price of electricity will cause imbalance between the supply and demand of electricity. A reasonable electricity price decision can improve the efficiency of the power system and realize the balance of power supply and demand. Borenstein^[6] proposed an opt-in time-varying residential pricing plan that would be equitable to both customers who opt in and those who don't. Chao^[7] presented an updated economic model of pricing and investment in restructured electricity market and uses the model in a simulation study for an initial assessment of renewable energy strategy and alternative pricing mechanisms. As for the problem of electricity price forecasting, Weron^[8] explained the complexity of available solutions, their strengths and weaknesses, and the opportunities and threats that the forecasting tools offer or that may be encountered, and also looks ahead and speculates on the directions EPF will or should take in the next decade or so. Iskin, et al.^[9] found that in an electric power system, demand fluctuations may result in significant ancillary cost to suppliers. Furthermore, in the near future, deep penetration of volatile renewable electricity generation is expected to exacerbate the variability of demand on conventional thermal generating units. Finn and Fitzpatrick^[10] studied the potential for the implementation of price based demand response by an industrial consumer to increase their proportional use of wind generated electricity by shifting their demand towards times of low prices. Tsitsiklis and Xu^[11] proposed a dynamic pricing mechanism that explicitly encourages consumers to adapt their consumption so as to offset the variability of demand on conventional units. Through a dynamic game theoretic formulation, their price mechanism achieves social optimality asymptotically, as the number of consumers increases to infinity. Sekizaki, et al.^[12] presented a novel electricity retail market model in which elastic demands of consumers in a distribution network are traded at flexible selling prices offered by a retailer. Kwon, et al.^[13] analyzed the effects of electricity-price policy on electricity demand and manufacturing output. South Korea's electricity demand in the manufacturing sector is used as a case study. Li, et al.^[14] conducted a life cycle cost (LCC) analysis of hydrogen systems using low-price electricity considering three of significant application paths. A sensitivity analysis of these hydrogen systems using either curtailed renewable electricity or VE in China is conducted.

The study of renewable energy investment has attracted more and more scholars' attention. In order to examine investment behavior under the most extensively employed support schemes, namely, feed-in tariffs and renewable energy certificate trading, Boomsma, et al.^[15] adopted a real options approach to analyze investment timing and capacity choice for renewable energy projects under different support schemes. Hu, et al.^[16] studied an organization's one-time capacity investment in a renewable energy-producing technology with supply intermittency and net metering compensation. Aflaki and Netessine^[17] analyzed incentives for investing in renewable electricity generating capacity by modeling the trade-off between renewable (e.g., wind) and nonrenewable (e.g., natural gas) technology. Mignon and Bergek^[18] showed that besides formal institutional demands, emerging investors were influenced by their task environment and by various informal demands which originated in investors' collective and internal contexts. Kök, et al.^[2] considered generation patterns and intermittency of solar and wind energy in relation to the electricity demand throughout a daytime. They investigate the impact of pricing policies (i.e., flat pricing versus peak pricing) on the investment levels of a utility firm in two competing energy sources (renewable and conventional), with a focus on the renewable investment level.

3 Model Preliminaries

3.1 Model Description

In the power system model, this paper considers that the power generation is made up of traditional power suppliers and renewable energy generators. We assume the marginal generation cost of renewable energy is lower. According to the so-called merit order dispatch rule, different types of power plants are brought online in the ascending order of their variable operating costs. Thus, in our model, the renewable energy source is dispatched into the grid first to satisfy the demand, followed by the conventional source (Kök, et al.^[2]). When renewable energy generation is insufficient, the traditional energy is added to the grid. Renewable energy has clean characteristics, so it has a carbon emission reduction effect and has certain energy advantages. However, although the marginal cost of renewable energy is low and has other advantages, their dependence on renewable energy for power generation in natural resources makes its power supply intermittent and cannot achieve stable power supply. Traditional power generation, although the marginal cost of electricity generation is higher, the supply of traditional energy is relatively stable. Because the power is a special commodity, electricity plays an important role in the real life, so the model does not consider the power shortage situation. In other words, the power supply is always greater than the demand for electricity. Meanwhile, since the power is greater than the electric power demand, a certain cost consumptive is needed of power system. The power supply is not the more the better. According to the principle of maximizing the welfare of the system, the power system determines the optimal daytime price and nighttime price, as well as the optimal installed capacity of renewable energy generators.

Figure 1 Hybrid power system

Full size|PPT slide

3.2 Model Parameters and Decision Variables

Table 1 Model parameters

Parameters variables
$D_{d}$	Daytime electricity demand	$δ$	Competitive demand price coefficient
$D_{n}$	Nighttime electricity demand	$L$	Unit carbon emission reductions
$α$	Cost coefficient of investment capacity	$β$	Price sensitive coefficient of supply
$q_{d}$	Daytime generation probability	$c_{1}$	Disposal cost coefficient
$q_{n}$	Nighttime generation probability	$c_{2}$	Disposal cost coefficient
$γ$	Demand price coefficient	$v$	Conventional energy unit generation costs

Table 2 Model decision

Decision variables
$p_{d}$	Daytime electricity price	$k_{r}$	Installed capacity of renewable energy sources
$p_{n}$	Nighttime electricity price

3.3 Model Function

1) Demand function of electricity market. For the demand of electricity market, because of the flexibility of electricity demand, the price of electricity in the daytime and the price of electricity at night have certain competition relation. The demand for electricity is not only related to the electricity price at current time, but also affected by the other time tariff. The higher current price, the smaller the demand for electricity, the higher the price of electricity at the other times, the greater the demand for electricity. The electricity demand for the daytime is:

D_{d} = a_{d} - γ p_{d} + δ p_{n}

, the power demand for the nighttime is:

D_{n} = a_{n} - γ p_{n} + δ p_{d}

. Among them, without loss of generality, we assume

a_{d} > a_{n}

, that is, the potential electricity demand during the daytime is greater than the potential electricity demand at nighttime. And the price of electricity at current time and the price sensitivity coefficients of electricity at other times were different, and

γ > δ

. That is, the impact of electricity prices on the immediate demand for electricity is more obvious, and the sensitivity to electricity demand is greater than the impact of electricity prices on electricity demand.

