# Safety of Nuclear Power Plants with Respect to the Fault Displacement Hazard
## 3 Concept Proposed for Definition of Design Basis PGD
Thus, for \(P(sr\
eq 0)\), the hazard level of seismic design basis holds:  
\[P(sr\
eq 0)<\mathrm{P}[\mathrm{sr}\
eq 0|Mw_{max}]\leq 1 \tag{3}\]  
The annual frequency of non-zero PGD is approximately equal to \(\mathrm{P}[\mathrm{sr}\
eq 0|Mw_{max}]\cdot\lambda(Mw_{max})\). Here, the \(\lambda(Mw_{max})\) is the annual frequency of \(Mw_{max}\) taken from the deaggregation. Figure 2: Distribution of the maximum potential (cut-off) magnitude in the PSHA model (truncated normal distribution with parameters: \(\mu=6.2\); \(\sigma=0.4\); Mw low = 5.8; Mw high = 6.4) and distribution of contribution of magnitudes to the \(2.24\times 10^{-4}/a\) and \(6.37\times 10^{-6}/a\) levels of hazard (Figure 2 was generated using data published by the author in [28].) The intensity measure, _IM_, of the seismic hazard is usually the maximum horizontal acceleration of design basis earthquake PGA\({}_{\text{DBE}}\). The rate of exceedance \(\lambda(\textit{IM}>\text{PGA}_{\text{DBE}})\) of PGA\({}_{\text{DBE}}\) will be:  
\[\lambda(\textit{IM}>\text{PGA}_{\text{DBE}})=\lambda(\textit{Mw}>\textit{Mw}_{ min})\int_{\textit{Mw}_{min}}^{\textit{Mw}_{max}}\int_{0}^{\textit{R}_{max}}P( \textit{IM}>\text{PGA}_{\text{DBE}}|m,r)p(m,r)drdm \tag{4}\]  
Since the double integral is larger than zero, the \(\lambda(\textit{Mw}_{max})<\lambda(\textit{IM}>\text{PGA}_{\text{DBE}})\). Consequently, the following inequality is also valid:  
\[\text{P}[\text{sr}\
eq 0|\textit{Mw}_{max}\,|\cdot\lambda(\textit{Mw}_{max})< \lambda(\textit{IM}>\text{PGA}_{\text{DBE}}) \tag{5}\]  
The PGD for seismic design basis can be calculated as the mean value of the permanent ground displacement, D that can be calculated formally as:  
\[<\text{PGD}_{\text{DBE}}>\ =\ \int\limits_{0}^{\text{PGD}_{\text{max}}}D\cdot \int_{\textit{Mw}_{min}}^{\textit{Mw}_{max}}p(\textit{D}|m)\cdot p(m)dm\;dD \tag{6}\]  
Instead of \(<\text{PGD}_{\text{DBE}}>\), it is reasonable to correlate the PGD design basis via the empirical relationship \(PGD=D(\text{Mw})\), selecting the PGD\({}_{\text{DBE}}\) at the maximum magnitude \(\textit{Mw}_{max}\) of the localized source from the deaggregation of seismic design basis using reference, for example, [28]. \[PGD_{\text{DBE}}=D(\textit{Mw}_{max}) \tag{7}\]  
Obviously, the \(\lambda(\text{PGD}_{\text{DBE}})=\lambda(\textit{Mw}_{max})\leq\lambda(\text{ PGA}_{\text{DBE}})\). Consequently, the selection of PGD\({}_{\text{DBE}}\) is conservative and can be used for the safe design. Since the magnitudes tend to be the maximum potential magnitude \(\textit{Mw}_{max}\,\), the PGD hazard can be screened out per annual frequency via comparison with the acceptable value for the early or large releases, ELR, frequency, if:  
\[\text{P}(\textit{sr}\
eq 0|\textit{Mw}_{max}\,^{*})\cdot\lambda(\textit{Mw}_{ max}\,^{*})<\lambda(\textit{ELR})\leq\frac{10^{-6}}{\textit{year}} \tag{8}\]  
Here, it is implicitly assumed that a PGD event will unavoidably cause early or large releases that is a rather conservative assumption that is an over conservative assumption, except if the site is directly located at a capable fault with very large maximum potential magnitude. The condition \(\lambda(\textit{Mw}_{max}\,^{*})\leq\lambda(\textit{Mw}_{max})\leq\lambda( \text{PGA}_{\text{DBE}})\) would also allow for considering:  
\[\text{PGD}=D(\textit{Mw}_{max}\,^{*}) \tag{9}\]  
as a beyond design basis effect that can be used for margin analysis. If the PGD hazard cannot be screened out via Equation (6), the hazard should be taken into account in the design and the safety should be evaluated and justified in accordance with the criteria for core damage and early or large releases. Following the logic of seismic design, the curve \(\lambda(\text{D}>\text{d})\) should be defined. For the design, the design basis ground displacement, PGD\({}_{\text{DBE}}\), can be defined on the mean \(\lambda(\text{D}>\text{d})\) at the hazard level for the seismic design basis, \(H_{\text{DBE}}\), that is usually \(10^{-4}\) to \(10^{-5}\) per annum.