Grid Scheduling Considering Energy Consumption Management and Quality of Service

Abstract

In this research, the grid scheduling problem has been investigated in order to maximize profit considering the dynamic voltage and frequency scaling technique, customer-centric quality of service and time-dependent energy pricing. Mixed-integer linear programming, constraint programming, a greedy heuristic algorithm along with a hybrid method of genetic algorithm and constraint programming are developed. Some techniques are proposed to improve the efficiency of the presented constraint programming model, and their effectiveness is investigated using a full factorial experiment. Parameters of the proposed hybrid algorithm have been set by Taguchi test. The hybrid meta-heuristic algorithm, with a short execution time, generates solutions of about 18% and 88% better than the best solution of the constraint programming model for large-scale problem instances. The results show that the final profit will be reduced by about 22% if the electricity prices are wrongly considered with a flat rate during the scheduling process.

Appendix 1: Proof of the Lemma

Assume that for the set of accepted tasksI^accepted, the value of the objective function is ${objective}^{I^{accepted}}$. Suppose task i is out of the set of accepted tasks and the other tasks are performed on their former resource, VF level and time. Define the value of the objective function as ${objective}^{I^{accepted\backslash \left\{i\right\}}}$ in this case. Here, the revenue from performing task i, which is equal to $\min \left\{{B}_i,{B}_i-{U}_i\left({t}_i+{ETT}_{i,{r}_i,{k}_i}-{D}_i\right)\right\}$, is deducted from the objective function, and the revenue from performing other tasks does not change. The power consumption of resource r_i in interval $\left[{t}_i,{t}_i+{ETT}_{i,{r}_i,{k}_i}-1\right]$ source changes from ${P}_{r_i,{k}_i}$ to ${P}_{r_i}^{sleep}$. So, the power consumption cost of this resource is reduced by $\sum_{s={t}_i}^{t_i+{ETT}_{i,{k}_i}-1}{C}_s\times \left({P}_{r_i,{k}_i}-{P}_{r_i}^{sleep}\right)$. The power consumption of other resources does not change and the related cost does not change as well. According to this, eq. (59) is established between ${objective}^{I^{accepted}}$ and ${objective}^{I^{accepted\backslash \left\{i\right\}}}$.

$${objective}^{I^{accepted}}-{objective}^{I^{accepted\backslash \left\{{i}_1\right\}}}=\min \left\{{B}_{i_1},{B}_{i_1}-{U}_{i_1}\left({t}_{i_1}+{ETT}_{i_1,{r}_{i_1},{k}_{i_1}}-{D}_i\right)\right\}-\sum_{s={t}_{i_1}}^{t_{i_1}+{ETT}_{i_1,{r}_{i_1},{k}_{i_1}}-1}{C}_s\left({P}_{r_{i_1},{k}_{i_1}}-{P}_{r_{i_1}}^{sleep}\right)={Profit}_{i_1,{r}_{i_1},{k}_{i_1},{t}_{i_1}}$$

(59)

According to (59), it can be easily seen that the following equation holds.

$$\left\{\begin{array}{c}{objective}^{I^{accepted}}-{objective}^{I^{accepted\backslash \left\{{i}_1\right\}}}={Profit}_{i_1,{r}_{i_1},{k}_{i_1},{t}_{i_1}}\kern4.75em \\ {}{objective}^{I^{accepted\backslash \left\{{i}_1\right\}}}-{objective}^{I^{accepted\backslash \left\{{i}_1,{i}_2\right\}}}={Profit}_{i_2,{r}_{i_2},{k}_{i_2},{t}_{i_2}}\kern2.5em \\ {}{objective}^{I^{accepted\backslash \left\{{i}_1,{i}_2\right\}}}-{objective}^{I^{accepted\backslash \left\{{i}_1,{i}_2,{i}_3\right\}}}={Profit}_{i_3,{r}_{i_3},{k}_{i_3},{t}_{i_3}}\kern1em \\ {}\vdots \\ {}{objective}^{I^{accepted\backslash \left\{{i}_1,{i}_2,\dots, {i}_{n-1}\right\}}}-{objective}^{I^{accepted\backslash \left\{{i}_1,{i}_2,\dots, {i}_{n-1},{i}_n\right\}}}\kern5.5em \\ {}\kern23.25em ={Profit}_{i_n,{r}_{i_n},{k}_{i_n},{t}_{i_n}}\end{array}\right.$$

(60)

By adding the expressions in (60), the following equation will be obtained.

$${objective}^{I^{accepted}}-{objective}^{I^{accepted\backslash \left\{{i}_1,{i}_2,\dots, {i}_{n-1},{i}_n\right\}}}=\sum_{i\in {I}^{accepted}}{Profit}_{i,{r}_i,{k}_i,{t}_i}$$

(61)

Note that ${objective}^{I^{accepted\backslash \left\{{i}_1,{i}_2,\dots, {i}_{n-1},{i}_n\right\}}}={objective}^{\varnothing }$ where objective^∅ is the value of the objective function in the mode of not accepting any tasks. Since no task is accepted in objective^∅, all resources will be idle, and the value of objective^∅ can be expressed as the following equation.

$${objective}^{\varnothing }=-\sum_{r\in R}{cost}_r^{idle}$$

(62)

By combining (61) and (62), Eq. (52) is proved.