## 3 Improved Model of TNEP with Uncertain Renewable Generation and Load Demand  
### Classical Multi-Scenario TNEP Model  
A scenario set that contains all the typical combinations of renewable power output and load demand is generated through the previous clustering method. Hence, the planning objective of TNEP under uncertainty can be simplified to find the most economical planning scheme that satisfies all typical scenarios. By incorporating the typical scenario set, the classical multi-scenario TNEP model is formulated by [6; 8]:  
\[min\;w=\sum\limits_{ij\in\Omega}c_{ij}n_{ij}+\beta\sum\limits_{k\in\Phi}\sum \limits_{i\in\Psi}\left(t_{i}^{k}+r_{i}^{k}\right) \tag{11}\]  
subject to  
\[S\times f^{k}+g_{C}^{k}+g_{R}^{k}+l^{k}=d^{k}+r^{k} \tag{12}\]  
\[f_  
\[0\leqslant n_{ij}\leqslant\overline{n_{ij}},\ n_{ij}\ \text{is integer} \tag{18}\]  
where \begin{tabular}{l l} \(c_{ij}\) & Cost of line that added to right-of-way \(i\)-\(j\) \\ \(n_{ij}^{0}\) & Number of lines in service on right-of-way \(i\)-\(j\) \\ \(n_{ij}\) & Number of lines in planning on right-of-way \(i\)-\(j\) \\ \(\Psi\) & Set of buses \\ \(\Omega\) & Set of candidate right-of-ways \\ \(\Phi\) & Set of typical scenarios \\ \(\beta\) & Penalty factor \\ \(\gamma_{ij}\) & Susceptance of right-of-way \(i\)-\(j\) \\ \(\theta\) & Voltage angle at node \(i\) \\ \(f\) & Vector of active power flow \\ \(g_{C}\) & Vector of conventional generation \\ \(g_{R}\) & Vector of renewable generation \\ \(S\) & Branch-node incidence matrix \\ \(l\) & Vector of curtailed load \\ \(d\) & Vector of load demand \\ \(r\) & Vector of renewable generation curtailment \\  & Denotes the max value of a quantity \\ \(k\) & Denotes the value of a quantity in scenario \(K\) \\ \end{tabular}   
The objective of Equation (11) is to find the minimum cost planning scheme that satisfies the load demand and renewable power consumption in all typical scenarios. It can be observed that a sufficiently large penalty factor \(\beta\) can effectively drive both the load shedding and renewable generation curtailment to zero. Constraint Equation (12) is the nodal power balance equation at each bus. Constraint Equation (13) is the DC power flow balance equation at each right-of-way. Constraints Equations (14)-(17) are operation limits of power system for the active power flow in the lines, the conventional generator output, the renewable generation curtailment, and the load shedding, respectively. Multi-Scenario Multi-Objective TNEP Model Considering Distribution Characteristics of Renewable Power Output  
The classical multi-scenario TNEP model caters to the rated capacity of renewable power plants under all typical scenarios, thus ensuring full consumption of renewable generation. However, according to previous research [10], the power output of renewable energy, especially wind energy, is relatively low in most scenarios, whereas the high level output seldom occurs. Therefore, the planning scheme, which satisfies the rated renewable power output, may lead to poor economic efficiency due to the huge cost of transmission investment in integrating renewable energy sources. Taking into account the distribution characteristics of renewable power, an appropriate curtailment of renewable generation is allowed in this paper to save the transmission investment. Obviously, the transmission investment conflicts with the curtailment level of renewable generation. A high investment in TNEP tends to reduce the curtailment of renewable generation, and _vice versa_. Consequently, a multi-objective TNEP model is proposed to obtain a Pareto optimal set of planning schemes with different levels of investment costs and renewable generation curtailments.