2) From this point onward, we assume each generator's (true) production cost is quadratic in its output:

C (q) = c q^{2} / 2, c > 0, q > 0

, which implies a (true) linear marginal cost:

C^{'} (q) = c q

. Hence, in a perfectly competitive market, generator would submit a supply function: Traditional power supply:

p = C^{'} (q) = c q

. Therefore, the traditional energy supply function can be obtained:

q = c^{- 1} p = β p

. Renewable energy has intermittent characteristics, so renewable energy cannot provide a stable supply of electricity, and the actual power supply of renewable energy is:

q_{i} k_{r}

. Among them, the probability of

q_{i}

denotes the possibility of power generation, the probability of

1 - q_{i}

stands for electricity generating failure, and

i \in {d, n}

d

represent daytime,

n

represent nighttime. Thus the total supply of electricity can be obtained:

S_{i} (p_{i}, p_{- i}) = β p_{i} + q_{i} k_{r}

3) Renewable energy power uses natural resources such as wind energy and solar energy to generate electricity, thereby reducing carbon emissions and alleviating environmental pollution. Assuming that the carbon emissions of conventional energy generation units are

L

, so the actual carbon emission reductions in the generation process are:

L k_{r} q_{d} + L k_{r} q_{n} = L k_{r} (q_{d} + q_{n})

4) The power consumption cost can be expressed as: Following the approach employed in practice, we model the extra costs for handling oversupply situations using a penalty function,

h (x) = c_{1} x + c_{2} x^{2} / 2

, which represents the extra cost when the total output exceeds the demand. For example, in the Texas electricity system, a penalty for violating the power balance constraint is included in the objective function of the security-constrained economic dispatch problem (Electric Reliability Council of Texas 2012, p.24).

5) Both the investment cost functions of conventional source and the renewable source are linear forms (Kök, et al.^[2]). Traditional power suppliers, with power plants already exist, do not require construction costs, but only the corresponding generation costs. We assume the unit generating cost is

v

. We also assume renewable energy generation cost function

C_{r} (q) = α k_{r}

. Here, the marginal cost of generating electricity for renewable energy is low and negligible for construction investment.

4 Power System Model

According to the so-called merit order dispatch rule, different types of power plants are brought online in the ascending order of their variable operating costs. Thus, in our model, the renewable energy source is dispatched into the grid first to satisfy the demand, followed by the conventional source. That is to say, due to the marginal cost of generating electricity for renewable energy sources is relatively low, according to the priority level of grid access, the first to use the power grid is renewable energy power, and then the traditional energy power to access the power grid. Because of the intermittent nature of the renewable energy power, the actual power generation is carried out with a certain probability to generate electricity at the rated installed capacity. Because renewable energy generation is related to time distribution, the probability of renewable energy generation is different in different time periods. Without loss of generality, it is assumed that renewable energy is more likely to generate electricity during the daytime:

q_{d} > q_{n}

. The supply of electricity varies between daytime and nighttime, and the demand for electricity varies between daytime and nighttime. As a result, the price of electricity during the daytime and the price of electricity at nighttime are not the same. Therefore, this paper divides the time of daytime into two periods (i.e., daytime and nighttime), and the demand of electricity can be adjusted appropriately in the two periods according to the difference of daytime and nighttime electricity price. The daytime price and the nighttime price have the opposite function to the electric power demand. We also assume that traditional energy generation unit carbon emission is

L

, so renewable energy power generation unit carbon emission reduction is also the same as

L

In order to ensure the stability of the power system of power supply in a timely manner (i.e., to ensure that no power shortage situation), the power supply in the power system should be no less than the demand of electric power. To ensure the safety of power, it is necessary to spend some money for extra power to electric power system of excess consumption. Assume the unit cost for traditional electricity generation is

v

. Besides, renewable energy installed capacity needs investment.

4.1 Objective Function of Power System

The objective function of power system is

\begin{aligned} max_{p_{n}, p_{d}, k_{r}} W (p_{n}, p_{d}, k_{r}) & = (p_{d} D_{d} + p_{n} D_{n}) + L k_{r} (q_{d} + q_{n}) - h (S_{d} + S_{n} - D_{d} - D_{n}) \\ - v (D_{d} - q_{d} k_{r} + D_{n} - q_{n} k_{r}) - α k_{r} . \end{aligned}

(1)

The target equation of power system consists of the following parts. The first part is the electricity sales revenue of the daytime and nighttime. The second part is the renewable energy carbon emission reductions. The third part is the power supply is greater than demand for electricity when the accommodation costs. The fourth part is the cost of traditional energy generation power. The fifth part is the renewable energy power the capacity of investment cost.

For the power system, the high price will lead to a decline in demand for electricity, but also lead to a rise in power supply. When the power supply is greater than the demand for electricity, the excess power consumptive cost increases. However, the low price of electricity will lead to the increase of power demand, and will lead to the decline of power supply, which cannot guarantee the supply of electricity is always greater than the demand for electricity, making the power system unstable. Therefore, the power system needs to decide the optimal daytime price and nighttime price.

For renewable energy providers, if renewable energy capacity is too large, the power supply is greater than the demand for electricity, the excess power consumptive cost increases. That is why the renewable power installed capacity is not the greater the better. When the installed capacity of renewable energy power is too small, it cannot guarantee that the supply of electricity is always greater than the demand of electricity, which makes the power system unstable.

Substitute the detailed functions above into the objective function of the system:

\begin{aligned} max_{p_{n}, p_{d}, k_{r}} W (p_{n}, p_{d}, k_{r}) & = p_{d} (a_{d} - γ p_{d} + δ p_{n}) + p_{n} (a_{n} - γ p_{n} + δ p_{d}) + L k_{r} (q_{d} + q_{n}) \\ - c_{1} (\begin{matrix} (β p_{d} + q_{d} k_{r}) + (β p_{n} + q_{n} k_{r}) \\ - (a_{d} - γ p_{d} + δ p_{n}) - (a_{n} - γ p_{n} + δ p_{d}) \end{matrix}) \\ - \frac{1}{2} c_{2} {(\begin{matrix} (β p_{d} + q_{d} k_{r}) + (β p_{n} + q_{n} k_{r}) \\ - (a_{d} - γ p_{d} + δ p_{n}) - (a_{n} - γ p_{n} + δ p_{d}) \end{matrix})}^{2} \\ - v ((a_{d} - γ p_{d} + δ p_{n}) + (a_{n} - γ p_{n} + δ p_{d}) - q_{d} k_{r} - q_{n} k_{r}) - α k_{r} . \end{aligned}

(2)

