For a given demand, the P+R lot choice procedure determines the distribution of this demand among the P+R lots. This distribution is used to calculate a parameter that is incorporated into the utility function of the P+R mode in the demand model. The paths resulting from the distribution can be visualized in the form of path sequences (Park + Ride leg split).
P+R lots are zones that are identified as P+R lots via an attribute. In this zone attribute, the capacity of the P+R lot is defined. P+R paths that start in a zone with a P+R lot cannot use that P+R lot, since such a path is essentially a pure public transport path. It may still make sense for paths to use the P+R lot located in the starting zone. This is especially the case if the zones are relatively large and thus there are actually such car trips to P+R lots. If you want to include these trips in parking capacity, the P+R lots must be modeled as separate zones with no production potential of their own.
The distribution of demand among P+R lots is basically equivalent to a capacity-based equilibrium assignment, where the choice of a path is equivalent to the choice of a P+R lot. The path impedances are derived from the impedances of the two path legs origin - P+R lot and P+R lot - destination, as well as the volume-dependent impedances of the P+R lots. In the procedure, an iterative process is used to achieve a demand split in which each path of an OD relation has the minimum total path impedance. For an OD relation, it is, therefore, possible that a conveniently located P+R lot is not used because its volume capacity ratio or its volume-dependent impedance is too high and another P+R lot can be used with less impedance.
The impedances of the trips to and from the P+R lots are defined by skim matrices, one each for the car trip to the P+R lot and one for the public transport trip from the P+R lot. The volume-dependent impedances of the P+R lots result from the VD functions selected for them. These generate increasing impedances with increasing utilization. The steeper the selected VD function, the more overloads are "punished" with high impedances. However, even very steep VD functions do not guarantee that overloads will not occur.
The iterative distribution procedure is finished when either the maximum number of iterations has been reached or the gap value goes below the defined threshold. Similar to an assignment, the gap corresponds to the ratio of actual to optimal total utility. The total utility is the sum of the utilities of all paths weighted by the volumes. In the case of optimal total utility, the optimal shortest path is used in the calculation instead of the path actually chosen. The gap utility calculation is carried out simultaneously across all OD pairs and demand strata.
From the distribution, the total impedance for the path origin - P+R lot - destination is calculated for each relation and stored in a skim matrix. There is no need to average over the impedances of different paths since the impedances of all paths of a relation are minimal and thus identical.
From a technical point of view, the incoming and outgoing skims of the procedure represent utility matrices, even though impedances are referred to here. The difference is essentially in the sign: while impedances typically have a positive sign (such as the travel time), utilities usually have a negative sign.
Mostly, the utility definitions of the corresponding modes (i.e. car and public transport) are taken from the mode choice model as incoming skims, which also already include a weighting of the different utility components. A negative weight must be defined in the procedure for the volume-dependent impedance of the P+R lots so that this component also represents a utility.
For the outgoing skim, it is important to note that this is also a utility matrix. Consequently, it typically enters the P+R mode utility function without a negative weighting parameter in the mode choice model. A utility of -999999 or lower is interpreted as unattainable in the procedure. To prevent certain OD pairs from using a P+R lot, you will need to manipulate the skim matrix data of their path legs accordingly. Since there is no P+R internal traffic, the value of the diagonal of the outgoing skim is also -999999.
