Mathematical Formulation
\label{RAS}
This section presents a brief overview of the mathematical modeling of the RAS and the solution algorithm for the optimization model of the RAS.
Mathematical Model
The objective of the proposed RAS is to reduce the wind energy curtailment, thereby avoiding overloads on the transmission line, while minimizing curtailment of renewable generation. This problem can be mathematically formulated as \cite{liu2016decentralized},
\begin{equation}
\label{eq1}Max\;f(X)=\sum\limits_{i=1}^{N}{{x_{i}}*{P_{Gen\_i}}}\\
\end{equation}
In this equation, the power generation for the \(i_{th}\) wind farm is represented by \(P_{Gen\_i}\). The number of wind farms installed is represented by N. The wind farm connection status for the \(i_{th}\) wind farm is represented by \(x_{i}\), \(i\in{\rm Z}\left({1,N}\right)\). Based on the different control methods in the wind farm substation, there are two different modes for this RAS.
- Switching Mode: If wind farm substation can only turn on/off the wind farm, the wind farm connection status \(x_{i}\) can only be zero or one. Zero means \(i_{th}\) wind farm is shed. One means \(i_{th}\) wind farm is connected to the grid.
- Generation Reduction Mode: If wind farm substation has the advance controller, which can accurately control the wind farm generation, the wind farm connection status \(x_{i}\) can be any real number ranging from zero to one.
The DC power flow equations can be:
\begin{equation}
\label{eq2}{P_{Gen}}-{P_{Load}}=B*\theta\\
\end{equation} \begin{equation}
\label{eq3}\theta={B^{-1}}({P_{Gen}}-{P_{Load}})\\
\end{equation} \begin{equation}
\label{eq4}{P_{ij}}={B_{ij}}({\theta_{i}}-{\theta_{j}})\leq L{R_{ij}}\\
\end{equation}
The line rating on the \(i_{th}\) transmission line is represented by \(LR_{i}\). \(P_{ij}\) is the flow in the line from bus \(i\) to bus \(j\). \(\theta_{i}\) is the angle at bus \(i\).
By converting the above equations into a standard linear programming format, this optimization problem can be solved. The optimization problem (\ref{eq1}) to (\ref{eq4}) is convex, since the objective function and the constraints are convex in nature. This is by virtue of the DC formulation. Although, the solution is straightforward, this formulation provides approximate solution. The bus voltages are assumed to be 1 p.u. However, in a power grid with considerable wind power penetrations, the voltages can vary significantly as the wind power varies. The inherent variability of wind power injections becomes particularly important for wind farms connected to weaker networks, as voltage deviations become more significant and voltage regulation more challenging \cite{wind1}. Hence, it is imperative to consider the voltage limits on transmission buses, while determining the optimal wind curtailment.
Considering the line flow limits and the upper limits and the lower limits on Voltage \(V_{i}\) (1.05 and 0.95, respectively, in this study), the objective function (\ref{eq1}) can be written as
\begin{aligned}
& \underset{x}{\text{max}} & & \;f(x)=\sum\limits_{i=1}^{N}{{x_{i}}*{P_{Gen\_i}}} \\
& & & {P_{ij}}\leq L{R_{ij}}\nonumber \\
& & & V_{i}^{min}\leq V_{i}\leq V_{i}^{max}\nonumber \\
\end{aligned}
Similar to the DC formulation, we can have switching mode and generation curtailment mode. Since we have the voltage constraints, we have to solve the power flow within each iteration of the optimization algorithm to determine the bus voltages and line flows. The power flow solution is non-linear and hence the constraint evaluation is implicit and non-convex. Hence, a global optimization technique is required to solve the problem. This problem is modeled as a non-convex mixed integer non-linear programming problem for switching mode and a non-linear programming problem for generation reduction mode. In this work, the branch and bound algorithm \cite{minlp2} has been used.
It is required to develop a decentralized, fast, fault tolerant RAS. The proposed RAS logic is designed to be implemented in multiple controllers at electric substations connected to data sources in a decentralized manner. This allows applications to perform calculations continually in an optimal layout for data delivery and initiating emergency control action. It provides further reliability over the failure of a single centralized calculation node. One of the main objectives of such an architecture is to ensure reliability, in case of any node/link failure in the distributed framework.
Depending on the distributed algorithm in development, a variety of different things can happen to the data after it’s made available to its local substation computer. First and foremost, data may be analyzed for importance, potentially shortening the caching period or deleting unnecessary and redundant data. This is not always possible to deduce on a local scale though, so some or all data may be forwarded onto a computational leader. The nodes selected to aggregate data from multiple (or all) other nodes are called computational leaders, they are often selected based on processing capability, but a variety of metrics can be used to varying effect, including physical grid placement, or mean network latency. These group leaders can work on data from their subsection of substations, working and share data with other leaders as necessary, to optimize the amount of stored data and work any one node is required to perform.
All data transfers along the group member, primary leader, and backup leader will pass through emulated communication network in real-time. The emulated communication network also needs to emulate the delays that occur in real world communication networks. NS-3 provides network processing delay, signal propagation delay, transmission delay, and queuing delay as a result of communication network emulation.
A set of algorithms can be used and adopted for different use cases to ensures replication of the data between multiple computational leader nodes, as well as handling their selection and coordination in failure conditions. The RAS program uses this data to monitor the system status and calculates the appropriate control action if an overload problem is forming or occurring. Control action will be sent through NS-3 simulated WAN communication network back to master PC. In the master PC, our self-designed communication program receives the control action and send it into RSCAD to control the breaker or wind farm output in the simulated power system. In a real-world implementation, these control actions would be routed directly to the appropriate windfarm.