Nicole Taheri

A Lagrangian decomposition approach for the pump scheduling problem in water networks

Dynamic pricing has become a common form of electricity tariff, where the price of electricity varies in real time based on the realized electricity supply and demand. Hence, optimizing industrial operations to benefit from periods with low electricity prices is vital to maximizing the benefits of dynamic pricing. In the case of water networks, energy consumed by pumping is a substantial cost for water utilities, and optimizing pump schedules to accommodate for the changing price of energy while ensuring a continuous supply of water is essential. In this paper, a Mixed-Integer Non-linear Programming (MINLP) formulation of the optimal pump scheduling problem is presented. Due to the non-linearities, the typical size of water networks, and the discretization of the planning horizon, the problem is not solvable within reasonable time using standard optimization software. We present a Lagrangian decomposition approach that exploits the structure of the problem leading to smaller problems that are solved independently. The Lagrangian decomposition is coupled with a simulation-based, improved limited discrepancy search algorithm that is capable of finding high quality feasible solutions. The proposed approach finds solutions with guaranteed upper and lower bounds. These solutions are compared to those found by a mixed-integer linear programming approach, which uses a piecewise-linearization of the non-linear constraints to find a global optimal solution of the relaxation. Numerical testing is conducted on two real water networks and the results illustrate the significant costs savings due to optimizing pump schedules.

A Dynamic Algorithm for Facilitated Charging of Plug-In Electric Vehicles

Plug-in electric vehicles (PEVs) are a rapidly developing technology that can reduce greenhouse gas emissions and change the way vehicles obtain power. PEV charging stations will most likely be available at home and at work, offering flexible charging options. Ideally, each vehicle will charge when electricity prices are relatively low, to minimize the cost to the consumer and maximize societal benefits. A demand response (DR) service for a fleet of PEVs could yield such charging schedules by regulating consumer electricity use during certain time periods, in order to meet an obligation to the market. We construct an automated DR mechanism for a fleet of PEVs that facilitates vehicle charging to meet the needs of the vehicles and satisfy a load scheduling obligation. Our dynamic algorithm depends only on the knowledge of driving behaviors from a previous similar day, and uses a simple adjusted pricing scheme to instantly assign feasible and satisfactory charging schedules to thousands of vehicles in a fleet as they plug-in. The charging schedules generated using our adjusted pricing scheme can ensure that a new demand peak is not created and can reduce the consumer cost by over 30% when compared to standard charging, which may also increase peak demand by 3.5%. In this paper, we present our formulation, algorithm, and results.

Conditions for Correct Sensor Network Localization Using SDP Relaxation

A Semidefinite Programming (SDP) relaxation is an effective computational method to solve a Sensor Network Localization problem, which attempts to determine the locations of a group of sensors given the distances between some of them. In this paper, we analyze and determine new sufficient conditions and formulations that guarantee that the SDP relaxation is exact, i.e., gives the correct solution. These conditions can be useful for designing sensor networks and managing connectivities in practice. Our main contribution is threefold: First, we present the first non-asymptotic bound on the connectivity (or radio) range requirement of randomly distributed sensors in order to ensure the network is uniquely localizable with high probability. Determining this range is a key component in the design of sensor networks, and we provide a result that leads to a correct localization of each sensor, for any number of sensors. Second, we introduce a new class of graphs that can always be correctly localized by an SDP relaxation. Specifically, we show that adding a simple objective function to the SDP relaxation model will ensure that the solution is correct when applied to a triangulation graph. Since triangulation graphs are very sparse, this is informationally efficient, requiring an almost minimal amount of distance information. Finally, we analyze a number of objective functions for the SDP relaxation to solve the localization problem for a general graph.

