Optimal Model for Electricity Tariff Calculation R. V. Menta, E. J. de Oliveira, L. W. Oliveira, B. H. Dias and A. L. M. Marcato All authors are from Federal University of Juiz de Fora Juiz de Fora - Brazil e-mail: [email protected] Abstract— This work proposes a time-of-use (ToU) tariff calculation based on Optimal Dynamic Tariff Method (ODTM) for both peak and off-peak periods. The proposed approach is formulated as a non-linear program where Shifted Demand (SD) from peak to off-peak is considered as an optimization variable as well as peak and off-peak tariffs. The proposed model takes into account consumer behavior when a change in the price of electricity for peak and off-peak hours occurs. In addition, the distribution network is included to evaluate changes in voltage profile and investment costs. Two distribution test systems are used to show the effectiveness of the ODTM to lead the system for a win-win energy market. Index Terms-- Non-linear program, time-of-use tariff, consumer behavior, win-win energy market, demand response.

I.

INTRODUCTION

Under smart grids concept both distributors and consumers have an opportunity to design an efficient mechanism to calculate the tariff in order to transform the energy market in a win-win opportunity with features to replace the current Flat Rate Model (FRM) of electricity. On distributors’ side the main interest is to avoid investment costs by moving partial amount of energy from peak to off-peak period. On the other hand, consumers only modify their consumption habits if they are attracted by lower energy prices in an off-peak period [1]. With advances in measurement technology and automation the restructuring of the distribution systems has become attractive. In this context the agents, directly or indirectly, use the network to interact in the electricity market by exchanging information with each other. Therefore, the market model that will guide the energy price formation is essential to lead investments in the supply side as well as the consumer’s side. The Time-of-Use (ToU) tariff is a type of demand response that uses two or more tariff values throughout the day. The consumer response is important in the structuring of the tariff model in smart grids since variations in electricity tariff affect the optimization of distribution systems expansion and operation [2].

12]. Reference [4] proposed Particle Swarm Optimization (PSO) to minimize the electricity cost considering the price elasticity of demand. Genetic Algorithm was used in reference [5] for smart home scheduling. Reference [6] proposed a neural networks approach to calculate tariff by using a model of consumers. A multi-objective problem to maximize the benefits of energy distributor and consumers using game theory was discussed in [7]. The papers [8] and [9] presented a method to calculate the tariff considering the quality of energy and a differential rate of use the network. The method described in [10] assesses the Time-of-Use (ToU) tariff considering the uncertainty in the price elasticity of demand. Consumer behavior is analyzed in [11] for the ToU tariff concepts. A novel model for time-of-use tariff is proposed in [12] based on Gaussian Mixture Model. Under this background, it can be observed that distribution network was not considered in tariff calculation, so this paper proposes a methodology to calculate ToU tariff and shifted demand using nonlinear programming based on interior point method with security barrier [13]. The active load is modeled to represent the participation of the consumers in the energy market. Also, the reductions in investment costs are considered as well as electrical radial distribution network. Case studies using distribution test systems show the effectiveness of the proposed approach. II.

=

So, based on tariffs the consumers decide to reduce or increase its energy consumption following their pattern of behavior. The use of smart meters is essential to ensure the efficiency of the differentiated tariffs model [3], through the exchange of information between the distributor and consumers. A number of previous works have been studied on solutions and proposed formulations for tariff calculation [4The authors would like to thank CAPES, CNPq, FAPEMIG and INERGE – UFJF for financial support.

PROPOSED METHODOLOGY

The proposed approach is developed to take into account two load levels: peak (p) and off-peak (np). The technique considers the objective of consumers (minimize energy bill) as well as the distributor point of view (maximize profits postponing investments). Hence, an optimal energy price minimizes the total costs and promotes a larger profit compared to conventional Flat Rate Model (FRM). Therefore, the proposed Optimal Dynamic Tariff Method (ODTM) can be written as follow: +

(1)

Subject to: ,

−

−

,

=0

,

(2)

∈ ,

=

,

.

− ,

,

−

.

,

,

.(

,

−

, ∈

+

=0

,

)

(3) (4)

=−

,

,

.(

+ −

)−

, ,

.

