**6. Monte Carlo simulation**

### **6.1. Basics**

### *6.1.1. Monte Carlo simulation*

Detailed basics of the MCS like random sampling, estimators, biasing techniques and performance characteristics (e.g. figure of merit / fom) are specified for example in [22] and [23].

In the references [9, 11, 19, 24] the MCS has been applied and verified successfully in order to estimate the probability of external explosion pressure waves.

#### *6.1.2. Estimators in use*

As the last event estimator (lee), introduced in [28], is used to predict the probability of an event (e.g. an explosion event), the observed frequency of explosions within the radius rP is determined. The sample mean probability is

$$
\hat{P}\_E = \frac{1}{N} \cdot \sum\_{i=1}^{N} P\_E(i) \tag{7}
$$

where PE(i) {0, 1} and N = number of trials.

An alternative method is to compute the theoretical probability of an explosion event within the radius rP in each scenario the wind direction will move the explosive gas mixture to the plant. The advantage over the lee is that each scenario gives a contribution to the probability of occurrence.

By analogy with transport theory, this procedure is called free flight estimator (ffe) also described in [25]. Depending on the accident coordinate (xi, yi) and the wind direction φi in trial i the probability of an explosion event within the radius rP is given by

$$\begin{split} P\_E(\mathbf{x}\_i, \phi\_i) &= \exp\left(-\boldsymbol{\lambda} \cdot \mathbf{1} / \,\boldsymbol{v}\_W \cdot d\_1(\mathbf{x}\_{i'}, \phi\_i)\right) \\ &- \exp\left(-\boldsymbol{\lambda} \cdot \mathbf{1} / \,\boldsymbol{v}\_W \cdot d\_2(\mathbf{x}\_{i'}, \phi\_i)\right) \end{split} \tag{8}$$

where d1(x, φ) and d2(x, φ) are the distances between the accident coordinate and the intersection of the wind direction and the plant area with radius rP.

The intersection coordinates (xI, yI) of the wind direction φi and the plant area with radius rP are determined by means of

$$\left(\mathbf{x}\_{I}^{2} + \left(y\_{i} + \tan(\phi\_{i}) \cdot (\mathbf{x}\_{I} - \mathbf{x}\_{i})\right)^{2} = r\_{P}^{2}\tag{9}$$

and

136 Nuclear Power – Practical Aspects

procedure in [21].

**installations** 

**6.1. Basics** 

[23].

**5.2. Ship accident statistics** 

**6. Monte Carlo simulation** 

*6.1.1. Monte Carlo simulation* 

*6.1.2. Estimators in use* 

determined. The sample mean probability is

where PE(i) {0, 1} and N = number of trials.

r radius of the nuclear power plant within which a) damages are expected in case of a detonation,

b) the drifting of a gas-air cloud (deflagration) has to be expected.

storage tanks, gas pipelines have to be taken into account according to [3].

gas mixture in the direction of the nuclear power plant are therefore excluded.

to estimate the probability of external explosion pressure waves.

Ship accidents (provided in Germany by the local Waterways and Shipping Directorate) are provided for a defined time period and the river-km and distinguished by the types of accidents. Information with respect to the participation of gas, liquid gas and ammunition shipments to the accident is usually given. The evaluation is performed according to the

**5.3. Occurrence frequency of accidents with explosive materials in stationary** 

In that context, installations such as industrial plants, loading and discharging stations,

In case of natural gas the formation of an explosive gas mixture is only assumed for the accident area because the specific gravity of natural gas is less than the air and drifts of the

Detailed basics of the MCS like random sampling, estimators, biasing techniques and performance characteristics (e.g. figure of merit / fom) are specified for example in [22] and

In the references [9, 11, 19, 24] the MCS has been applied and verified successfully in order

As the last event estimator (lee), introduced in [28], is used to predict the probability of an event (e.g. an explosion event), the observed frequency of explosions within the radius rP is

> 1 <sup>1</sup> <sup>ˆ</sup> ( ) *N E E i P Pi N*

(7)

$$\mathbf{y}\_I = \left( y\_i + \tan(\phi\_i) \cdot (\mathbf{x}\_I - \mathbf{x}\_i) \right)^2. \tag{10}$$

The sample mean probability is

$$\hat{P}\_E = \frac{1}{N} \cdot \sum\_{i=1}^{N} P\_E(\mathbf{x}\_{i'}, \phi\_i) \tag{11}$$

where N = number of trials.

#### *6.1.3. Biasing techniques in use*

If the forced transition method is used (see, e.g., [26]), the next transition is forced to take place within the area (wind direction, distance, time etc.) of interest.

The modified conditional cdf is

$$\begin{split}P(X \le \mathbf{x} \mid \mathbf{x}\_1 < X \le \mathbf{x}\_2) &= F(\mathbf{x} \mid \mathbf{x}\_1, \mathbf{x}\_2) \\ &= \frac{F(\mathbf{x}) - F(\mathbf{x}\_1)}{F(\mathbf{x}\_2) - F(\mathbf{x}\_1)}. \end{split} \tag{12}$$

The weight associated to this bias is

$$\mathbf{w}^\* = F(\mathbf{x}\_2) - F(\mathbf{x}\_1) \,. \tag{13}$$
