Design and Analysis of Dynamic Pricing Strategies
3.9 Profitability Robustness Analysis
This section analyzes the optimal ToU tariffs obtained to ensure their profitability if some model parameters are known with uncertainty. To carry on with this study, one consumer consumption pattern is considered in the following analysis. However, the same ideas can be generalized for all consumer categories.
The price elasticity parameters are considered to be known with uncertainty. The optimal ToU tariffs profitability is analyzed by calculating the gains of the consumer and the UC with respect to the uncertain parameters of the consumer behavior model.
Moreover, the difference caused by the optimal ToU tariff in the profit of the consumer and the UC is influenced by the weighting factors introduced in the cost-based objective function (3.25). Hence, the effect of changing the weighting values is also analyzed.
3.9.1 Price-elasticity uncertainties
Let us recall that the price sensibility parameters of the matrixΞi in the consumer behavior model (3.20) are supposed to be known to find the optimal ToU tariff for each consumer category. If the real price sensibility is different, then the ToU tariff may not be optimal.
Figure 3.22: Average consumption curve for the three categories after and before the imple-mentation of the optimal ToU electricity tariffs in City B in winter season.
Figure 3.23: Total consumption of the eligible categories medium-low: 4, average: 3, medium-high: 2, and high: 1 and the consumption distribution before (left) and after (right) implementing the optimal ToU tariffs for City B in winter season.
Moreover, it is useful to analyze the profitability robustness of the tariff against the uncer-tainty of the elements of Ξi.
For the sake of simplicity, it is assumed that two price periods ToU electricity tariffs are designed and applied to one consumer category. The price sensibility used for the calculation of the ToU tariffs is a 2×2 matrix so that the nominal values are denoted by Ξ0= [ξij,0], (i,j∈ {1,2}).
The diagonal elements of the price sensibility matrix significantly impact the loads shifting from one period to another, determining the gains on UC and consumer sides. Hence, the off-diagonal elements are going to be ignored for this analysis. It is assumed that the diagonal elements ofΞ0 areξ11,0=−0.7 and ξ22,0=−0.5 and remark that the other elements can be obtained by the consumption conservation assumption (3.26).
The goal is to prove the robustness of the optimal ToU tariff against the price sensibility uncertainties that may be introduced while performing measurements or identifying the latter. The optimal ToU tariffs are calculated for Ξ0 where the on-peak price is 1.49, and the off-peak price is 0.8174. The consumer and UC profits computed using the optimal ToU tariffs are 0.2133 and 0.2415, respectively.
To check the robustness of ToU tariffs profitability for both the UC and the consumers, using the optimal ToU tariffs, the consumer behavior model (3.20) is used to calculate the new loads for different parameters in the price sensibility matrix denoted byΞthat is different from the nominal matrix Ξ0. Ξ incorporates parameters values considering an uncertainty interval for each parameter: ξij ∈[ξij,0−ξij,ξij,0+ξ¯ij].
In our example: ξ11 = 0.04, ξ¯11 =0.25, ξ22 =0.03 and ξ¯22 =0.03. Considering the category 3 of City A, the resulting UC and consumer gains, considering the uncertain price sensibility parameters, are shown in Fig. 3.24 and Fig. 3.25, respectively.
It can be noticed that at the value of the price sensibility parameters indicated in Fig. 3.24 by a circle, the UC’s gain becomes negative, indicating the start of the losses on the UC side; however, it remains positively profitable for the consumer. Hence, one can conclude that the profitability remains robust even if the tariff is no longer optimal.
3.9.2 Cost-function Components Weighting Analysis
The objective of the UC is to minimize its costs in order to increase its profit by implementing the DSM program. Therefore, the optimal ToU electricity tariffs are designed to verify such an objective. On the other hand, the UC can only encourage the consumers to participate if they will also receive a certain gain from their participation. Hence, the optimal ToU tariffs should ensure a positive gain for both parties. From the results presented above, it can be noted that the designed ToU tariffs result in a gain margin that ends up distributed over the consumers and the UC thanks to the cost function (3.25).
The cost function (3.25) comprises two components that indicate the profits of the UC and consumers, respectively. As discussed before, the weights w1 andw2 determine how the benefit of the reduced overconsumption is shared among the UC and consumers. Hence, analysing the effects of the changes of the weighting factors on the benefits of the UC and the consumer is necessary and presented here.
For this analysis, two periods of the optimal ToU tariffs for the consumer category 3
Figure 3.24: Dependence of UC gains on price sensibility parameter variations.
Figure 3.25: Dependence of consumer gains on price sensibility parameter variations.
Table 3.10: UC gain calculated by changing the values ofw1 and w2.
