## 1. Introduction

The introduction of communication and information technologies in the current power infrastructure has accommodated the digital connection of electricity market participants [

1] and has enabled their active participation in various activities pertained to grid operation. The concept of an energy Internet as introduced in [

2] is a characteristic example of the integration of digital connectivity with power systems. Through active participation, consumers play a significant role in determining the shape of the final load demand, mainly by responding to price signals with their individual demand [

3]. In addition, consumers are now exposed to a high volume of information that is utilized for making optimal decisions and fulfilling their goals. Notably, the goals of electricity consumers encompass the maximum fulfillment of energy needs and the minimum possible cost [

4].

On one hand, the introduction of digital connectivity in the power grid has leveraged the role of consumers resulting in the enhanced grid stability, minimization of power losses and decreased cost of operation [

5]. On the other hand, it came with a reduced degree of consumer privacy given that information sharing and exchange may reveal details about the private life of the consumers. More specifically, load profiles, if shared, can be used to infer the consuming activities of a specific customer. For instance, the use of a washer may be inferred by identifying the consumption pattern of the washer in the load profile. Such information may be utilized by third parties for advertising reasons or for nefarious purposes, where burglars may infer the presence or not of the owner and break into the house. Therefore, shared load information comprises of a point of vulnerability that may be used to compromise privacy of consumers in the smart cities of the future [

6].

In a market, it is anticipated that consumers are interested in purchasing products of the highest quality at the lowest possible cost. With regard to electricity consumption, electricity consumers care about satisfying their maximum demand while attaining the lowest cost; in other words, they would like to minimize their electricity bill, and fully perform their planned activities [

4]. In general, the decision-making process of consumers, including electricity consumers, is a cost driven approach, where the consumer has as a first priority the minimization of the overall cost [

7]. In that case, the consumer is prone to morphing his/her load demand in order to retain the cost at a comfortable level. In smart grids, load demand morphing refers to the actions of either cancelling, or shifting the load demand [

3] with respect to an initial plan. Cancelling load refers to abandon the scheduling of specific consuming actions, while shifting refers to postponing the consuming actions at a later time, usually at times where the price is lower [

8]. The response of consumers with their load demand to prices set by the market operator is known with the general term of “demand response” [

9].

There are several approaches that deal with the demand response of consumers aiming at minimizing the cost of consumption. For instance, in [

10] an approach that optimizes the electric appliances scheduling for demand response is presented, while in [

11] an approach that reduces the load variation limits to minimize consumption costs is introduced. The concept of Virtual Budget as an efficient method for optimizing the electricity cost of demand scheduling using anticipation is proposed in [

3], while a sliding window driven method, which utilizes streamed big data for real time electricity consumption optimal adjustment, is proposed and tested in [

12]. Furthermore, an optimization algorithm for residential consumption pattern flattening by identifying the time-of-use tariff that minimizes the overall consumption cost is introduced in [

13]. In [

14], methods that assume the use of interruptible tasks are proposed, whereas in [

15] an informatics solution that is based on the synergism of three models in optimizing household appliance management. Furthermore, an autonomous system for demand response that achieves minimal cost for participating in demand response programs consumers is introduced in [

16], and a similar approach that assumes consumers response to utility signals prior to any decision making is presented in [

17]. Moreover, an approach for scheduling the electricity consumption of a residential community based on the aggregated payment is introduced in [

18]. A coordinated approach considering a community of prosumers is discussed is in [

19], while the distributed coordination of a grouped consumption using the alternative directions method of multipliers is introduced in [

20]. Overall, the variety of demand response approaches aim at securing the operation of the grid [

21], while the consumers target to minimize their consumption expenses [

22]. However, there are demand response methods that rather focus on consumers’ privacy preserving than the cost-minimizing. An example is presented in [

23], where a privacy preserving method employs data encryption in the form of a homomorphic encryption of the group aggregated demand. Further, a method focusing on incentive-based demand response of consumers using cryptographic primitives, such as identity-committable signatures and partially blind signatures, is presented and tested in [

24]. Data exchange architectures to ensure the privacy of electricity consumption profiles are presented in [

25] and [

26], where in the first case trusted-platform modules for advanced metering infrastructure (AMI) are used, and in the second case a set of Internet-of-Things tools are put together. In [

27], a secure algorithm tailored for bidding driven markets is proposed utilizing cryptographic primitives without any explicit third trusted party. Lastly in [

28], a study that involves the use of privacy threat models together with attributed based encryption is provided.

The proposed demand response methods are focusing on single residents and are proposed within the framework of optimizing the power grid infrastructure. Furthermore, they do not exploit the ubiquitous digital connectivity that will be the backbone of the smart cities and smart grids. In this work, we propose an approach that builds upon that connectivity driven future. Notably, we see that both the digital connectivity and the smart grid technologies are the enablers for smart cities. In particular, we assume that consumers, which are also residents of a smart city, are able to connect and form “virtually connected groups” [

29]. This grouping of the residents/consumers is performed at the cyber level and subsequently allows consumers to form energy partitions within the smart grid [

30]. Hence, we will exploit digital connectivity of smart grids to allow consumers within the same partition to collaborate and formulate the partition’s demand response. The partition demand response is a single load pattern that coincides with the morphed aggregation of the individual demand patterns of the consumers within the partition, an idea that has been proposed in [

31].

