## 1 Introduction  
A dam is a barrier constructed across a river to store water or arrest the flow of water. Before erecting a dam, a temporary diversion structure should be installed to provide dry work conditions for main dam construction. A diversion structure consists of an upstream/downstream cofferdam and diversion tunnel. By constructing a cofferdam, the river water can be diverted into a diversion tunnel, which makes a dry workspace between the upstream and downstream cofferdams for the main dam construction.  
Although a diversion structure is used temporarily, it has a large proportion of total EPC (Engineering, Procurement, Construction) cost. Therefore, an optimal size of a diversion structure should be determined to optimize projects in terms of economic feasibility and safety. Theoretically, the design flood frequency of a diversion structure is determined by considering characteristics of flood frequency, basin size, construction period, the type of main dam, and damage or loss caused by flooding during construction. However, in practice, the return period presented by the design standard is normally used as the design -flood frequency for the diversion structure, which is normally related to the type of main dam. However, the return period for a diversion structure is presented differently in design standards; for example, ICOLD proposes 10 years as a return period for a diversion structure when the type of main dam is a concrete gravity dam [1], whereas the Korean dam design standard proposes 1-2 years [2].  
[PERSON] et al. [3] classified the evolution of design approaches for hydro-system infrastructure into three principal stages: (1) the historical event-based approach, (2) the return-period approach, and (3) the risk-based approach [3]. According to this classification, the practical method to determine the design flood frequency for a diversion structure can be classified as the return-period approach. The risk-based approach is an advanced procedure that evaluates different alternatives by considering the trade-off between the investment cost and the expected economic losses due to failure. Therefore, the risk-based approach can be utilized to study or evaluate the optimal design flood frequency for a diversion structure.  
In previous studies on the design flood frequency of a diversion structure, an optimization model based on permanent hydraulic structures was applied to the diversion structure in a dam construction project. In this modeling, the failure of the diversion structure is defined such that the flood water level resulting from an extreme rainfall event is higher than the design flood water level. The results of optimization are presented using the relations graph of the failure risk, construction period, expected damage, and loss for the flood water level [4; 5]. The failure of a cofferdam should, however, be evaluated considering the overall force relationship, which lies in the scope of stability analysis after calculating the loadings and resistances acting on a cofferdam.  
This study explores the design optimization methodology for a diversion structure with a concrete cofferdam using a risk-based approach to quantify the practical hazards and risks of dam construction. The proposed methodology is based on a risk-based least-cost design process and risk analysis and is applied to a case study of the Gulpur hydropower project in Pakistan. Risks for the design optimization methodology include the structural failure of the upstream cofferdam as a structure risk and the overflow of the upstream cofferdam as a functional risk. The failure probability was calculated using the limit state function and Monte Carlo Simulation (MCS). The overflow probability was calculated from the inverse of the return period. Finally, the optimization result was presented as the lowest value of Expected Monetary Value (EMV), which considers construction cost, re-construction cost and delay LD (Liquidated Damage) with the failure probability and the recovery cost with the overflow probability.