H HH
HH H
w2
w1
0.01 0.02 0.1 0.2 0.5 1 10 100
0.01 0.2417 0.2414 0.241 0.2418 0.2408 0.241 0.2409 0.2412 0.02 0.2414 0.2417 0.241 0.2408 0.2403 0.2403 0.2405 0.2396 0.1 0.2411 0.241 0.2417 0.2416 0.2418 0.2415 0.2413 0.2418 0.2 0.2415 0.2409 0.2408 0.2411 0.2415 0.2412 0.2409 0.2403 0.5 0.241 0.241 0.2411 0.2406 0.241 0.2413 0.2415 0.2408 1 0.2415 0.2406 0.2406 0.2408 0.2407 0.2415 0.241 0.241 10 0.2408 0.2404 0.2402 0.2412 0.2414 0.2415 0.2401 0.2406 100 0.2409 0.2415 0.2408 0.2409 0.2409 0.2365 0.2418 0.2412 Table 3.11: Consumer gain calculated by changing the values of w1 and w2.
H HH
HH H
w2 w1
0.01 0.02 0.1 0.2 0.5 1 10 100
0.01 0.213 0.2133 0.2136 0.2129 0.2141 0.2138 0.2138 0.2136 0.02 0.2134 0.213 0.2137 0.214 0.2146 0.2146 0.2145 0.2153 0.1 0.2137 0.2138 0.213 0.2131 0.2129 0.2133 0.2134 0.2129 0.2 0.2132 0.2139 0.2141 0.2137 0.2132 0.2135 0.2139 0.2146 0.5 0.2138 0.2138 0.2135 0.2143 0.2138 0.2134 0.2133 0.214
1 0.2132 0.2139 0.2143 0.2141 0.2141 0.2132 0.2138 0.2138 10 0.2139 0.2145 0.2148 0.2136 0.2134 0.2133 0.2148 0.2142 100 0.2138 0.2132 0.2141 0.2139 0.214 0.2195 0.2129 0.2135 in city A are considered as an example. Starting from different values of these weighting factors, the ToU tariff is optimized and the new gains are calculated.
Tables 3.10 and 3.11 contain the gains calculated for different weighting values at the UC side and the consumer side, respectively. It can be noticed that the distribution of the positive gain resulting from the ToU tariff implementation over the UC and consumer is different from one weighting value to the other, which makes it complicated to extract a variation pattern of the gains based on the changes of the weighting factors. The maximum value of the UC gain is 0.2418 and recorded in the following weighting values: w1=0.2,w2=0.01, w1=0.5,w2=0.1, w1=100,w2=0.1 and w1=10,w2 =100, where the consumer gain is systematically the minimal 0.2129. On the other hand, the maximum value of the consumer gain is 0.2195 results in the minimum gain value for the UC 0.2365 and can be noted at the weighting factors w1=1,w2=100. Hence, the resulting ToU tariffs robustness can be further optimized to guarantee a balanced distribution of the positive gain. For this purpose, the weighting factors can be optimized.
The optimization of the weighting factors is incorporated into the optimization algorithm presented in section 3.6. Since the cost function (3.25) seeks to maximize the gain of both the UC and the consumers, by setting the weighting factor to be a free variable, the algorithm will chose the values to achieve the gain equilibrium. The weighting values range in this example
is set to be from [0.01,100]. Therefore, some additional constraints can be introduced to the optimization algorithm on the range w1 and w2 that the algorithm can search. The constraints can be like in (3.28), or the upper bound can be open to search all possible values for the weighting factors.
0< w1≤100
0< w2≤100 (3.28)
The resulting gains after simulating the optimization are 0.2412 and 0.2135 for the UC and the consumers, respectively. Such gains can be obtained by a set of optimal weighting factors, from which are shown in Tables 3.10 and 3.11, where the maximum weighting values results in the gains above are w1=100 and w2=81.50.
Consider the weighting factors w1 and w2 are parameterized as the following:
w1=ab,
w2=a(1−b), (3.29)
where a varies in the same interval used earlier defined by [0.01,0.02,...,100] and b is a parameter defined as b∈[0,1].
Based on 3.25, ifb=0, theIncome@U C component is ignored and only the Incentives on the consumer side are being considered. Similarly, ifb=1, the priority is given to the UC’s profit. Tables 3.12 and 3.13 show the distribution of the gain resulting from the optimal ToU tariffs calculated by different values of a and b. It can be seen that the parameter b is responsible for distributing the gain on the UC and the consumer. The smaller is the value of b; the less weight is exercised on the UC’s income, and the higher weight is on the consumer’s incentive; hence, the consumer gains more than he gains when b=1. Moreover, as the value ofa increases, the total gain increases. Thus, the aparameter is responsible for the gain elevation.
Table 3.12: UC gain calculated by changing a and b.
HH HH
HH
a
b 0 1
0.01 0.2141 0.2414 0.02 0.2408 0.2411 0.1 0.2408 0.2416 0.2 0.2402 0.2406 0.5 0.241 0.241
1 0.2416 0.2417 10 0.2407 0.2415 100 0.2407 0.2411
Table 3.13: Consumer gain calculated by changing a and b.
HH HH
HH
a
b 0 1
0.01 0.2134 0.2133 0.02 0.2142 0.2138 0.1 0.2141 0.2131 0.2 0.2148 0.2144 0.5 0.2139 0.2136
1 0.213 0.213
10 0.2139 0.2133 100 0.2142 0.2138