)

,

.(

,

Behavior of consumer at peak period Behavior of consumer at off-peak period ≥0 ( , ) ≥

≤ ≤ 0≤

,

The active and reactive power balances for each bus are (5) represented by equations (2) to (5). This enlarged set of equations is necessary to accommodate the demand in a peak (6) and off-peak period, respectively to x=p and x=np. The reactive power is considered in this paper to improve the (7) precision of active power loses. The constraints (6) and (7) represent the consumers’ behavior facing tariff variation. In other words, they represent how much the consumer should shift their demand from peak (9) to off-peak attracted by price. Therefore, this paper proposes a nonlinear function to relate the variation between Shifted (10) Demand (SD) and tariff for each period, as described below: (8)

( ,

)

( , )

≤

≤

≤

(11)

( )

−

( )

−

1 1+

( , )

( , )

− 0,5

( )

=0

(6)

− 0,5

( )

=0

(7)

(12) 1 1+

( ,

)

( ,

)

Where:

This Objective Function (OF) parcel is related with the profit of the Electric Power Distributor;

Where: ( ) ( , )

Represents the discount in the consumer energy bill; x

Represents the load level condition: peak (x=p) and off-peak (x=np);

,

Active power generation at bus j at load level x;

,

Reactive power generation at bus j at load level x;

,

Active power flow in branch i-j at load level x;

,

Reactive power flow in branch i-j at load period x;

,

Active load demand at bus j at load level x;

,

Reactive load demand at bus j at load level x;

Ω

Set of the buses connected to bus i through distribution branches; Conductance of branch ij; Susceptance of branch ij; Shunt susceptance of branch ij; ,

,

Phase angle between buses i and j at load level x; Voltage profile at bus i at load level x; Lower voltage limit at buses; Upper voltage limit at buses.

Shifted Demand from peak to off-peak; Tariff for each consumer at load level x; Represents the tariff to flat rate model (FRM);

( , )

and ( ,

( )

)

Represents the consumer sensitivity facing the tariff variation. For example, more sensitive consumers have a large amount of demand shifted for small change in the tariff. The opposite occurs for low sensitivity consumers to the same change in price; This parameter is related with the saturation of shifted demand where variations in the tariff are not able to change the actions of consumers.

Figures 1 and 2 show a consumer behavior curve based on proposed equations (6) and (7) considering the following values: Tf = 0.4814 $/kWh, a(p) = 7 kWh/$, a(np) = -25.5 kWh/$ and ( ) = 0.894 kW. From Figure 1, it can be observed that the high tariff induces the consumer to shift their demand. On the other hand, Figure 2 shows that the shifted demand increases when low tariff is practiced in off-peak period. In both figures, if tariff is equal to Tf, no demand will be shifted (SD = 0). Although these parameters have no real values, they represent suitable consumer behavior for the analysis proposed in this paper. However, the real values of these parameters can be considered by appropriate survey, but this issue is not addressed in this paper. The constraint (8) guarantees that the discount to consumer will be null or greater than zero. In other word, the consumer energy bill for ToU tariff will never exceed the value of the flat rate model (FRM) [14]. Constraint (9) ensures that the peak demand will always be greater than off-peak demand avoiding the new formation of peaks in system load curve. Limits of

tariff and voltage profile are ensured by (10) and (11), respectively. In equation (12), the maximum value of the shifted demand was considered equal to 40% of the peak demand for fixed tariff model.

The revenue comes from the sale of energy. In equation (14), the optimization variables: ( , ), ( , ) and SD(j) represent the ToU tariff to consumer j in peak (p), off peak (np) and shifted demand (SD), respectively. The parameters ( , ) and ( , ) represents the well-known average demands by applying the Flat Rate Model (FRM) to consumer ( )

j. The parameter α is equal to

(

and it corresponds to the

)

ratio between the peak (t(p)) and off-peak (t(np)) periods to take into account the shifted energy. The cost of distributor (SE) in equation (13) is composed by three parcels: 1) Total energy Cost: Represents the electricity cost purchased by SE on market in order to supply the consumers connected in the grid, see equation (15). =

( ) ( )

( )

+

(

(

)

) (

)

(15)

Where parameters ( ) and ( ) represent the fixed costs of kWh during the peak and off-peak periods, respectively. These prices represent an average energy cost. The variables ( ) and ( ) are the average demands and they vary with optimization variable SD as follow:

Figure 1. Behavior of consumer at peak period.

