Mathematical Optimization of Unbalanced Networks with Smart Grid Devices

Abstract

Electric distribution networks should be prepared to provide an economic and reliable service to all customers, as well as to integrate technologies related to distributed generation, energy storage, and plug-in electric vehicles. A proper representation of the electric distribution network operation, taking into account smart grid technologies, is key to accomplish these goals. This chapter presents mathematical formulations for the steady-state operation of electric distribution networks, which consider the unbalance of three-phase grids. Mathematical models of the operation of smart grid related devices present in networks are discussed (e.g., volt-var control devices, energy storage systems, and plug-in electric vehicles). Furthermore, features related to the voltage dependency of loads, distributed generation, and voltage and thermal limits are also included. These formulations constitute a mathematical framework for optimization analysis of the network operation, which makes it possible to model decision-making processes. Different objectives related to technical and/or economic aspects can be pursued within the framework; in addition, the extension to multi-period and multi-scenario optimization is discussed. The presented models are built based on mixed integer linear programming formulations, avoiding the use of conventional mixed integer nonlinear formulations. The application of the presented framework is illustrated throughout control approaches for the voltage control and the plug-in electric vehicle charging coordination problems.

Appendices

Appendix 1: Piecewise Linearization Technique

The piecewise linearization is a technique in which a nonlinear function is approximated using a set of piecewise linear functions [37]. Widely used in engineering, this technique is often employed to cope with quadratic nonlinearities, helping to reach LP models. Typically, a function $f$ is defined in order to calculate the square value of a variable $\sigma$, represented as $\sigma^{ + } + \sigma^{ - }$ and limited by the interval $[0,\bar{\sigma }]$. This type of function has a general structure, as

$$f\left( {\sigma ,\bar{\sigma },\Lambda } \right) = \sum\limits_{\lambda = 1}^{\Lambda } {\phi_{\sigma ,\lambda } \Delta_{\sigma ,\lambda } }$$

(3.168)

$$\sigma = \sigma^{ + } - \sigma^{ - }$$

(3.169)

$$\sigma^{ + } + \sigma^{ - } = \sum\limits_{\lambda = 1}^{\Lambda } {\Delta_{\sigma ,\lambda } }$$

(3.170)

$$\begin{array}{*{20}l} {0 \le \Delta_{\sigma ,\lambda } \le \bar{\sigma }/\Lambda } \hfill & {\forall \lambda \in\Lambda } \hfill \\ \end{array}$$

(3.171)

$$\begin{array}{*{20}l} {\phi_{\sigma ,\lambda } = (2\lambda - 1)\bar{\sigma }/\Lambda } \hfill & {\forall \lambda \in\Lambda } \hfill \\ \end{array}$$

(3.172)

The parameter $\phi_{\sigma ,\lambda }$ is calculated to compute the contribution of $\Delta_{\sigma ,\lambda }$ in each step of the discretization. The parameter $\bar{\sigma }$ represents the maximum value of $\sigma$, while $\Lambda$ is the number of discretizations used in the linearization.

It is important to remark that this approach is limited to maximizing strictly concave functions or minimizing convex functions. If the application of this technique under different conditions is desired, the inclusion of binary variables and additional constraints is mandatory.

Appendix 2: Multi-period and Multi-scenario Extension

Typically, optimization analyses in electric distribution network operation are done along a time window in which several control actions have be defined and they may be dependent among them; this is known as multi-period optimization. For example, the day-ahead operation planning is typically divided in one-hour time windows, and the decisions from one hour may or may not affect the decisions regarding the next time intervals. Thus, mathematical formulations for the distribution network operation should be able to handle multi-period optimization analyses. In this regard, the LP formulations presented can be easily adapted to handle several time intervals. Hence, a new index associated to the time interval is added to the variables that represent the distribution network operation (e.g., voltages, currents, and power flows).

Furthermore, adaptability to multi-scenario optimizations is also required in an optimization framework for electric distribution network to model the uncertainty in the grid. The multi-scenario optimization is a method usually employed to solve stochastic programming problems in which some of the variables or parameters are of uncertain nature (e.g., EV behavior, renewable DG availability, and demand variations). The uncertainties are represented through a set of scenarios and each one with an associated probability, i.e., a multi-scenario model will provide an optimal solution on average, considering all the scenarios simultaneously. Analogue to the multi-period case, a new index is added to the uncertain variables associated to each scenario. Thereby, the objective function of the problem is calculated as the expected value due to the inclusion of the probabilities related to each scenario.

Appendix 3: Estimated Steady-State Operation Point

As mentioned in Sect. 3.2.3, the accuracy of the presented three-phase formulations relies on the precision of the estimated operation point. High quality estimations will minimize the error corresponding to some approximations in voltage magnitudes and some linearization techniques (e.g., Taylor’s linearization). In order to obtain a suitable estimated operation point, the followings techniques might be employed:

• 1. A two-stage approach, in which a first stage solves the LP model using a flat start (e.g., assuming nominal voltages and disregarding power). Later, the solution of the first stage is used to initialize the second stage in which the LP model is once again solved from the already calculated operating point.

• 2. Using historical data, where historical data is used in order to determine the estimated values. Typically, the operator’s knowledge and experience are crucial to select previous operating points which have occurred under similar loading and generation scenarios.

• 3. Using the previous time interval operating point is another technique for the estimation of the operating point. This approach is commonly used on small time interval optimization approaches in which abrupt changes in the demand are not expected (e.g., EVCC problems).

It is important to remark that the estimation of the operation point is an important issue to be taken into account when applying the presented formulations. Furthermore, the technique chosen to determine the estimated operation point will depend on the information available and the characteristics of the problem that is being tackled.