4.2 Optimal Daytime Price

Optimal daytime price for power system decision making:

\begin{aligned} \frac{\partial W (p_{n}, p_{d}, k_{r})}{\partial p_{d}} & = a_{d} - γ p_{d} + δ p_{n} - γ p_{d} + δ p_{n} - c_{1} (β + γ - δ) - v (- γ + δ) \\ - c_{2} (\begin{matrix} (β p_{d} + q_{d} k_{r}) + (β p_{n} + q_{n} k_{r}) \\ - (a_{d} - γ p_{d} + δ p_{n}) - (a_{n} - γ p_{n} + δ p_{d}) \end{matrix}) (β + γ - δ) \\ = a_{d} - 2 γ p_{d} + 2 δ p_{n} - c_{1} (β + γ - δ) - v (- γ + δ) \\ - c_{2} (\begin{matrix} β p_{d} + γ p_{d} - δ p_{d} + β p_{n} - δ p_{n} \\ + γ p_{n} + q_{d} k_{r} + q_{n} k_{r} - a_{d} - a_{n} \end{matrix}) (β + γ - δ), \end{aligned}

(3)

\begin{aligned} \frac{\partial^{2} W (p_{n}, p_{d}, k_{r})}{\partial p_{d}^{2}} & = - 2 γ - c_{2} {(β + γ - δ)}^{2} < 0. \end{aligned}

(4)

Therefore, it is concluded that the welfare function of the power system is a concave function of the daytime electricity price. The optimal solution can be obtained under the FOC

(f i r s t o r d e r c o n d i t i o n)

\begin{aligned} (c_{2} {(β + γ - δ)}^{2} + 2 γ) p_{d} & = a_{d} + 2 δ p_{n} - c_{1} (β + γ - δ) - v (- γ + δ) \\ - c_{2} (β p_{n} - δ p_{n} + γ p_{n} + q_{d} k_{r} + q_{n} k_{r} - a_{d} - a_{n}) (β + γ - δ) \\ p_{d} & = \frac{(\begin{array}{c} a_{d} + 2 δ p_{n} - c_{1} (β + γ - δ) - v (- γ + δ) \\ - c_{2} (\begin{array}{l} β p_{n} - δ p_{n} + γ p_{n} \\ + q_{d} k_{r} + q_{n} k_{r} - a_{d} - a_{n} \end{array}) (β + γ - δ) \end{array})}{c_{2} {(β + γ - δ)}^{2} + 2 γ} . \end{aligned}

(5)

That is, the optimal daytime price of electricity makes the welfare of power system reach the maximum.

4.3 Optimal Nighttime Price

Optimal nighttime price for power system decision making:

\begin{aligned} \frac{\partial W (p_{n}, p_{d}, k_{r})}{\partial p_{n}} & = δ p_{d} + a_{n} - γ p_{n} + δ p_{d} - γ p_{n} - c_{1} (β - δ + γ) - v (δ - γ) \\ - c_{2} (\begin{matrix} β p_{d} + q_{d} k_{r} + β p_{n} + q_{n} k_{r} - a_{d} \\ + γ p_{d} - δ p_{n} - a_{n} + γ p_{n} - δ p_{d} \end{matrix}) (β - δ + γ), \end{aligned}

(6)

\begin{aligned} \frac{\partial^{2} W (p_{n}, p_{d}, k_{r})}{\partial p_{n}^{2}} & = - 2 γ - c_{2} {(β - δ + γ)}^{2} < 0. \end{aligned}

(7)

Therefore, the welfare function of the power system is a concave function of the nighttime price. The optimal solution can be obtained according to the FOC:

\begin{aligned} (\begin{array}{l} 2 δ p_{d} + a_{n} - 2 γ p_{n} \\ - c_{1} (β - δ + γ) - v (δ - γ) \end{array}) & = c_{2} (\begin{matrix} β p_{d} + γ p_{d} - δ p_{d} \\ + q_{d} k_{r} + q_{n} k_{r} - a_{d} - a_{n} \end{matrix}) (β - δ + γ) \\ + c_{2} (β p_{n} + γ p_{n} - δ p_{n}) (β - δ + γ), \end{aligned}

(8)

that is,

\begin{aligned} (\begin{array}{l} c_{2} (β + γ - δ) \\ (β - δ + γ) + 2 γ \end{array}) p_{n} & = 2 δ p_{d} + a_{n} - c_{1} (β - δ + γ) - v (δ - γ) \\ - c_{2} (β p_{d} + γ p_{d} - δ p_{d} + q_{d} k_{r} + q_{n} k_{r} - a_{d} - a_{n}) (β - δ + γ) \\ p_{n} & = \frac{(\begin{array}{c} 2 δ p_{d} + a_{n} - c_{1} (β - δ + γ) - v (δ - γ) \\ - c_{2} (\begin{array}{l} β p_{d} + γ p_{d} - δ p_{d} \\ + q_{d} k_{r} + q_{n} k_{r} - a_{d} - a_{n} \end{array}) (β - δ + γ) \end{array})}{c_{2} (β + γ - δ) (β - δ + γ) + 2 γ} . \end{aligned}

(9)

That is, the optimal nighttime price of electricity makes the welfare of power system reach the maximum.

4.4 Optimal Installed Capacity of Renewable Energy Sources

Optimal installed capacity of renewable energy for power system decision making:

\begin{aligned} \frac{\partial W (p_{n}, p_{d}, k_{r})}{\partial k_{r}} & = L (q_{d} + q_{n}) - c_{1} (q_{d} + q_{n}) + v (q_{d} + q_{n}) - α \\ - c_{2} (\begin{matrix} β p_{d} + q_{d} k_{r} + β p_{n} + q_{n} k_{r} - a_{d} \\ + γ p_{d} - δ p_{n} - a_{n} + γ p_{n} - δ p_{d} \end{matrix}) (q_{d} + q_{n}) \end{aligned}

(10)

\begin{aligned} \frac{\partial^{2} W (p_{n}, p_{d}, k_{r})}{\partial k_{r}^{2}} & = - c_{2} {(q_{d} + q_{n})}^{2} < 0. \end{aligned}

(11)

Therefore, the welfare function of the power system is a concave function of the installed capacity of renewable energy sources. The optimal solution can be obtained according to the FOC, so:

\begin{aligned} (L - c_{1} + v) (q_{d} + q_{n}) - α = c_{2} (\begin{matrix} β p_{d} + q_{d} k_{r} + β p_{n} + q_{n} k_{r} - a_{d} \\ + γ p_{d} - δ p_{n} - a_{n} + γ p_{n} - δ p_{d} \end{matrix}) (q_{d} + q_{n}), \\ k_{r} = \frac{(L - c_{1} + v)}{c_{2} (q_{d} + q_{n})} - \frac{(\begin{matrix} β p_{d} - (a_{d} - γ p_{d} + δ p_{n}) \\ + β p_{n} - (a_{n} - γ p_{n} + δ p_{d}) \end{matrix})}{q_{d} + q_{n}} - \frac{α}{c_{2} {(q_{d} + q_{n})}^{2}} . \end{aligned}

(12)

That is, the optimal renewable energy installed capacity leads to the maximum welfare of the power system.

