## 2 Fundamentals of Pricing Schemes: Uniform, Zonal, and Nodal Pricing
### Market-Clearing Models
#### 2.1.2 Transmission Flow Constraints  
The key differentiating factor among electricity markets is the treatment of transmission flow constraints. These constraints are modeled using various approaches, ranging from simplified representations to more complex methods. In zonal pricing, ATC and FBMC are commonly used. ATC is simpler but less reflective of network interconnectivity, whereas FBMC offers higher efficiency, albeit with greater complexity [6; 27; 28]. In nodal pricing, the DC-OPF approximation of the AC optimal power flow (AC-OPF) is typically applied due to computational limitations in case of practical market operations [28]. These  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Market-Clearing Models** & **Pricing** \\  & **Mechanisms** \\ \hline Objective Function & Minimize the total system cost or maximize social welfare & \\ \hline \multirow{4}{*}{Constraints} & Generator output limits & LMP \\  & Generator ramp rate constraints & ZMP \\  & Generator minimum up- and down-time requirements & ELMP \\  & Power balance constraints & SMP \\  & Transmission flow constraints & \\  & Other system operational constraints & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Overview of electricity market operation.  
variations in constraint modeling can significantly affect the market outcomes, including dispatch decisions and pricing.
* ATC: This is a simplified approach that models power exchanges through zonal interfaces with predefined transfer limits, without considering the network reactance or other electrical characteristics of the system. This can be mathematically expressed as Equation (2). \[F_{l}^{min}\leq f_{l,l}^{ATC}\leq F_{l}^{max},\ \ \ \ \forall l\in L,t\in T\] (2) where \(F_{l}^{max}\) represents the predefined transfer upper limit, which takes positive values for interface \(l\), and \(F_{l}^{min}\) denotes the predefined transfer lower limit, which takes negative values for interface \(l\).
* FBMC: This approach considers the impact of power flows on multiple lines simultaneously using power transfer distribution factors (PTDFs). The zonal PTDF matrix is applied only to calculate flows on critical branches (CBs), facilitating a focused consideration of key network constraints while preserving the zonal market structure. The FBMC approach is characterized by Equations (3) and (4). \[f_{k,l}^{FB}=\sum_{z\in Z}ZPTDF_{k,z}\times NEX_{z},\ \ \ \ \ \forall k\in CB,t\in T\] (3) \[\left|f_{k,l}^{FB}\right|\leq TL_{k},\ \ \ \ \ \forall k\in K,t\in T\] (4) where \(ZPTDF_{k,z}\) is the zonal PTDF matrix for branch \(k\) and zone \(z\), and \(NEX_{z}\) is the net exchange in zone \(z\). The parameter \(TL_{k}\) indicates the thermal limit of branch \(k\), and the set \(CB\) refers to all the CBs within the network. Generally, \(ZPTDF_{k,z}\) is calculated based on node-level PTDF matrix; further details can be found in [27; 28].
* DC-OPF: This is the simplified linear model that approximates the AC-OPF for market clearing and is expressed as Equations (5) and (6). \[f_{k,l}^{DC}=\sum_{i\in I}PTDF_{k,i}\times P_{i},\ \ \ \ \ \forall k\in K,t\in T\] (5) \[\left|f_{k,l}^{DC}\right|\leq TL_{k},\ \ \forall k\in K,t\in T\] (6) where \(PTDF_{k,i}\) is the node-level PTDF matrix for line \(k\) and node \(i\), and \(P_{i}\) is the net injection at node \(i\).