In this work, we push further the envelope in load morphing by assuming that the partition demand response results from the concurrent consideration of the partitions consumption expenses and privacy. The paper introduces a new approach where citizens collaborate in order to concurrently attain low cost and high privacy as a group. A premature version of this method was presented in [

32], where its high potential for smart grids was highlighted. As compared to [

32], this paper presents in more details the morphing method, while it applies it in a higher variety of smart meter data taken from the power grid of Ireland [

33]. Notably, the proposed approach considers the morphing problem as a multi-objective problem whose solution is located by an evolutionary algorithm. In this work, the non-dominated sorting genetic algorithm-II (NSGA-II) is adopted to provide a solution to the final morphing of the demand [

34]. NSGA-II is a genetic algorithm that utilizes the Pareto optimality theory to identify a solution that optimizes both the cost and privacy of the grid partition [

35].

At this point it should be noted, that the current work significantly differs from the works in [

29,

30,

31]. Those works focus only on enhancing the privacy of the group of the consumers without considering the cost of the final morphed consumption pattern, whereas the current work also explicitly considers the cost of consumption. Furthermore, in References [

29,

31] a genetic algorithm is adopted to solve a single objective optimization problem, while in the current work the genetic algorithm is selected for locating a solution to a multi-objective problem.

The innovation in the current work is the concurrent consideration of both privacy and consumption cost in a grid partition, and their concurrent optimization through Pareto optimality, which finds the optimal tradeoff between the two objectives, i.e., cost and privacy. This work aspires to show that the morphing of partition consumption driven by those two objectives is a complex optimization problem with multiple constraints, that can be solved by genetic algorithms. Notably, genetic algorithms have the ability to always identify a global optimal solution (or near optimal) to complex problems independently of the number of objectives and constraints.

The roadmap of this paper is as follows: In the next section a brief introduction to evolutionary computing and Pareto optimality is given, while in

Section 3 the morphing methodology is presented. In

Section 4, the results on a set of real-world datasets are presented and discussed, while

Section 5 concludes the paper.

## 3. Grid Partition Load Morphing Methodology

In this section, the load morphing of smart grid partitions methodology is presented. It should be noted that the way the partitions are formed in the grid is not of interest to the current manuscript. For instance, one way to form partitions is via grid partitioning as discussed in [

39]. The methodology presented in this manuscript formulates a multi-objective problem composed of a dual of objective functions [

8]. The two objectives, which express the consumption cost and the degree of privacy in the grid partition respectively, are minimized by seeking an optimal solution to a multi-objective problem. The NSGA-II algorithm is applied to identify an optimal solution. The block diagram of the proposed methodology is presented in

Figure 3, where its individual steps are clearly provided.

Initially, the consumers within the grid partition anticipate their day ahead load demand. The anticipation is conducted at an hourly level, i.e., that means a set of 24 load values for the next day is provided. The overall load anticipation for a whole day is denoted as given below:

where

L_{i} is the anticipated load at hour

i. In addition to the anticipated load, the consumers are also providing their hourly upper and lower bounds of their anticipated load denoted as

U_{i} and

F_{i} for the hour

i. These bounds coincide with the maximum and minimum values between which the initial anticipation can be morphed for a specific hour, and we call them the “morphing bounds”. To make it clearer, when the anticipation for the hour

i is

L_{i}, then the anticipation can be morphed within the tube [

L_{i} −

F_{i},

L_{i} +

U_{i}].

In the next step, the individual consumers’ anticipated loads are aggregated and a single anticipated load demand signal is obtained. Thus, the aggregated pattern expresses the anticipated load of all the partition consumers and is expressed as:

where

${L}_{j}^{i}$ is the load of consumer

j at hour

i, and

N is the population of consumers. Likewise, the individual morphing bounds are also aggregated providing the respective aggregation morphing bounds:

where

${U}_{i}^{A}$ and

${F}_{i}^{A}$ are the upper and lower aggregation morphing bounds for the hour

i. At this point, we introduce the morphed aggregated load, which is denoted as:

with

${A}_{i}^{M}$ being the morphed aggregated load at hour

i, and

${\alpha}_{i}$ the morphing factors for the hour

i. The morphing factors express the degree of morphing, where: (i)

${\alpha}_{i}$ = 1 denotes no morphing, (ii)

${\alpha}_{i}$ < 1 denotes decreasing of the initial anticipation, and (iii)

${\alpha}_{i}$ > 1 denotes increasing of the initial anticipation. Notably, for

${\alpha}_{i}$ = 1 Equation (6) drops down to Equation (3).