( )

=

( ,

)

−

)

+ .

(16)

( )

∈

(

)

=

( ,

(17)

( )

∈

2) Total demand Costs: Represents the total value paid by distributor as described by equation (18), where parameters ( ) and ( ) denote fixed demand costs. = Figure 2. Behavior of consumer at off-peak period.

On distributors side the profit is given by: = (

−

)

(13)

=

( )

( ,

)

∈

−

+

(

)

( ) ( ,

∈

+ .

( , )



(14)

) ( )

( ,

)

( )

+

(

)

(

)

(18)

3) Total investment costs: Represents SE investments in power grid. In this paper, it is considered that those costs vary according to the average demand as described in equation (19). In this equation, parameters ( ) and ( ) are adopted as investment costs to ensure the maintenance of equipment and the expansion of networks. =

Where:

( )

( ) ( )

+

(

) (

)

(19)

Although the marginal investment costs, tariff for power distributor buying electricity on market and monthly off-peak and peak times were considered as constants, the proposed approach is suitable to take in account the variation of these parameters due to the adoption of the nonlinear model. The second part of the objective function (1) represents the discount for the consumer. Therefore, the consumer’s point of view is to maximize the difference between ToU (BillToU) and

TABLE I. COST OF DISTRIBUTOR (SE)

FRM (Bill0), see equation (20). Equation (21) shows the (BillToU) formulation for consumer j, which varies according to SD, where p0 and pn0 denote peak and off-peak for flat rate model, respectively. On the other hand, equation (22) represents the Bill0 determination for flat rate tariff. =

−

Costs ($/kWh)

(20)

+

)

(

Peak (p)

0.2736

0.4058

($/kW)

10.34

27.26

($/kW)

17.00

17.00

: energy cost; : demand cost; : investment cost.

Where: ( ,

Off-Peak (np)

)

=

( )

( ,

( ,

)

) +

.

−

( )

( , )

(21) ( ,

( )

)

In this study the ToU tariff can vary between 0.5 and 1.0 Tf. Also the voltage profile limits range from 0.85 to 1.0 pu. The results are compared with the FRM to verify the effectiveness of the proposed approach. A. Results: 33-Bus System

( , )

=

( ,

) ( )

+

( ,

) (

)

.

(22)

Analytically, the problem may be simplified by noting that when inserting equation (14) and (21) into the objective function, the revenue terms cancel out. So, the objective function can be replaced by Equation (23). Although this equation varies with only SD(j), it is connected with Tarcons(i,p) and Tarcons(j,np) by equations (6), (7) and (8) ensuring enough discount for consumer side.

The 33-bus system shown in reference [16] is composed of 37 branches, which nominal voltage is 12.66 kV and the total load corresponds to 3,715.0 kW and 2,300.0 kVAr that is considered as the peak load. In order to test the proposed approach, the simulation is performed considering five different types of consumers, which have their behavior parameters established in Table II. The second column of the table shows the group of buses where consumers present the same behavior. TABLE II. PARAMETER FOR CONSUMER BEHAVIOR CURVE.



=

+

+

(23)

Type

Buses

a(p) kWh/$

a(np) kWh/$

After solution of problem (1) - (12) by using interior point method [13], the distributor obtains the tariff value for both peak and off-peak period as well as the expected amount of shifted demand for all consumers connected in the electrical network.

1

4, 5, 8, 9, 10, 14, 15, 16, 21, 22, 25, 26, 27 and 32

7.0

-25.5

2

2, 11, 12, 13, 19, 20, 28 and 30

5.0

-24.0

3

1, 3, 17, 18, 29 and 31

5.5

-24.5

4

6 and 23

6.0

-26.0

The development of this work considering only two load level makes the model easier to implement in real distribution systems due to less volatility in tariffs [15]. However, the proposed methodology can be extended to consider more than two load levels in active load curve.

5

7 and 24

7.0

-27.0

III.

