# Residential Demand Response Scheduling with Consideration of Consumer Preferences
## 2 Proposed Model  
### Model Outline  
There are several frameworks used in the literature for demand scheduling. In this work, we follow a commonly-accepted concept similar to the ones proposed in [14; 15; 16; 17; 18; 19]. The general idea is that the consumers (households) give time slots in which individual appliances can operate on the following day. Based on these requests, the grid controller will perform day-ahead scheduling of the operating times for all of the appliances. In this work, we focus on the benefits that can be gained for both the consumer and utility company by the use of \"preference levels\" in the submitted requests. In return, the reduced production cost could be reflected in customer bills, _i.e._, many demand response programs offer customers fixed discounts for participation.  
The basic setup of the proposed model is as follows. The scheduling will be performed for \(N\) households (consumers) with \(M\) different types of appliances. The appliances will be grouped into different categories specified by the pair \((t,l)\), corresponding to the \(t\) time and \(l\) energy consumption. For simplicity, we will assume that each appliance works uninterruptedly and that each consumer can have only one appliance of each type. Each consumer will submit for each appliance an array \(c_{i}\), indicating the preference level for each time period, in which the appliance can work. The cost of electricity is defined as a function \(f\) of the total load (consumption of all households) \(l\) distribution over all of the time periods. Finally, a limit \(L\) is given as the peak load of the system, indicating the maximal allowed demand at each time period. A simplified illustration of the proposed scheduling problem is given in Figure 1.  
Figure 1: Illustration of the problem setup in the model. The upper part of the image shows the requests submitted by three households, each having two types of appliances (s,w). Boxes with different patterns are used to indicate different households. For individual appliances, we use the color code: blue, green and red corresponding to the maximal, medium, and minimal preference level, respectively. The lower part of the image shows the scheduled times for all of the appliances.  
In the proposed model, two types of goal functions are employed. The first one focuses on the utility company standpoint. More precisely, the objective function is used to minimize the total cost of electricity generation. The second part of the objective function, on the other hand, is dedicated to the satisfaction of the households preferences. Although, initially, it seems that the natural goal is to maximize the average satisfaction level for all of the households, this approach may not be enough. In practice, the electric utilities try to maximize the number of consumers that will participate in the demand response program. The incentive for joining such a system is the promise of a lower monthly cost with minimal strain on consumer daily activities. The main deterrent for participating in such programs is the anxiety of experiencing discomfort for a minimal financial gain. Because of this, a second objective function for the satisfaction of preferences needs to be defined for the maximization of the minimal level of satisfied preferences of all of the participating households.