Once the aggregated values have been computed, then the objective functions are formulated. According to

Figure 3, two objectives are formulated, namely, the cost and privacy objectives. The cost objective expresses the daily cost of purchasing the anticipated load and is formulated as:

with

P_{i} being the day ahead forecasted electricity price for the hour

i.

Formulation of the second objective, which expresses the degree of privacy, is more complex than that of the cost. It should be noted that as a measure of privacy we assume the degree of variance in the load pattern. On one hand a highly varying pattern is a carrier of information that can be easily extracted and subsequently lead to inferences about consumption activities. In other words, variability can be a source of information–the peaks and valleys of the pattern can be associated with consumption activities. On the other hand, a constant load pattern exhibits no variance and thus, inference making becomes challenging. In this work, we aim at deriving a constant load pattern by morphing the aggregated demand. To that end, we adopt a target constant pattern that is equal to the mean value of the aggregated value given below:

where

A_{i} is the aggregated value for hour

i computed by Equation (3). Furthermore, to quantify the degree of difference between the aggregated load and the mean aggregated value, an error measure is adopted. More specifically, the mean square error is utilized as the objective for expressing the degree of privacy as the distance of the aggregated demand to the mean value. Thus, the privacy objective takes the following form:

where

M is taken by Equation (8) and

A_{i} by Equation (2).

The objective functions as defined by Equations (7) and (9) are accompanied with a set of constraints. The constraints, which are in the form of box constraints, express the morphing bounds of the aggregated load and are formulated as:

with

${A}_{i}^{M}$ being the morphed aggregated load (see Equation (6)). As shown in Equation (10), there are 24 box constraints, i.e., one for every hour of the day.

At this point, the two objectives and the respective constraints have been fully defined and therefore, we are able to define the multi-objective problem utilized for load morphing of the grid partition. The particular form of the multi-objective problem is given below:

where the optimization process takes the form of a minimization of the two objectives.

The multi-objective minimization problem in Equation (11) is solved utilizing evolutionary computing and more specifically the NSGA-II algorithm. The NSGA-II seeks for non-dominated solutions, which satisfy by default the Pareto optimality criterion. The identified solution which is comprised of a set of 24 optimal morphed values, i.e., optimal

${\alpha}_{i}^{opt}$,

i = 1,…,24, is the final solution of the problem. Each morphed factor expresses the degree of morphing of the load for the hour

i. Having computed the morphed values, then the final morphed pattern is obtained by plugging-in the identified solution to Equation (6):

where the superposition of the morphed values provides the final load curve of the smart grid partition. To conclude, consumers by working all together and exploiting the digital connectivity, may minimize their overall cost and enhance their privacy.

## 5. Conclusions

In this paper, we have presented a new methodology for morphing the load pattern of a smart power grid partition. The partitions are assemblies of consumers that exploit the smart grid communications to collaborate and pursue common goals. Their goals entail minimization of their consumption expenses as well as enhancement of their consumption privacy.

The presented methodology achieves this set of goal by formulating a multi-objective problem. The problem comprises of two objectives, namely, the cost and the privacy objective. The first objective expresses the anticipated cost consumption for a day ahead of time, and the second measures the distance of the final load pattern to the initial consumers’ anticipated load. The two objectives are minimized by an evolutionary algorithm and more specifically the NSGA-II that identifies a solution using the Pareto optimality theory. Furthermore, the presented work is compared to two single objective optimization approaches: The first approach handles only the cost objective, whereas the second approach handles the privacy objective. Comparison exhibited that our approach provides equal or better performance as compared to the single objective cases. It should be noted that for the privacy measure all three approaches provided very close values, proving that our approach enhances the degree of privacy as much as the privacy objective approach does. With regard to cost, the multi-objective morphing approach provided the lowest cost values in all tested cases. By combining the above observations, we conclude that the lumping of the two objectives in a single formulation did provide a better performance as compared to single objective problems. Therefore, we conclude that NSGA-II utilizing the Pareto theory attained to concurrently secure low cost and to enhance privacy.

However, the current study exhibits some limitations. More specifically, the optimization problem is based on the box constraints for each hour of the day. In this work, we assumed narrow intervals within the morphed values may lie. In practice, these intervals may be totally different than the ones assumed in this work, and depend on the characteristics of the grid and the consumers. Furthermore, we assume that the consumers are able to form a partition via direct communication links, and that the consumers trust each other (which is not always the case). Lastly, a limitation of our study has to do with the number of customers contained in the partition: We assume that the number of partition consumers remains constant at least for a whole day (since this is the anticipation horizon).

In the current work, the morphing methodology is tested on a set of real-world data taken from smart meters deployed in the Irish power grid. Testing entails four scenarios where grid partitions are comprised of six, seven, 10 and 15 consumers. Obtained results clearly demonstrate the effectiveness of the methodology in decreasing the partition’s consumption costs (both anticipated and real cost), and enhancing the privacy of the individual consumers. Future work will focus on identifying and expanding the set of objectives that a grid partition may pursue.