4.5 Optimal Decision Combination of Power System

According to the system welfare maximization principle, the power system determines the optimal daytime price, nighttime price and installed capacity of renewable energy.

Theorem 1 The objective equation of the power system is the joint concave function of the day price, the night price, and the investment of the renewable energy capacity

(

see Appendix

1)

The welfare function of three order Hesse matrix about price of daytime and nighttime, and the investment of the installed capacity of renewable energy. Besides, the matrix is negative definite. So the power system's welfare function is combined with a concave function of daytime price, nighttime price and the renewable energy installed capacity.

Therefore, the optimal daytime price, nighttime price and installed capacity of renewable energy sources are respectively

(

see Appendix

2)

\begin{aligned} {\begin{cases} p_{d} = \frac{(\begin{matrix} α (β - δ + γ) (δ + γ) + (q_{d} + q_{n}) \\ (a_{n} δ + a_{d} γ - (δ + γ) ((- δ + γ) L + β (l + v))) \end{matrix})}{(2 (q_{d} + q_{n}) (γ^{2} - δ^{2}))}, \\ p_{n} = \frac{(\begin{matrix} α (β - δ + γ) (δ + γ) + (q_{d} + q_{n}) \\ (a_{d} δ + a_{n} γ - (δ + γ) ((- δ + γ) L + β (l + v))) \end{matrix})}{(2 (q_{d} + q_{n}) (γ^{2} - δ^{2}))}, \\ k_{r} = \frac{(\begin{matrix} 2 α (- δ + γ + c_{2} {(β - δ + γ)}^{2}) \\ + (q_{d} + q_{n}) (\begin{matrix} a_{d} c_{2} (β + δ - γ) + a_{n} c_{2} (β + δ - γ) \\ - 2 (\begin{matrix} c_{1} (δ - γ) + (- δ + γ + c_{2} {(β - δ + γ)}^{2}) L \\ + (- δ + γ + β c_{2} (β - δ + γ)) v \end{matrix}) \end{matrix}) \end{matrix})}{(2 c_{2} {(q_{d} + q_{n})}^{2} (δ - γ))} . \end{cases} \end{aligned}

(13)

5 Property Analysis

Property 1 The price of electricity during the daytime is greater than the price at nighttime

(

see Appendix

3)

The difference between the price of electricity at the daytime and the price of electricity at nighttime can be expressed as

\begin{aligned} p_{d} - p_{n} = \frac{a_{d} - a_{n}}{2 (γ + δ)} . \end{aligned}

(14)

Since the potential electricity demand during the daytime is greater than the potential electricity demand at nighttime, the difference between the price of the daytime and the price of electricity at nighttime is a positive value, that is, the price of electricity in the daytime is greater than the price at nighttime. And the greater the difference between the potential electricity demand of the daytime and the nighttime, the greater the difference between the daytime price and the nighttime price. Because the difference between the potential electricity demand of the daytime and the nighttime is greater, the demand for electricity mainly occurs during the daytime, so the difference between the price of electricity during the daytime and the price of electricity at nighttime will increase. At the same time, as the demand for electricity during the daytime and the sensitivity of nighttime electricity demand increases, the difference between the daytime price and the nighttime price will be reduced. This is because the demand for electricity will be fully distributed between daytime and nighttime when the sensitivity of power demand increases, resulting in a reduction in difference of the price of the daytime and the price of electricity at nighttime.

Property 2 The average price of daytime and nighttime electricity price is negatively correlated with the generation probability of daytime and nighttime

(

see Appendix

3)

\begin{aligned} \bar{p} & = \frac{1}{2} (p_{d} + p_{n}) \\ = \frac{α (β + γ - δ)}{2 (γ - δ) (q_{d} + q_{n})} + \frac{(a_{d} + a_{n} - 2 (β + γ - δ) L - 2 β v)}{4 (γ - δ)} . \end{aligned}

(15)

The upper model shows the average electricity price during the daytime and nighttime is fixed regardless the peak pricing or the flat pricing. The average price of the day is inversely related to the sum of the generation probabilities of the day and the night. When the generation probability of the day and the night is greater, the actual power generation in the electricity market is larger, and the larger power supply will lead to the decline in average electricity price.

In addition, the average price of electricity is related to the demand sensitive coefficient of current price minus the demand sensitive coefficient of the other period price. If the demand sensitive coefficient for the current price is much larger than the other period, the average price of all day long is smaller, because consumers will always reduce the power consumption of current. So the power system for electric power consumption, the best way is to use lower prices to increase power consumption of current consumers.

Property 3 The installed capacity of renewable energy is inversely related to the unit installed cost, and is positively related to the unit carbon emissions and the traditional energy unit generation costs. In addition, with the increase of the price sensitivity of traditional energy supply, the installed capacity of renewable energy has dropped first and then increased

(

see Appendix

4)

\begin{aligned} \frac{\partial k_{r}}{\partial α} & = \frac{δ - γ - c_{2} {(β + γ - δ)}^{2}}{c_{2} (γ - δ) {(q_{d} + q_{n})}^{2}} < 0, \end{aligned}

(16)

\begin{aligned} \frac{\partial k_{r}}{\partial L} & = \frac{γ - δ + c_{2} {(β + γ - δ)}^{2}}{c_{2} (γ - δ) (q_{d} + q_{n})} > 0, \end{aligned}

(17)

\begin{aligned} \frac{\partial k_{r}}{\partial v} & = \frac{γ - δ + β c_{2} (β + γ - δ)}{c_{2} (γ - δ) (q_{d} + q_{n})} > 0, \end{aligned}

(18)

\begin{aligned} \frac{\partial k_{r}}{\partial β} & = \frac{4 α (β + γ - δ) + (q_{d} + q_{n}) (a_{d} + a_{n} - 4 β (L + v) + 2 (δ - γ) (2 L + v))}{2 (δ - γ) {(q_{d} + q_{n})}^{2}} . \end{aligned}

(19)

The renewable energy marginal cost of generating power can be ignored, so the cost of renewable energy power generation is mainly the unit installed capacity of renewable energy. If the unit cost is higher, the installed capacity of renewable energy with the increase of unit cost decline.

