The Smart Grid is the system that always tries to satisfy the equilibrium between energy supply and demand in order to maintain the operations of the system in the most efficient and safe way. This objective is achieved thanks to the two major technological advancements introduced into the grid, which make it smart:
- The use of electronic devises (smart/digital meters) which collect data on energy demand from each user connected to the grid.
- Analysis of obtained data with subsequent application of statistical machine learning techniques to predict future energy demand, in order to optimize production and distribution of energy.
Introduction of distributed renewable energy production (such as roof solar panels) and energy storage devices (batteries) brings new opportunities to improve the efficiency and reliability of the Smart Grid.
One route power producing companies try to reduce their own expenses, as well as to help their consumers save on their electric bills, is the variable day tariffs on electricity (or the time-of-use tariff). The price for electricity increases during high demand hours and decreases during low demand hours, which is supposed to encourage consumers to adjust their energy usage in proportion with pricing, as it would be economically beneficial. While this approach can definitely benefit both parties, power companies and their customers, there are obvious disadvantages to it. It is usually not easy for a typical household to change their energy consumption behavior in order to save on electricity bills. Most peoples' demand is pretty typical during a day, which is high demand during mornings before leaving home for work or school, followed by low demand during the mid-day with subsequent increase of demand in evening continuing into night hours until midnight. That is where the home-based battery pack (for example, the Tesla Power-Wall) would be a great solution to this problem. Having a battery allows a household to store low-cost energy in order to use it when the price is high without altering its energy consumption behavior. I developed an algorithm which optimizes charging-standby-discharging cycles of battery pack operations in an intelligent way, so that it minimizes consumers' electric bill on a daily basis.
In order to minimize daily electric bills, in the best possible way, one has to consider two sources of data:
- Variation of energy cost within a day.
- Local user's energy consumption behavior.
The first is assumed to be provided by the power generating company on daily basis, thus it is just a given input. The second is what has to be predicted for at least a day in advance. Once we have both, the consumer's predicted energy demand for a day and variable pricing for the same day, we can solve an optimization problem with respect to battery pack's cycles of operation to minimize the consumer's daily electrical bill. The most challenging part of the algorithm is the predictive model of the user's energy consumption behavior. The model should produce a forecast of the energy demand based on the energy demand of the consumer/household in the past.
There are three phases of operation of the battery pack:
- Charging from the grid.
- Standby.
- Discharging to power household appliances.
The output of the algorithm is a daily schedule of these three phases. A day is split into equal time intervals, then the algorithm tells the battery which phase it should be in with each time interval.
I used a large collection of smart meter data obtained by UK Power Networks. The data contained a sample of 5,567 London households between November 2011 and February 2014. Smart meter readings of energy consumption (in kWh) were taken from half hour time intervals every day starting from 00:00 morning to 23:30 night of the same day making 48 records in one day. The data are anonymous, thus each household is given a unique household id number and Acorn group it belongs to, but no other information (such as which particular neighborhood in London the household is located in, how many rooms are in the household, or what the square footage is in each household). There is also a pricing curve of the time-of-use tariff plan in London with the following costs for electricity: high (67.20p/kWh), low (3.99p/kWh), and normal (11.76p/kWh).
This type of data is time series, below is a graph displaying a typical household with two days of energy consumption:
Fig.1. A sample of smart meter readings for two days from a typical household.
The particular type of time series of households energy demand has a very distinguished daily seasonality. I have found that training a single ARMA model for this type of time series on a number of past days does not produce a good forecast for the energy demand for one day into the future. However, if I train a single ARMA model for a particular time interval in a day, let us say for 2:00pm-2:30pm, over past days, then this model gives a far better prediction for the energy demand during this specific time interval, ( i.e. 2:00pm-2:30pm ). This is no surprise, considering the seasonality of this time series mentioned above. It can be summarized as follows: past energy demand for a time interval in a day over a number of previous days is a better predictor of the future energy demand for the same time interval in the day than all times over previous days. This fact led me to develop a complex model which consists of a number of ARMA sub-models. Below is outline of its architecture:
- I split the day into a number of time intervals and trained a separate sub-model for each time interval over a number of previous days.
- Each sub-model produces a single prediction for one day in the future, such that, all sub-models together produce a forecast for the entire day.
