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In order to optimize the ladder-pricing scheme in Shanghai, we present a multi-objective optimization model (MOOM). To build this model, first we use price elasticity theory; divide the ladder pricing into peak electricity bill and valley electricity bill in the time dimension to model the single-user demand response. Second based on the single-user demand response model, combined with the overall users’ electricity distribution density function, we build an all-users demand response model. The proposed model has two objectives: minimize energy consumption and maximize residents’ satisfaction. Simulation results confirm that the proposed model can optimize the ladder-pricing scheme.

Over a longer period of time, as a result of a unitary low price has been implemented in China which had a problem of cross-subsidization [

Ladder pricing is based on Ramsey rule [

The discussions and analysis in this paper are based on the above literature. In this paper, we construct a single-user demand response model for users with different stalls in ladder pricing. Then combined the overall users’ electricity distribution density function, an all-users demand response model is established. Finally, we propose a multi-objective optimization model whose objectives are to minimize energy consumption and maximize residents’ satisfaction.

The rest of this paper is organized as follows. We introduce elasticity in Section 2. The single-user demand response model and all-users demand response model are formulated in Section 3. The multi-objective optimization model is presented in Section 4. In Section 5, simulation results are shown.

In microeconomics, the elastic theory is mainly used for researching the measurement of how an economic variable is to change in another [

Price elasticity of demand is one of elasticity, commonly referred to as the price elasticity.

Price elasticity of demand primarily used to represent a period of time, the extent of the relative change in the demand for commodity reactions with the relative changes in the price of the commodity itself. Price elasticity of demand usually represented by the following formula:

where ε is the price elasticity of demand coefficient,

Under normal circumstances, the demand for a commodity is not only concerned with its own price, but also related to the price of similar products. Also in the electricity market, as in the TOU conditions, the energy in the time dimension of peak, valley, flat can be seen as three different goods, user demand for electricity usually depends not only on the flat period price, but also related to peak and valley time price [

Cross elasticity of demand mathematical expression is as follows:

where

Defines price elasticity matrix:

where

Changes in user requirement matrix can be formulated as follows:

where

Take (1)-(3) into (4), we can get:

where

Ladder pricing in Shanghai is divided into three levels by user electric energy consumption, assume the lowest level consumption in the (

where

From (6), we can see the first-tier user only responds to the first-tier pricing changes, the second-tier user needs to responds to the first-tier and second tier pricing changes and the third-tier user needs to responds to all the three tiers pricing changes.

In this paper, we suppose that all the users do not shift from one tier to another after they respond to the pricing changes.

The pricing changes, and the user has regulated their demand, the electricity bill can be obtained as:

where

We have formulated single-user demand response, now the question is: How to get the all-users demand response? To answer this question, we applied the overall users electricity distribution density function. Define f(x) as the peak electric energy consumption f(y) as the valley electric energy consumption and:

Given f(x), f(y), the electric energy consumption and electricity bill can be derived as:

where

The satisfaction of the residential customer can be modeled as:

where

Clearly, the lower C^{*} is the higher satisfaction the customer will get. If C^{*} is equal to C, satisfaction will approach the upper bound 1. If C^{*} is twice than C, satisfaction will be close to 0.

Implementation ladder pricing policy, an important goal is to guider resident users to reduce electricity consumption, improve power efficiency, to increase the utilization of electric power system [

where

On considering only price influence of factors on residential electricity consumption condition, when the electricity price increases, the overall residential users’ electricity consumption should show a negative trend. It is reasonable to assume that we always have the following constraint:

So far, we are ready to formulate the ladder pricing optimization problem as the following multi-objective optimization problem:

From (14), ②③④ guarantee the gross proceeds of the electric power company not reduced, but also guarantee electricity bill growth after user response within a certain range. ⑤⑥ refer to guidance of the National Development and Reform Commission. ⑦⑧ refer to the current Ladder Pricing Scheme in Shanghai, that valley electricity bill is half of the peak electricity bill.

In the above optimization problem, the two objectives conflict with each other, we cannot find the optimal solution to meet these two objectives. So in this article, we will use genetic algorithm to find the Pareto set of this model.

According to the guidance of NDRC, this paper supposes that the first-tier electricity price will not change, the second-tier peak electricity price relative to the first-tier peak will increase Δp_{p}_{,2} RMB/kWh, the second-tier valley electricity price will increase Δp_{p}_{,2}/2 RMB/kWh, the third-tier peak electricity price relative to first-tier electricity peak price will increaseΔp_{p}_{,3} RMB/kWh, and the third-tier valley electricity price will increaseΔp_{p}_{,3}/2 RMB/kWh. The electricity energy consumption standard for each tier in Shanghai will remain the same.

The experimental data are actual consumption data of 1487 residents in an area of Shanghai in 2012. From these data, statistical results show that:

1) There are 1242 residents belongs to first-tier user. Average electricity consumption per month is 167 kWh. Ratio between peak and valley is 0.692:0.308;

2) There are 217 residents belongs to first-tier user. Average electricity consumption per month is 293 kWh. Ratio between peak and valley is 0.691:0.309;

3) There are 28 residents belongs to first-tier user. Average electricity consumption per month is 380 kWh. Ratio between peak and valley is 0.694:0.306.

Taking the limited electricity historical data into account, the data of electric power elasticity matrix references that in [

By using R to test the distribution of peak and valley power consumption for each tier customer, the result shows that they are all belongs to lognormal distribution. The lognormal distribution is as follows:

The parameters of these density functions can be worked out through the history power data of 1473 residents, which are shown in

We use Matlab to solve the multi-objective optimization model (13) by genetic algorithm. Finally the result of this optimization problem is shown as follows:

From the result, we can see when

If we take the current ladder pricing policy of Shanghai into (7), the result shows the 1478 residents can conserve 502.378 kWh energy the whole year, the extra electricity bill is 4561.142 RMB and the resident satisfaction is 0.997.

Now if the ladder pricing decision makers want to save more energy relatively, when

User | Period | Price Elastic | Cross-Price Elastic |
---|---|---|---|

First-tier Resident | Peak | −0.0654 | 0.0138 |

Valley | −0.0606 | 0.0207 | |

Second-tier Resident | Peak | −0.0903 | 0.0294 |

Valley | −0.0849 | 0.0396 | |

Third-tier Resident | Peak | −0.0513 | 0.0159 |

Valley | −0.0447 | 0.0234 |

Density Function | ||
---|---|---|

7.150205 | 0.4095554 | |

6.30344 | 0.5020718 | |

7.758021 | 0.2832774 | |

6.920627 | 0.3824835 | |

8.025475 | 0.2658047 | |

7.163866 | 0.4158537 |

In this paper, we have first analyzed the single-user demand response and all-users demand response. Based on these and combined with the density function of energy consumption, we have proposed a multi-objective optimization model. Through the optimization model, the design makers can formulate different ladder pricing scheme for various purpose in different period.

Here, we have just focused on optimal the price of ladder pricing. A higher research on optimal both the price and the tier range of ladder pricing may be done in the future.

Xiu Cao,Haiyong Jiang,Lei Huang,Xueping Wang,Xuqi Zhang, (2016) Study on Demand Response of Residential Power Customer. Journal of Power and Energy Engineering,04,1-7. doi: 10.4236/jpee.2016.47001