In addition, because renewable energy generation has the characteristics of clean and environmental protection, renewable energy power can reduce carbon emissions, alleviate environmental pollution, and have an energy substitution effect, which can increase the environmental greenness. Therefore, the greater the unit emission reduction of renewable energy sources, the greater the installed capacity of renewable energy sources.

Similarly, if the traditional energy unit generation cost increases, the power output of the traditional power reduces while the electricity market demand remains relatively unchanged. Therefore, the electric power market demand for renewable energy will increase and the installed capacity of renewable energy will also experience a corresponding increase.

Finally, the supply price coefficient of traditional energy and electricity has a two-way influence on the installed capacity of renewable energy. On the one hand, when the traditional energy supply price coefficient is low, the traditional energy supply for the price sensitivity is low. When the electricity price increases, the traditional energy power supply increases slowly. Renewable power generator will increase the installed capacity, so as to earn more profits brought by higher prices. On the other hand, when the traditional energy power supply price coefficient is high, the traditional power generator is too sensitive for the price. The traditional energy power supply changes greatly, even if prices drop a little, the traditional energy supply will fall sharply and the installed capacity of renewable energy will increase a lot.

Conclusion 1 For power system managers, because the difference of daytime price and nighttime price was positively correlated with the difference of the potential demand of the daytime and nighttime, and negatively correlated with the sum of the demand price sensitive coefficient, if the electric potential demand is significant different between daytime and nighttime, then the power system can make the daytime price significantly greater than the nighttime price. When the sum of the demand price sensitive coefficient is large, which means the consumer is sensitive to price changes, then the managers of power system should reduce the difference of daytime price and nighttime price. The average price is inversely proportional to the sum of the generation probabilities of the daytime and the nighttime, and there is no correlation with the probability of a single daytime or nighttime, so the power system managers optimize the average daytime and the nighttime price under peak pricing is equal to that of under flat pricing. If the generation probabilities of the daytime and the nighttime is large, the power system managers should lower the average price of all day long.

Conclusion 2 The installed capacity of renewable energy is negatively related to the unit cost of renewable energy, positively related to the unit carbon emission reductions of renewable energy sources, and is positively related to the traditional electricity generation costs of energy units. Therefore, power system managers should increase investment in research and development of renewable energy, reduce the unit cost of power generation, increase renewable energy per unit of carbon emission reductions, to further expand the renewable energy power for the traditional cost advantage.

With the increase of the supply price sensitivity coefficient of traditional energy, renewable energy installed capacity decreased firstly and then increased, so the power system manager should do accurate adjustment to reduce the supply price sensitivity coefficient of traditional energy, or to maintain the basic price to increase the supply price sensitivity coefficient of traditional energy. These measures can increase the installed capacity renewable energy.

6 Numerical Example

We investigate the supply and demand information in Shanghai, in April, 2017. Shanghai's total social electricity consumption is 10 billion 550 million kW

\cdot

h, with an increase of 1.5%. We can get the whole society daily consumption in Shanghai is about

D = 4 \times 10^{5}

thousand kW

\cdot

h. The electricity demand during the daytime is about

D_{d} = 3.0 \times 10^{5}

thousand kW

\cdot

h and the electricity demand of the nighttime is about

D_{n} = 1.0 \times 10^{5}

thousand kW

\cdot

h. The price is about 500 yuan / thousand kW

\cdot

h. The renewable energy unit carbon emission reduction is

L

=0.5, the traditional energy unit generating cost is about

v = 0.2

yuan/kW

\cdot

The welfare function of power system is a concave function of daytime price and nighttime price. So the optimal daytime price and nighttime price make the welfare of power system reach the maximum. As can be seen from Figure 2, when the electricity price remains the same, the welfare of the electric power system rises first and then decreases with the increase of the electricity price during the daytime. Similarly, when the electricity price remains the same, the welfare of the electric power system rises first and then decreases with the increase of the nighttime price.

Figure 2 The effect of daytime price and nighttime price on system welfare

Full size|PPT slide

This shows that in the real life, daytime and nighttime electricity price should not be as high as possible. If the price is too high, the electricity demand will decline, so the high price of power cannot bring high profit.

As can be seen in Figure 3, the installed capacity of renewable energy is decreasing with the rise in electricity prices during the daytime and at nighttime. When the price rises, either daytime or nighttime electricity price rises will lead to reduced demand for electricity. In order to achieve the balance of supply and demand of power market, power supply of renewable energy will be decreased, so the renewable energy installed capacity will decline. If the price rises and lead to a decline in demand for electricity, the oversupply power will take more disposal cost, so the high price will not bring benefits to the power system and the installed capacity of renewable energy will decline.

Figure 3 The impact of daytime electricity price and night price on installed capacity

Full size|PPT slide

The supply price coefficient of traditional energy power has a two-way influence on the installed capacity of renewable energy sources, as shown in Figure 4. On the one hand, when the traditional energy supply price coefficient is low, the traditional energy supply for the price sensitivity is low. When the electricity price increases, the traditional energy power supply increases slowly, renewable power generator will increase the installed capacity, so as to make this part of the profits brought by higher prices. On the other hand, when the traditional energy power supply price coefficient is high, the traditional energy power supply changes greatly even if prices drop a little, the traditional energy supply will fall sharply, the renewable energy the installed capacity will increase a lot.

Figure 4 Impact of power supply sensitivity coefficient on installed capacity

Full size|PPT slide

Figure 5(a) and Figure 5(b) show the relationship between the current price and the price sensitivity coefficient, and the relationship between other time and price sensitive coefficient and price. It can be seen from the figure, whether it is in the current period or at any time during the daytime, the price is higher than the price of the nighttime. In addition, both the daytime electricity price and the nighttime electricity price decrease with the increase of the current price sensitive coefficient, and increase with the increase of other time sensitive coefficient. This indicates that the electricity consumers are more concerned about the current price, the more consumers pay attention to the current electricity price, the lower the price of the power system decision-making should be. On the contrary, if the consumer price for other time is more sensitive to current prices but not very sensitive, for the power system, the best strategy is to improve the electricity price to increase the welfare level of the power system.