I also made an option to allow for training of the model for weekdays and weekends separately, which yielded better results. I developed two models, in the first model I used existing day split in the data into 48 time intervals, thus this model consists of 48 sub-models. The second model splits the day into a number of time intervals equal to the number of time intervals in the pricing curve provided by the power generating company. Then I calculate average energy demand within each of these time intervals and trained the models to predict average energy demand for each time interval for the future day. Since there are 4 time intervals in the London pricing curve, the second model consists of 4 sub-models. Below are the plots of each of the models performance for the same household. The first model is Hourly ARMA 48 sub-models and the second model is Price Correlated ARMA 4 sub-models:
Fig.2. Weekdays 48 sub-models forecast for one day. The model was trained on 350 days. The scores for this model are: MSE = 0.0108 and R^2 = 0.7405
Fig.3. Weekdays 4 sub-models forecast for one day. The model was trained on 350 days. The scores for this model are: MSE = 0.0022 and R^2 = 0.8951
The objective is to predict for each time interval how much energy the battery pack should charge from the grid or discharge to power household appliances or standby with its current energy content. Mathematically this problem expresses as:
here in the Eq.(1) index i denotes a time interval in a day, thus herein all terms indexed by i represent values of quantities for a single time interval. The term 
is the bill, 
is the price per unit of energy (usually it is money value/kWh), 
is the energy consumed from the grid due to the demand, 
is the amount of energy the battery pack charged from the grid, and 
is the amount of energy the battery pack discharged to power household appliances. It is assumed that battery pack can only be in a single phase within each time interval, thus and cannot simultaneously be not equal to zero. Equation (2) expresses minimization of the daily bill over and variables; here capital N is the total number of time intervals in a day.
The Eq.(2) is the subject to the following constraints:
Here constants 
, C, and e are charging rate (C-rate), battery capacity, and efficiency for a roundtrip of charge-discharge combined with AC-DC conversion respectively. The Eqs.(3) and (5) simply state that amount of energy the battery pack can charge and discharge is not negative. The Eq.(4) restricts how much energy can battery charge in a single time interval. The Eq.(6) restricts discharge from exceeding demand. The Eq.(7) implies that total amount of discharged energy from the battery over a number of previous time intervals cannot exceed the amount of energy the battery has charged for the same number of preceding time intervals. Finally, the Eq.(8) restricts the amount of energy the battery contains after a number of preceding time intervals when the battery underwent charging and discharging phases by its capacity.
The Eqs.(1)-(8) define a linear programming or linear optimization problem. Once the optimization problem is solved and values of and obtained for all time intervals we can compute what is the energy content of the battery pack in each time interval. The difference between energy content of the battery pack between neighboring time intervals tells us what phase the battery pack should be in within each time interval. The energy content of the battery pack is computed as follows:
Below are the solutions of the Eq.(9) shown in the graph for the battery pack charging-standby-discharging cycles for the same household whose predictions of the energy demand obtained with two different models were shown in Fig.2 and Fig.3 above.
Fig.4. Forecast of the battery pack charging-standby-discharging cycles (solid red line) for one day. This forecast is based on predictions of demand (green line) obtained from weekdays 48 sub-models and provided pricing (black line). The projected savings on electric bill with the 2 kWh battery (dashed red line) for this day is 29.8%.
Fig.5. Forecast of the battery pack charging-standby-discharging cycles (solid red line) for one day. This forecast is based on predictions of demand (green line) obtained from weekdays 4 sub-models and provided pricing (black line). The projected savings on electric bill with the 2 kWh battery (dashed red line) for this day is 29.9%.
Despite the fact that for this particular household, the Price Correlated Model predicted slightly higher savings than Hourly Sliced Model; however it is not enough to conclude that one model is better than the other. I have found that predicted savings of each model vary from household to household along with the number of days each model was trained on. Thus, each model can predict higher savings than the other for one household and vice-versa for another household. The amount of savings also depends on battery pack capacity, efficiency of charge discharge cycles, charging rate , and pricing curve. The results shown in the Fig.2 and Fig.3 were obtained with the efficiency assumed to be 80% and charging rate =0.25. Below is the graph of savings as a function of battery capacity with the same assumptions made regarding efficiency and charging rate:
Fig.6. Forecast of potential savings for one day as a function of battery capacity for the same household as on Figs(1)-(4). It shows that increase in battery capacity leads to increase in savings on daily electric bill until it reaches a plateau in savings beyond which increase in battery capacity does not result in significant increase in savings.
The calculations presented in Fig.6 can be used to estimate the maximal size of the battery pack for a household based on its energy usage history and a particular pricing offered by a power company. Below is similar plot of savings vs battery capacity for a different household:
Fig.7. Forecast of potential savings for one day as a function of battery capacity for a particular household. It shows that there is a battery capacity below which savings become negative.
In Fig.7 one can see that savings can become negative if the battery is too small. This is because efficiency of the battery pack system is not 100% efficient and there are always losses associated with the use of the battery. The battery capacity and the gap in price between low-cost hours and high-cost hours should be large enough in order for the battery pack system to become economically justified.