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I I I I I I I I I I I I I I I I I I I Un.r\r I U concern. It is common to adopt a single interval to represent flows in excess of the maximum design inflow (usually the PMP Design Flood), where the probability assigned to inflows in this uppermost interval is equal to the exceedance probability of the inflow event at the lower boundary of the interval, and the representative flow magnitude for the interval is notionally increased over the maximum design value. . Discrete probability distribution of outflows. Depending on the degree of non-linearity of the spillway rating curve, outflows may be discretised into class intervals of equal magnitude, or else intervals can be selected to provide more accuracy in the region of interest (e.g. for flows just above and below the spillway capacity). . Probabifity distribution of initial slorage volume. The analysis of a time series of storage level or storage volume is used to define the probability distribution of initial storage volume. The time series of reservoir storage volume could be derived directly from the historical record, but in most cases a synthetic time series of storage volume, derived from simulation ibeilaviour analysis) studies, would be more appropriate. In the latter approach the current operating rules can be applied to the historic hydroclimatic sequence, thus providing a long stationary series relevant to the system under consideration. The usual time interval for behaviour analyses is one month, which allows the within-season variation of storage volume to be taken into account in the frequency analysis. If the time series of reservoir storage volume is derived directly from the historicai record, or else if the synthetic results are derived using a daily time-step model, then the correlation between inflows and storage volume can be implicitly accounted for by using the storage level prior to the commencement of any flood which occurred during the month. This is of course not possible if the results are obtained from a monthly simulation model, but in general a synthetic series is to be preferred as it is assumed that greater benefits are to be had from consideration of a long time series than from incorporation of the dependence between inflows and storage contents. (c) Deterministic relationship between inflow, storage volume and outflow The conditional probability of a specified outflow event occurring, given that the conditioning event is in a specific class interval, can be determined using a deterministic relationship between inflows, outflows, and storage volume (the 1-8-Q relationship). The I-S-Q relationship has to be determined for a range of peak inflows (corresponding to a range of design rainfalls for selected exceedance probabilities) and for a set of initial storage values. The process of computing the conditional probability of a specified outflow event is illustrated in Figure 9 for the case where the reservoir inflow was chosen as the conditioning event and the initial storage volume as the secondary variable. From this figure it is seen that the conditional probability of a specified outflow event is evaluated as the width of the storage volume probability interval (P[Q;II;]) that translates an inflow in the interval I; into an outflow in the interval A.. As different design rainfall durations result in different 1- S-Q relationships, the computed value of the storage volume probability interval will also depend on the rainfall duration used. The critical rainfall duration to be used in the analysis. is the one that translates into the highest outflow; this also produces the largest estimate of conditional outflow probability. Unfortunately the critical rainfall duration varies with reservoir drawdown, and in some cases it is CUUK VI - t:.5UIIli:lLtUlI VI L.dlyt::: lU CAlIt::HIl::: rluuu:::, necessary to compute separate I-S-Q relationships for different durations, and to derive an outflow frequency curve as the envelope of frequency curves derived for different durations. Another complication is that the above formulation assumes that the two distributions of storage volume and inflows are independent. This may not be the case, but if such correlation is found to be significant then the calculations must be based on the appropriate conditional selection of input variables. The evaluation of the 1-8-Q relationship is the most time consuming element of the process. Many tens of individual runs are required to define the I-S-Q relationship in sufficient detail, though it is possible to automate the processing of different initial starting levels. The computation of the conditional probabilities is readily undertaken using spreadsheet software and is not resource intensive; it can also be undertaken manually. (d) Computation of outflow probabilities The derivation of the outflow frequency curve by Laurenson's (1974) joint probability approach involves the calculation of a transition probability matrix. Each element in this matrix represents the conditional probability of an inflow within the given inflow interval resulting in an outflow in a specified interval. The total probability of an outflow in that interval can then be obtained as the sum of the probabilities over all the inflow intervals, i.e. all the inflow and initial storage combinations that produce an outflow in the specified range. Outflow AEPs are then computed as the cumulative probability over all outflow ranges exceeding the flood magnitude of interest. An example of the application of this approach is given in Section 6.4. 5,3 Concurrent Tributary Flows 5,3.1 Overview In some design situations it is desirable to determine the flow in an adjacent catchment that is likely to coincide with design floods in the stream of interest. The most common requirement for this is the assessment of the incremental impact of reservoir failure, where it is desirable to identify separately the inundation due to the direct consequences of dam failure and the floods generated from adjacent catchments. There are a number of methods available for the assessment of concurrent flov c' (see, for example, Green, 1996). In the context of risk analysis it is important to focus on those methods that yield AEP-neutral estimates. In essence, the issue of concurrent flooding is another joint- probability problem, and the method of Laurenson (1974) described in Section 5.2 can be applied directly to the joint occurrence of floods in tributaries and adjacent catchments. With the analysis of concurrent flows, the deterministic I-S-Q relationship referred to in Section 5.2 is replaced by the relationship between total flows downstream of the confluence and the joint occurrence of upstream flows of differing magnitudes, and the marginal distnbution of storage volume is replaced by the probability distribution of flows in the adjacent tributary. Careful consideration needs to be given to the specification of the marginal distribution of tributary inflows as the two flow distributions will be correlated. Also, peak discharges are unlikely Illl coincide. While application of Laurenson's (1974) method should satisfy the AEP-neutral requirement, it does involve considerable resources. In some design situations the