Figure 5 Effect of price sensitivity coefficient on electricity price at different time

Full size|PPT slide

7 Concluding Remarks

In a hybrid power system composed of renewable energy and traditional energy, the marginal cost of renewable power is low and can be ignored, and the renewable energy power generation mainly depends on natural conditions and has the characteristics of intermittent. At the same time, renewable energy has energy substitution, can reduce carbon emissions in the process of power generation, increase the environmental green degree, and reduce environmental pollution. For traditional energy power, the marginal cost of traditional energy power generation is higher. However, traditional power generation relies mainly on fossil fuels such as coal, so it can achieve stable power supply. For the power system, the electric power is a kind of special goods, related to all aspects of production and life, it is necessary to ensure a steady supply of power, and can't be short of power. We consider the necessity for safe handling of power. Through this study, we find that the greater the difference in potential electricity demand between daytime and nighttime, the greater the difference between the daytime price and the nighttime price. Because the difference between the potential electricity demand of the daytime and the nighttime is greater, the demand for electricity mainly occurs during the daytime, so the difference between the price of electricity during the daytime and the price of electricity at nighttime will increase. At the same time, as the demand for electricity during the daytime and the sensitivity of nighttime electricity demand increases, the difference between the daytime price and the nighttime price will be reduced. This is because the demand for electricity will be fully distributed between daytime and nighttime when the sensitivity of power demand increases, resulting in a reduction in the price of the daytime and the price of electricity at nighttime. The sum of electricity prices during the daytime and nighttime is fixed regardless the one price system or the time-sharing pricing strategy is adopted. The average electricity price is negative related to the sum of the generation probabilities of the daytime and the nighttime. When the probability is greater, the actual electricity generation in the electricity market is larger, and the larger power supply will lead to an average electricity price decline.

The installed capacity of renewable energy is inversely related to the installed cost of renewable energy units, positively related to the carbon emission reductions of renewable energy sources, and is positively related to the traditional electricity generation costs of energy units. The supply price coefficient of traditional energy and electricity has two-way influence on the installed capacity of renewable energy. On the one hand, when the traditional energy supply price coefficient is low, the traditional energy supply for the price sensitivity is low, when the electricity price increases, the traditional energy power supply increases slowly, this renewable power will increase the installed capacity, so as to make this part of the profits brought by higher prices. On the other hand, when the traditional energy power supply price coefficient is high, the traditional energy supply for the price sensitive, the traditional energy power supply changes greatly. When prices drop a little, the traditional energy supply will fall sharply, the renewable energy generator will increase the installed capacity.

For future research, one can consider the general probability distribution of renewable energy power generation. One can also consider the dynamic pricing problem of multi period state. Besides, different renewable energy penetration condition and the installed capacity of renewable energy investment decision sound interesting.

Appendix 1

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

\begin{aligned} \frac{\partial^{2} W (p_{n}, p_{d}, k_{r})}{\partial p_{d}^{2}} & = - 2 γ - c_{2} {(β + γ - δ)}^{2} < 0, \\ \frac{\partial^{2} W (p_{n}, p_{d}, k_{r})}{\partial p_{n}^{2}} & = - 2 γ - c_{2} {(β - δ + γ)}^{2} < 0, \\ \frac{\partial^{2} W (p_{n}, p_{d}, k_{r})}{\partial k_{r}^{2}} & = - c_{2} {(q_{d} + q_{n})}^{2} < 0, \\ \frac{\partial^{2} W (p_{n}, p_{d}, k_{r})}{\partial p_{d} \partial p_{n}} & = 2 δ - c_{2} {(β - δ + γ)}^{2}, \\ \frac{\partial^{2} W (p_{n}, p_{d}, k_{r})}{\partial p_{d} \partial k_{r}} & = - c_{2} (q_{d} + q_{n}) (β + γ - δ), \\ \frac{\partial W (p_{n}, p_{d}, k_{r})}{\partial p_{n} \partial k_{r}} & = - c_{2} (q_{d} + q_{n}) (β + γ - δ) . \end{aligned}

Three order Hesse matrix:

\begin{array}{lll} [\begin{array}{c} \frac{\partial^{2} W (p_{n}, p_{d}, k_{r})}{\partial p_{d}^{2}} & \frac{\partial^{2} W (p_{n}, p_{d}, k_{r})}{\partial p_{d} \partial p_{n}} & \frac{\partial^{2} W (p_{n}, p_{d}, k_{r})}{\partial p_{d} \partial k_{r}} \\ \frac{\partial^{2} W (p_{n}, p_{d}, k_{r})}{\partial p_{n} \partial p_{d}} & \frac{\partial^{2} W (p_{n}, p_{d}, k_{r})}{\partial p_{n}^{2}} & \frac{\partial W (p_{n}, p_{d}, k_{r})}{\partial p_{n} \partial k_{r}} \\ \frac{\partial^{2} W (p_{n}, p_{d}, k_{r})}{\partial k_{r} \partial p_{d}} & \frac{\partial W (p_{n}, p_{d}, k_{r})}{\partial k_{r} \partial p_{n}} & \frac{\partial^{2} W (p_{n}, p_{d}, k_{r})}{\partial k_{r}^{2}} \end{array}] \\ = [\begin{array}{c} - 2 γ - c_{2} {(β + γ - δ)}^{2} & 2 δ - c_{2} {(β - δ + γ)}^{2} & - c_{2} (q_{d} + q_{n}) (β + γ - δ) \\ 2 δ - c_{2} {(β - δ + γ)}^{2} & - 2 γ - c_{2} {(β - δ + γ)}^{2} & - c_{2} (q_{d} + q_{n}) (β + γ - δ) \\ - c_{2} (q_{d} + q_{n}) (β + γ - δ) & - c_{2} (q_{d} + q_{n}) (β + γ - δ) & - c_{2} {(q_{d} + q_{n})}^{2} \end{array}] . \end{array}

The three order determinant for Hesse matrices:

| \begin{matrix} - 2 γ - c_{2} {(β + γ - δ)}^{2} & 2 δ - c_{2} {(β - δ + γ)}^{2} & - c_{2} (q_{d} + q_{n}) (β + γ - δ) \\ 2 δ - c_{2} {(β - δ + γ)}^{2} & - 2 γ - c_{2} {(β - δ + γ)}^{2} & - c_{2} (q_{d} + q_{n}) (β + γ - δ) \\ - c_{2} (q_{d} + q_{n}) (β + γ - δ) & - c_{2} (q_{d} + q_{n}) (β + γ - δ) & - c_{2} {(q_{d} + q_{n})}^{2} \end{matrix} | .

