## Results

Our optimization and pricing scheme reduces the peak-to-ratio average by approximately 20%, which translates to a 20% decrease in the cost of electricity. This allows the average user to save $0.25 per day, a savings of around $90 per year, while still using the same amount of electricity!

## Individual Schedules

Through the graph below, one can visually see how our optimization works. The optimization shifts the electrical usage around during the day, by finding a time when electricity is the cheapest. This eliminates sharp peaks of electrical usage, making a more balanced and efficient electrical schedule. One can see that the curves of electrical use, for each of the four users, are considerably smoother using the optimized loads than the non-optimized loads.

## Savings

The four box plots below illustrate the amount of savings each type of user saves through our optimization.

By looking at the box plots, one can see that through our optimization, we have created a system in which over all users in our simulation, every user saves money by entering into our "game", meaning savings are positive and never negative. Through the principles of game theory, our optimization achieves the Nash Equilibrium by creating a scheme in which every player benefits by contributing to the system. We see in the plot, that users who own PEVs (Type 1 & 3 Users) benefit the most from the optimization, with above average savings of $0.30-$0.35 per day. This is because these users can use their PEV as a way to store energy. They will draw energy from the grid when prices are low and then can either sell it back to the grid or use the energy at a later time (and thus avoiding higher prices).

By looking at the box plots, one can see that through our optimization, we have created a system in which over all users in our simulation, every user saves money by entering into our "game", meaning savings are positive and never negative. Through the principles of game theory, our optimization achieves the Nash Equilibrium by creating a scheme in which every player benefits by contributing to the system. We see in the plot, that users who own PEVs (Type 1 & 3 Users) benefit the most from the optimization, with above average savings of $0.30-$0.35 per day. This is because these users can use their PEV as a way to store energy. They will draw energy from the grid when prices are low and then can either sell it back to the grid or use the energy at a later time (and thus avoiding higher prices).

## Optimal Schedule and the Peak-to-Average Ratio

Here is a graph of the total hourly consumption of energy for both the final optimized and non-optimized loads. The mean for the electrical load is also shown to better illustrate the peak-to-average ratio.

One can see that the optimized load schedule is much closer to the mean than the non-optimized load schedule. This results in a smaller peak-to-average ratio and thus a more efficient electrical grid.

One can see that the optimized load schedule is much closer to the mean than the non-optimized load schedule. This results in a smaller peak-to-average ratio and thus a more efficient electrical grid.

## Convergence

To prove that our optimization gives the optimal solution, we had to observe that our optimization truly converged. By looking at the graph below, we observed that the algorithm converges after four passes through the algorithm. One can see that the optimization produces a smaller and smaller error after every iteration of the optimization, up until the fourth iteration where the error is within the tolerance of convergence.

(*Mohsenian-Rad, Wong, Jatskevich & Wong prove the convergence of our algorithm within their paper, Optimal and Autonomous Incentive-based Energy

(*Mohsenian-Rad, Wong, Jatskevich & Wong prove the convergence of our algorithm within their paper, Optimal and Autonomous Incentive-based Energy

*Consumption Scheduling Algorithm for Smart Grid*)## A Look into the Future

Our findings further advance the feasibility study into the benefit of smart grid technology and its implementation into the households of the future. In attempts to make our model more realistic, we could perform some further research into smart meter appliances and how they perform using slow charging or smaller amounts of electrical usage. We could also implement more economic principles to build upon the principles of game theory, such as price elasticity of demand, as well as look into the future market of energy sale using distribution generators. Implementing these changes in our model would make the model more representative of the world that we live in and thus provide even more insight into the future of demand response management.