for i of c subscript i t end subscript p subscript i t end subscript end cell end table end cell row cell sum for t of left parenthesis p with not stretchy bar on top subscript i t end subscript minus p subscript i t end subscript right parenthesis greater or equal than d subscript t end cell row cell 0 less or equal than p subscript i t end subscript less or equal than p with not stretchy bar on top subscript i t end subscript end cell end table"/> (1.5)
where p¯it is the power required by the load i at time t; pit is the amount of power that is reduced due to the demand-side management model; cit is the cost of disconnecting one unit of power; and dt is the minimum demand. This is only the basic optimization model, which can be modified, in order to include more type of loads and other aspects of the operation of the system.
Some loads can be moved in time, for example, the washing machine in a residential user. These loads, known as shifting loads, can be optimized by defining the load’s optimal starting time. This optimization model is binary but tractable as presented in Chapter 13.
A demand-side management model can also include a model for tertiary control in microgrids or a model for charging electric vehicles. The latter is usually called vehicle-to-grid or V2G. In these cases, the optimization model requires to be executed in real-time by an aggregator as depicted in Figure 1.4.
Figure 1.4 Vehicle-to-grid concept with an aggregator that centralizes control actions. Dashed lines represent a communication architecture with the aggregator.
An aggregator is a crucial component in modern smart distribution networks. This device receives information of the final users – in this case, the electric vehicles – and gives the control actions in order to obtain a smart operation. However, the intelligent part of this system is not in the hardware but in the optimization required to solve the problem efficiently and in real-time; therein lies the importance of understanding the optimization model.
A V2G strategy can be unidirectional or bidirectional. In unidirectional V2G, an aggregator controls the electric vehicles’ charge similarly as shifting loads. In bidirectional, the electric vehicle can inject power into the grid if required for improving the operation. In any case, the model can become stochastic since the state of charge of the vehicles can be unknown, and the aggregator does not control the arrival/departing time of the vehicles1. The aggregator can also incorporate economic dispatch and OPF models to manage other distributed resources such as local batteries, solar panels, and wind turbines. Chapter 11 examines these problems.
1.2.7 Energy storage management
Modern power systems can integrate renewable energy and energy storage devices through a virtual power plant (VPP), an entity that group and centralize the operation of distributed resources to be dispatched by the power system operator. A VPP can encompass an entire region with different renewable sources and energy storage devices. It can also group other microgrids along a distribution feeder.
There are at least two moments where optimization models are required: day-ahead dispatch and real-time operation. Day-ahead dispatch corresponds to the optimization model executed the day before the operation as an economic dispatch model (see Section 1.2.1). This model must include the availability of generation and consider a forecast of the primary resource (inflows, wind, and solar radiance). Moreover, it gives the value power that the VPP operator undertakes on the day of the operation. During the operation, the VPP requires satisfying operative constraints and correcting errors in forecasting the primary resource. Again, a real-time algorithm is necessary for energy storage management.
1.2.8 State estimation and grid identification
The problem of state estimation is classic in power systems. It is also a key component in Supervisory Control And Data Acquisition (SCADA) systems. The problem consists in determining the most probable state of the system from redundant measurements and knowledge of the topology and electrical relations of the grid. When the variables to be measured are active and reactive powers, a non-convex problem is obtained with the same degree of complexity as the load flow. Modern technologies such as the phasor measurement units (PMUs) allow to include direct measures of voltages and angles.
The problem can be also formulated in power distribution networks and microgrids, both AC and DC. Figure 1.5 shows, for example, a microgrid with a centralized control. Each active element of the network can have both voltage and current measurement. We can use these measurements in order to find the most likely state of the system based on the least squares model as shown below:
Figure 1.5 Example of a microgrid with a centralized control/measurement in the aggregator.
where J, U are measurements of current and voltage, respectively; I, V are the corresponding estimations and M, N are diagonal matrices that represent the weight of each measurement. The state estimation problem is closely related to the optimal power flow. In fact, some authors call this problem as the inverse power flow problem. The problem is studied in more detail in the second part of the book (Chapter 12).
Another operation problem, closely linked to the state estimation, is the identification of the network. In this case, we have measurements of both voltages and currents at different operating points. Our goal is to estimate the value of the nodal admittance matrix from these measurements. In this case, the optimization model is the following:
The decision variable is the nodal admittance matrix Y, and the objective function is the norm of error between measurements and estimations2.
The model can include information about the structure of the matrix Y. For example, we already know that the matrix is symmetric and, some of its entries are zero. In that case, the optimization model is the following:
Both AC and DC grids may handle this type of estimation. In this case, we only presented the DC case because it is easier to develop. The entire model must be implemented in an aggregator structure, as depicted in Figure 1.5.
1.3