Pump Scheduling for Uncertain Electricity Prices

Water utilities have optimized pump schedules to take advantage of day/night electricity pricing plans for several decades. As intermittent renewable energy sources such as solar and wind power provide an increasingly large share of the available electricity, energy providers are moving to dynamic pricing schemes where the electricity price is forecast 24 hours in advance on 30-minute time steps. The customer only knows the actual price several days after the electricity is used. Water utilities are uniquely positioned to take advantage of these dynamic prices by using their existing infrastructure for pumping and storage to respond to changing costs for power. This work develops an operational technique for generating pump schedules and quantifying the uncertainty in the cost of these schedules. With information about the pumping schedules and the distribution of possible costs, a system operator can pump according to her desired level of risk. To develop this information, a representative sample of electricity price forecasts covering nearly the full range of possible price curves must be created. Forecasts from the energy supplier and historical data on actual prices are used to condition stochastic sampling of daily energy price trajectories using covariance decomposition methods. From this ensemble of realizations, electricity price profiles are classified into a handful of scenario classes. The optimal pumping schedule for each price class is then computed. Once the pumping schedule is known, the price of that schedule is evaluated against all other price classes to determine the robustness of the schedule. The method is applied on a simple real-world network in Ireland. In this application, electricity prices vary every half hour and range from 5 to 262 €/mWh. Optimizing the pumping schedule proved to be the slowest step in the process so selection of proper price scenarios on which to generate the schedule was critical to obtaining results in an operational time-frame.

A fair assignment of drivers to parking lots

Searching for a parking spot wastes time and energy, and becomes more prevalent with electric vehicles. This waste can be reduced by assigning drivers to parking lots based on their destination and arrival time. In such a system, drivers could request a parking spot in advance and be alerted (e.g., via their phone or vehicle) of their assignment to a specific parking lot or available spot. In this paper, a parking assignment system is described to allocate parking spaces in a fair and equitable manner. Heuristics are developed to solve the underlying large scale optimization problem. The efficacy of the system is demonstrated by applying our algorithms to real data sets.

Robustness to Time Discretization Errors in Water Network Optimization

Water network optimization problems require modeling the progression of flow and pressure over time. The time discretization step for the resulting differential algebraic equation must be chosen carefully; a large time step can result in a solution that bears little relevance to the physical system, and small time steps impact a problem’s tractability. We show that a large time step can result in meaningless results and we construct an upper bound on the error in the tank pressures when using a forward Euler scheme. We provide an optimization formulation that is robust to this discretization error; robustness to model uncertainty is novel in water network optimization.

Scenario Generation for Network Optimization with Uncertain Demands

The demand imposed on utility networks is a fundamentally uncertain quantity that varies in both space and time. Utilities have long dealt with this uncertainty by maintaining additional capacity. But, technological advances in measuring and especially transmitting water usage information allow further improvement in operational efficiency. Unlike the monthly or annual values obtained by automatic meter reading, smart water meters measure consumption at sub-daily intervals. This increased resolution, combined with measurements of bulk flows and network pressures, presents an opportunity to characterize the spatio-temporal distribution of demand. Once this representation of demand uncertainty is available, it can be used within optimization models to support decisions in network operations and planning. Several methods are available for dealing with uncertain parameters in optimization problems including sensitivity analysis, stochastic programming and robust optimization. This work deals with robust optimization, which characterizes uncertainty by scenarios or uncertainty sets. The robust solution to an optimization problem gives the best objective value in the worst case, which is feasible over the range of possible values. The choice of scenarios is a critical step in obtaining meaningful solutions with the right compromise between system performance (value of the objective function) and robustness to variations in uncertain parameters. This paper examines several techniques for generating scenarios of uncertain demands. Based on the spatial distribution of demands modeled as a multi-variable Gaussian with a mean vector and covariance matrix calculated at each time step, a technique is proposed to generate scenarios of demand which (a) cover a given probability level and (b) adhere to the spatial correlation of demands between nodes. The technique is compared with Monte Carlo sampling using the same co-variance matrix. Particular emphasis is given to the interpretation of scenarios and their effect on the solution to the operational problem of finding valve set points for a water