CASE STUDIES

The case studies were performed using the 33 and 119 buses test systems, where the standard data can be found in references [16] and [17], respectively. In addition, it is assumed that the substation (distributor) provides average demand to peak and off-peak. During peak period the active and reactive power loads are equal to standard data, while for off-peak the loads are reduced by a factor of 0.625. The peak period is composed by 90 hours per month whereas off-peak has 630 hours. The tariff to FRM ( ) is considered equal to 0.4814 $/kWh, which ensures the payback already established. The energy and demand costs as well as investment cost per month are given in Table I.

After setting all parameters’ values, the proposed problem (1) - (12) can be solved using any nonlinear programming package. This paper has adopted the nonlinear programming based on interior point method with security barrier [13] which is a gradient-based method. This package works well on the proposed problem because the objective function, as well as the constraints, has continuous first derivatives. In this paper the convergence considering the load flow problem is better evaluated. Table III shows the ToU tariff results related to peak and off-peak for each type of consumer. It can be emphasized that the peak tariff is greater than Tf as well as off-peak is lower for all types of consumers. These optimized results are attractive to the consumer to shift the demand from peak to off-peak.

TABLE III . TARIFF FOR PEAK AND OFF-PEAK: 33-BUS SYSTEM.

Peak

Off-Peak

1

0.7442

0.4093

Table V shows the results for both FRM and ToU considering the consumer side. ToU introduces a considerable benefit for all types of consumers as shown in the fourth row of the table which varies according to consumer behavior curve.

2

0.9255

0.3889

TABLE V. RESULTS FOR CONSUMERS ($/MONTH): 33-BUS SYSTEM.

3

0.8635

0.3956

Type

FRM

ToU

Discount (%)

4

0.8062

0.4035

1

206,096.40

195,538.14

5.12

5

0.8225

0.3896

2

181,644.26

174,464.30

3.95

3

188,630.57

180,376.10

4.38

4

144,383.90

137,943.00

4.46

5

144,383.90

134,406.74

6.91

Types of Consumers

Tariff ($/kWh)

Figure 3 shows the percentage of optimal Shifted Demand (SD) for each type of consumer. The percentage is different because the proposed approach has used different model for each type of consumer. It should be emphasized that in a practical application this sensitivity values should be obtained through monitoring the real behavior of consumers face to energy price fluctuations. The optimized values of tariffs and shifted demand can be used to determine other interesting results for the market analysis considering both distributor and consumer sides.

Besides ToU tariff introduces a win-win market as described in tables IV and V, other benefits have been obtained due to shifted demand from peak to off-peak. For example, the voltage profile increased in all buses because the peak load decreases. The most significant improvements in voltage profile occurred at busbar 17 that has increased from 0.91 to 0.94 pu due shifted demand. In addition, the active power losses were reduced by 3.92%. B. Results: 119-Bus System The system of 119 bus bars [17] is composed by 133 branches, which nominal voltage is 11.0 kV and active and reactive load given in standard data are considered peak load. The simulation is performed considering five different types of consumers, which have behavior parameters established in Table VI. The second column of the table shows the group of buses where the consumers present the same behavior. It should be highlighted that the parameters a(p) and a(np), related with consumer behavior are the same used in 33-Bus system analysis. TABLE VI. PARAMETER FOR CONSUMER BEHAVIOR CURVE.

Figure 3. Shifted Demand: 33-Bus System.

Table IV presents a comparison between FRM and ToU under distributors point of view. It can be observed a substantial reduction in a total investment cost as well as energy and demand costs when ToU was adopted. On the other hand, the revenue was reduced because the consumers shifted their demand for cheaper period (off-peak). TABLE IV. RESULTS FOR DISTRIBUTION ($/MONTH): 33-BUS SYSTEM. Total Costs

T

ProfitSE:

FRM

ToU

107,344.80

92,111.04

556,196.63

543,115.59

131,578.38

104,703.58

865,139.00

822,728.28

70,019.19

82,798.06

Type

Buses

a(p) kWh/$

a(np) kWh/$

1

From 105 to 123

7.0

-25.5

2

From 45 to 65

5.0

-24.0

3

From 29 to 44

5.5

-24.5

4

From 66 to 103

6.0

-26.0

5

From 2 to 27

7.0

-27.0

After setting all parameter values, the proposed problem (1) - (12) has been solved. Table VII shows the optimal results obtained for each type of consumers. It can be observed different values for tariff, which are dependent of consumer behavior. Figure 4 shows the results to shifted demand. In this case, consumers of type-1 had the greater value of SD because their off-peak tariff was the cheapest as well as peak tariff was the highest.

