**1. Introduction**

The Fukushima reactor disaster in 2011 made the question of nuclear safety relevant again. Similar accidents are known to have happened in the Soviet Union in 1986 (Chernobyl) and in the USA in 1979 (Three Mile Island). These core melt accidents are the most severe ones in nuclear reactors. When the rods containing the nuclear fuel and the fission products melt, a huge amount of radioactivity is set free within the reactor and possibly into the atmosphere.

But the rate of such accidents seemed much higher than previously claimed. So, we tried to study the probability of such events empirically by looking at the real events.

This a posteriori approach differs from the a priori approach of Probabilistic Risk Assessment (PRA) which is done during the design phase of a reactor. PRA determines failure probability

prior to accidents by analysing possible paths towards a severe accident, rather than using existing data to determine probability empirically.

fluctuate randomly. The Poisson distribution allows us to calculate the probability of a given

If the time of reference, *T*, is 1 year, then *λ* is the expected number of incidents within 1 year. If *λ* is much smaller than one, then it is also the probability of one incident within 1 year and of at least one incident per year. Analysing not only one but many reactors, the expected total number of accidents is simply the sum of the expected number for each single reactor, and, as long as the reactor incidents are independent of each other, the actual number of accidents

In analysing real systems, the number of (statistically fluctuating) incidents *x* is known, and *λ* has to be determined. Then, the best estimate for *λ* is simply this empirical value *x*. However, this estimate is not necessarily the true value of *λ* because the incidents occur randomly. Poisson statistics allow us to compute an interval that contains the true value of *λ* with a confidence level α (typically 90, 95 or 99%), the so-called confidence interval. This is determined

interval, we choose *λ*<sup>1</sup> <sup>&</sup>lt; *<sup>x</sup>* such that the probability of observing *x* or more events is 2.5% and *<sup>λ</sup>*<sup>2</sup> <sup>&</sup>gt; *<sup>x</sup>* such that the probability of observing *x* or fewer events is 2.5%. Then, the interval *λ*<sup>1</sup>

 is a 95% confidence interval. This means that if we study many cases, then in 95% of these cases, the true value of *λ* lies within this interval. The more cases we observe the narrower the

As an example, suppose that the empirical number of events is *x* = 4. Then, the Poisson distribution with a value for *λ* equal 1.090 gives the probability that the number of events is greater than or equal to 4 to be 2.5%. If *λ* is 10.242, then the probability that the number of events is less than or equal to 4 is also 2.5%. Thus, for the empirical value of *x* = 4, we say that the true

A similar measure of the probable distance between the estimated empirical value and the true value is the standard error. In large samples the probability that the distance between the

The International Atomic Energy Agency in Vienna publishes data on all power reactors worldwide [2]. The same and additional information about connection to the grid, shut down, operator, manufacturer and fuel supplier can be found in several Wikipedia entries [3, 4].

It was 1952 when the Soviet Union connected the first nuclear power reactor worldwide to the grid. Two years later the UK followed with Calder Hall. The number of reactors increased

confidence interval will be and the closer the estimate of *λ* will be to the true value.

estimated and the true value is less than the standard error is approximately 68%.

, for a given number of incidents *x*. For the 95% confidence

*<sup>x</sup>*! (1)

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Severe Nuclear Accidents and Learning Effects http://dx.doi.org/10.5772/intechopen.76637

to

Probability of *<sup>x</sup>* incidents if *<sup>λ</sup>* incidents are expected <sup>=</sup> *<sup>e</sup>* <sup>−</sup>*<sup>λ</sup> <sup>λ</sup><sup>x</sup>* \_\_\_\_

and *λ*<sup>2</sup>

value for *λ* is between 1.090 and 10.242 with 95% confidence.

number *x* of events within *T*:

is Poisson distributed.

by calculating two values, *λ*<sup>1</sup>

**3. Data acquisition**

**3.1. How many reactors?**

*λ*2

After an accident very often 'learning from experience' is claimed. The luckily low number of severe accidents does not allow for testing this claim. But reactor operators should be interested in reducing all incidents and accidents; so, their frequency should decrease with increasing operating experience. We use the total time reactors are operating, the reactor-years, as a measure of experience, analyse the accidents as a function of this experience with generalised linear models and compare a frequentist and a Bayesian approach.

Accidents can and did happen in several areas of nuclear energy, e.g. military use for weapons or submarine propulsion, medical use or fundamental research. Discussing the risks of nuclear energy involves very different arguments in all these areas. We restricted the study to accidents in nuclear reactors for power generation.

According to our analysis, we have to expect one core melt accident in 3700 reactor-years with a 95% confidence interval of one in 1442 reactor-years and one in 13,548 reactor-years. In a world with 443 reactors, with 95% confidence we have to expect between 0.82 and 7.7 core melt accidents within the next 25 years.

Analysing all known accidents, we can show a learning effect. The probability of an incident or accident per reactor-year decreased from 0.01 in 1963 to 0.004 in 2010. Furthermore, there is an indication of a slightly larger learning effect prior to 1963.

It is well known that the actual number of all incidents and accidents is much higher than the numbers published in scientific journals. Therefore, we studied whether the known incidents and accidents are distributed randomly over the reactors using countries. While the data are random for most of the countries, this is not the case for the USA. From the present data, we cannot decide whether this is due to higher incident rates or to more effective sampling.

After this introduction the second section will explain some basics of the Poisson distribution. In Section 3 we present the data acquisition and its problems. Section 4 contains the discussion of core melt accidents and predictions for future events. The learning effect analysis is presented in Section 5.

While some of the results have been published already elsewhere [1], the underlying statistical work is presented here.