1) The first order principal:

- 2 γ - c_{2} {(β + γ - δ)}^{2} < 0.

2) The second order primary principal:

\begin{array}{lll} | \begin{array}{c} - 2 γ - c_{2} {(β + γ - δ)}^{2} & 2 δ - c_{2} {(β - δ + γ)}^{2} \\ 2 δ - c_{2} {(β - δ + γ)}^{2} & - 2 γ - c_{2} {(β - δ + γ)}^{2} \end{array} | \\ = {(2 γ + c_{2} {(β + γ - δ)}^{2})}^{2} - {(2 δ - c_{2} {(β - δ + γ)}^{2})}^{2} \\ = (2 γ + c_{2} {(β + γ - δ)}^{2} + 2 δ - c_{2} {(β - δ + γ)}^{2}) (2 γ + c_{2} {(β + γ - δ)}^{2} - 2 δ + c_{2} {(β - δ + γ)}^{2}) \\ = 4 (γ + δ) (γ - δ + c_{2} {(β + γ - δ)}^{2}) > 0. \end{array}

3) The third order primary principal:

\begin{array}{lll} | \begin{array}{c} - 2 γ - c_{2} {(β + γ - δ)}^{2} & 2 δ - c_{2} {(β - δ + γ)}^{2} & - c_{2} (q_{d} + q_{n}) (β + γ - δ) \\ 2 δ - c_{2} {(β - δ + γ)}^{2} & - 2 γ - c_{2} {(β - δ + γ)}^{2} & - c_{2} (q_{d} + q_{n}) (β + γ - δ) \\ - c_{2} (q_{d} + q_{n}) (β + γ - δ) & - c_{2} (q_{d} + q_{n}) (β + γ - δ) & - c_{2} {(q_{d} + q_{n})}^{2} \end{array} | \\ = 4 c_{2} (δ + γ) (δ - γ) {(q_{d} + q_{n})}^{2} \\ = 4 c_{2} (δ^{2} - γ^{2}) {(q_{d} + q_{n})}^{2} \\ = - 4 c_{2} (γ^{2} - δ^{2}) {(q_{d} + q_{n})}^{2} < 0. \end{array}

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.

\begin{aligned} {\begin{cases} p_{d} = \frac{(\begin{array}{c} a_{d} + 2 δ p_{n} - c_{1} (β + γ - δ) - v (- γ + δ) \\ - c_{2} (\begin{array}{l} β p_{n} - δ p_{n} + γ p_{n} \\ + q_{d} k_{r} + q_{n} k_{r} - a_{d} - a_{n} \end{array}) (β + γ - δ) \end{array})}{c_{2} {(β + γ - δ)}^{2} + 2 γ}, \\ p_{n} = \frac{(\begin{array}{c} 2 δ p_{d} + a_{n} - c_{1} (β - δ + γ) - v (δ - γ) \\ - c_{2} (\begin{array}{l} β p_{d} + γ p_{d} - δ p_{d} \\ + q_{d} k_{r} + q_{n} k_{r} - a_{d} - a_{n} \end{array}) (β - δ + γ) \end{array})}{c_{2} (β + γ - δ) (β - δ + γ) + 2 γ}, \\ k_{r} = \frac{(L - c_{1} + v)}{c_{2} (q_{d} + q_{n})} - \frac{(\begin{matrix} β p_{d} - (a_{d} - γ p_{d} + δ p_{n}) \\ + β p_{n} - (a_{n} - γ p_{n} + δ p_{d}) \end{matrix})}{q_{d} + q_{n}} - \frac{α}{c_{2} {(q_{d} + q_{n})}^{2}} . \end{cases} \end{aligned}

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

{\begin{cases} p_{d} = \frac{(α (β - δ + γ) (δ + γ) + (q_{d} + q_{n}) (a_{n} δ + a_{d} γ - (δ + γ) ((- δ + γ) L + β (l + v))))}{(2 (q_{d} + q_{n}) (γ^{2} - δ^{2}))}, \\ p_{n} = \frac{(α (β - δ + γ) (δ + γ) + (q_{d} + q_{n}) (a_{d} δ + a_{n} γ - (δ + γ) ((- δ + γ) L + β (l + v))))}{(2 (q_{d} + q_{n}) (γ^{2} - δ^{2}))}, \\ k_{r} = \frac{(2 α (- δ + γ + c_{2} {(β - δ + γ)}^{2}) + (q_{d} + q_{n}) (\begin{matrix} a_{d} c_{2} (β + δ - γ) + a_{n} c_{2} (β + δ - γ) \\ - 2 (\begin{matrix} c_{1} (δ - γ) + (- δ + γ + c_{2} {(β - δ + γ)}^{2}) L \\ + (- δ + γ + β c_{2} (β - δ + γ)) v \end{matrix}) \end{matrix}))}{(2 c_{2} {(q_{d} + q_{n})}^{2} (δ - γ))} . \end{cases}

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:

\begin{aligned} p_{d} - p_{n} & = \frac{(\begin{matrix} (q_{d} + q_{n}) (a_{n} δ + a_{d} γ - (δ + γ) ((- δ + γ) L + β (l + v))) \\ - (q_{d} + q_{n}) (a_{d} δ + a_{n} γ - (δ + γ) ((- δ + γ) L + β (l + v))) \end{matrix})}{2 (q_{d} + q_{n}) (γ^{2} - δ^{2})} \\ = \frac{(q_{d} + q_{n}) (a_{n} δ + a_{d} γ - a_{d} δ - a_{n} γ)}{2 (q_{d} + q_{n}) (γ^{2} - δ^{2})} \\ = \frac{(a_{n} δ + a_{d} γ - a_{d} δ - a_{n} γ)}{2 (γ^{2} - δ^{2})} \\ = \frac{(a_{d} - a_{n}) (γ - δ)}{2 (γ^{2} - δ^{2})} \\ = \frac{a_{d} - a_{n}}{2 (γ + δ)} . \end{aligned}