TABLE VII. TARIFF FOR PEAK AND OFF-PEAK: 119-BUS SYSTEM. Type

Type

FRM

ToU

Discount (%)

Peak

1

1,175,619.55

1,022,881.76

13.0

694,624.63

662,025.03

4.7

Tariff ($/kWh) Off-Peak

TABLE IX. RESULTS FOR CONSUMERS($/MONTH): 119-BUS SYSTEM.

1

0.3283

1.0392

2

2

0.3786

0.9750

3

1,072,654.27

1,030,907.68

3.9

3

0.4018

0.8361

4

1,718,711.89

1,641,367.76

4.5

4

0.4030

0.8081

5

626,966.80

586,068.53

6.5

5

0.3936

0.8077

Although consumer behavior parameters were the same for both case studies (33 and 119-Bus systems), the results for the tariff as well as shifted demand have been quite different as shown in figures 3 and 4. These aspects emphasize that the tariff changes according to the physical conditions of the network and not only demand response. So, the entire demand and supply represented into one single node can be not suitable to evaluate ToU tariff. Table VIII and IX show a comparison between FRM and ToU under distributor and consumers point of view, respectively. As occurred with the 33-Bus system, it can be verified a reduction in investment and energy purchased by the distributor. So, despite the revenue decrease, profit has increased substantially. The same situation has occurred with consumers that have obtained reduction with electricity cost. These facts show the positive feature of win-win process in which no participant wish to leave the pool.

As in the previous case study, other positive aspects of ToU can be highlighted: (i) the voltage profile increased in all buses because the peak load decrease. The most improvements in voltage profile occurred at busbar 80 that increased from 0.86 to 0.91 pu due shifted demand and (ii) the active power losses were reduced to about 4.0%. Although these benefits have not been included in the objective function they were achieved by the system. A sensibility analysis has been conducted for this system in order to show the robustness of the win-win market. Figure 5 shows the variation between the profit of the distributor and the average discount provided to consumers. It can be highlighted the optimal profit obtained by the proposed methodology corresponding to 6.53% of average discount to consumer. In addition, 7.09% of average discount correspond to a FRM profit. Then it should be noted that even around the optimal point there are other interesting points that remain attractive for both distributor and consumers. Therefore, these points also guarantee a profit under possible uncertainties of consumer behavior.

Figure 4. Shifted Demand: 119-Bus System. TABLE VIII. RESULTS FOR DISTRIBUTION ($/MONTH): 119-BUS SYSTEM. Total Costs

T

ProfitSE:

FRM

ToU

657,480.93

558,836.27

3,405,027.21

3,320,134.27

806,115.29

632,113.67

5,288,577.14

4,943,250.76

419,953.71

432,166.55

Figure 5. Sensitivity of Profit and Discount: 119-Bus System.

IV.

CONCLUSION

This work has investigated the electricity ToU tariff considering consumer behavior modeled by nonlinear curves. The consumers’ response was established for peak load level as well as off-peak. From the results obtained, some points can be highlighted:

•

The proposed optimization model was efficient to provide attractive prices for consumers leading to shift peak demand;

[5]

•

The optimized values of ToU tariff have provided a reduction in a total investment cost as a consequence the distributor profit has increased;

[6]

•

The sensitivity analysis performed between discount and profit has shown the robustness of optimal point because the distributor can offer a greater discount to consumers and still obtain an attractive profit when compared to flat rate model;

[7]

•

The proposed approach for optimal ToU calculation has shown that the ToU tariff was able to maintain the win-win market even around the optimal market point;

•

The results have shown that the distribution companies can improve both voltage profiles and active power losses by shifting the demand from peak to off-peak period;

•

The tariff changes according to demand response as to the physical conditions of the network.

Although the results are presented for two load levels, the proposed methodology can be extended to other cases. This aspect shows that the proposed approach is a promising technique for application in smart grid environments. V. [1] [2] [3]

[4]

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