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

\begin{aligned} \bar{p} & = \frac{1}{2} (p_{d} + p_{n}) \\ = \frac{(α (β - δ + γ) (δ + γ) + (q_{d} + q_{n}) (a_{n} δ + a_{d} γ - (δ + γ) ((- δ + γ) L + β (l + v))))}{4 (q_{d} + q_{n}) (γ^{2} - δ^{2})} \\ + \frac{(α (β - δ + γ) (δ + γ) + (q_{d} + q_{n}) (a_{d} δ + a_{n} γ - (δ + γ) ((- δ + γ) L + β (l + v))))}{4 (q_{d} + q_{n}) (γ^{2} - δ^{2})} \\ = \frac{2 α (β - δ + γ) (δ + γ) + (q_{d} + q_{n}) (a_{n} δ + a_{d} γ + a_{d} δ + a_{n} γ - 2 (δ + γ) ((- δ + γ) L + β (l + v)))}{4 (q_{d} + q_{n}) (γ^{2} - δ^{2})} \\ = \frac{(2 α (β - δ + γ) + (q_{d} + q_{n}) (a_{d} + a_{n} - 2 (β - δ + γ) L - 2 β v))}{4 (γ - δ) (q_{d} + q_{n})} \\ = \frac{α (β + γ - δ)}{2 (γ - δ) (q_{d} + q_{n})} + \frac{(a_{d} + a_{n} - 2 (β + γ - δ) L - 2 β v)}{4 (γ - δ)} . \end{aligned}

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 = (4 α (γ - δ) + (q_{d} + q_{n}) (a_{d} + a_{n} + 2 (δ - γ) (2 L + v))) / (4 ((q_{d} + q_{n}) (L + v) - α))

\begin{aligned} \frac{\partial k_{r}}{\partial α} & = \frac{δ - γ - c_{2} {(β + γ - δ)}^{2}}{c_{2} (γ - δ) {(q_{d} + q_{n})}^{2}} < 0, \\ \frac{\partial k_{r}}{\partial L} & = \frac{γ - δ + c_{2} {(β + γ - δ)}^{2}}{c_{2} (γ - δ) (q_{d} + q_{n})} > 0, \\ \frac{\partial k_{r}}{\partial v} & = \frac{γ - δ + β c_{2} (β + γ - δ)}{c_{2} (γ - δ) (q_{d} + q_{n})} > 0. \end{aligned}

1) When

β < θ

\begin{aligned} \frac{\partial k_{r}}{\partial β} & = \frac{4 α (β + γ - δ) + (q_{d} + q_{n}) (a_{d} + a_{n} - 4 β (L + v) + 2 (δ - γ) (2 L + v))}{2 (δ - γ) {(q_{d} + q_{n})}^{2}} \\ = \frac{4 β ((q_{d} + q_{n}) (L + v) - α) - 4 α (γ - δ) - (q_{d} + q_{n}) (a_{d} + a_{n} + 2 (δ - γ) (2 L + v))}{2 (γ - δ) {(q_{d} + q_{n})}^{2}} \\ < 0. \end{aligned}

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

\begin{aligned} \frac{\partial k_{r}}{\partial β} & = \frac{4 α (β + γ - δ) + (q_{d} + q_{n}) (a_{d} + a_{n} - 4 β (L + v) + 2 (δ - γ) (2 L + v))}{2 (δ - γ) {(q_{d} + q_{n})}^{2}} \\ = \frac{4 β ((q_{d} + q_{n}) (L + v) - α) - 4 α (γ - δ) - (q_{d} + q_{n}) (a_{d} + a_{n} + 2 (δ - γ) (2 L + v))}{2 (γ - δ) {(q_{d} + q_{n})}^{2}} \\ > 0. \end{aligned}

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.

References

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Funding

the National Natural Science Foundation of China(71273091)

the National Natural Science Foundation of China(71272015)

the Postgraduate Innovation Fund Project of SUFE(CXJJ-2016-327)

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Abstract
Key words
Cite this article
1 Introduction
2 Literature Review
3 Model Preliminaries
3.1 Model Description
Figure 1 Hybrid power system
3.2 Model Parameters and Decision Variables
Table 1 Model parameters
Table 2 Model decision
3.3 Model Function
4 Power System Model
4.1 Objective Function of Power System
4.2 Optimal Daytime Price
4.3 Optimal Nighttime Price
4.4 Optimal Installed Capacity of Renewable Energy Sources
4.5 Optimal Decision Combination of Power System
5 Property Analysis
6 Numerical Example
Figure 2 The effect of daytime price and nighttime price on system welfare
Figure 3 The impact of daytime electricity price and night price on installed capacity
Figure 4 Impact of power supply sensitivity coefficient on installed capacity
Figure 5 Effect of price sensitivity coefficient on electricity price at different time
7 Concluding Remarks
Appendix 1
Appendix 2
Appendix 3
Appendix 4
References
Funding

Received	Accepted	Published
2017-08-18	2017-12-07	2018-06-25
Issue Date
2018-06-25

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Abstract

Key words

Cite this article

1 Introduction

2 Literature Review

3 Model Preliminaries

3.1 Model Description

Figure 1 Hybrid power system

3.2 Model Parameters and Decision Variables

Table 1 Model parameters

Table 2 Model decision

3.3 Model Function

4 Power System Model

4.1 Objective Function of Power System

4.2 Optimal Daytime Price

4.3 Optimal Nighttime Price

4.4 Optimal Installed Capacity of Renewable Energy Sources

4.5 Optimal Decision Combination of Power System

5 Property Analysis

6 Numerical Example

Figure 2 The effect of daytime price and nighttime price on system welfare

Figure 3 The impact of daytime electricity price and night price on installed capacity

Figure 4 Impact of power supply sensitivity coefficient on installed capacity

Figure 5 Effect of price sensitivity coefficient on electricity price at different time

7 Concluding Remarks

Appendix 1

Appendix 2

Appendix 3

Appendix 4

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References

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Content to export

Abstract

Key words

Cite this article

1 Introduction

2 Literature Review

3 Model Preliminaries

3.1 Model Description

Figure 1 Hybrid power system

3.2 Model Parameters and Decision Variables

Table 1 Model parameters

Table 2 Model decision

3.3 Model Function

4 Power System Model

4.1 Objective Function of Power System

4.2 Optimal Daytime Price

4.3 Optimal Nighttime Price

4.4 Optimal Installed Capacity of Renewable Energy Sources

4.5 Optimal Decision Combination of Power System

5 Property Analysis

6 Numerical Example

Figure 2 The effect of daytime price and nighttime price on system welfare

Figure 3 The impact of daytime electricity price and night price on installed capacity

Figure 4 Impact of power supply sensitivity coefficient on installed capacity

Figure 5 Effect of price sensitivity coefficient on electricity price at different time

7 Concluding Remarks

Appendix 1

Appendix 2

Appendix 3

Appendix 4

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References

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Footnotes

Funding