Turvey–Shapley Method for Network Cost Allocation

• The Turvey–Shapley method is a cost-reflective approach combining a probabilistic LRMC model and cooperative game theory to assign network costs based on customer demand impacts.
• It employs advanced statistical modeling, including Weibull-based tail event estimation and hierarchical clustering, to achieve granular and computationally tractable allocations.
• Case studies reveal that this method correlates closely with actual peak demands, reducing cross-subsidies compared to traditional energy-based allocation approaches.

The Turvey–Shapley Method is a cost-reflective, forward-looking approach for allocating distribution network costs to residential electricity customers. By combining a probabilistic long-run marginal cost (LRMC) model based on the Turvey perturbation principle with cooperative game theoretic Shapley value allocation, it aims to assign costs in a manner that aligns with each customer’s causal impact on future network investment requirements. The method incorporates advanced statistical modeling of tail events, exact and approximate Shapley calculations, and hierarchical clustering for computational tractability, yielding granular, scalable, and causally interpretable cost allocations (Azuatalam et al., 2019).

1. Turvey Perturbation Foundations for Network LRMC

The Turvey perturbation method (Turvey 1969) yields a forward-looking LRMC associated with capacity-driven network investments by considering the incremental cost of a small, permanent increase in demand. The canonical expression is

$\text{LRMC} = \frac{PV(\Delta C)}{PV(\Delta D)}$

where $\Delta C$ denotes the present value of capacity cost required for a permanent increment $\Delta D$ in peak load, and $PV(\cdot)$ applies discounting across the planning horizon.

For distribution networks with uncertain annual peaks, the Turvey–Shapley method employs a probabilistic 50% probability of exceedance (“50 POE”) framework, associating “small” demand growth (typically a 1% annual increment) with increments in the 50 POE annual peak. The peak load for a customer coalition $S$ is modeled as a stochastic variable, with the critical event “peak exceeds line limit $x$” assumed to follow a fat-tailed Weibull distribution:

• $X \sim \text{Weibull}(\alpha, \beta)$, with $\beta=1.5$
• CDF $F_{\alpha,\beta}(u) = 1 - \exp[-(u/\alpha)^\beta]$
• $\alpha$ is calibrated so that mean $\Delta C$0 (the forecasted coalition’s 50 POE peak).

The incremental (forward-looking) cost assigned to coalition $\Delta C$1 is thus

$\Delta C$2

with $\Delta C$3 denoting the cost of an augmentation (e.g., line upgrade). In operational terms, the LRMC per unit increment is $\Delta C$4 (Azuatalam et al., 2019).

2. The Turvey LRMC Computation

The computation involves the following:

• $\Delta C$5: forecasted 50 POE annual peak for coalition $\Delta C$6
• $\Delta C$7: line (emergency) capacity threshold
• $\Delta C$8: cost to augment by the next block
• $\Delta C$9: mean of $\Delta D$0, set as the 1% annual growth target

Assuming the distribution of $\Delta D$1 is Weibull, and with a negligibility threshold (e.g., $\Delta D$2), the core steps are:

1. Calculate $\Delta D$3
2. Approximate $\Delta D$4
3. Model only one augmentation per calculation block; ignore negligible tail probabilities.

Key modeling assumptions include permanent 1% annual 50 POE peak growth and single augmentation events.

3. Shapley Value Allocation and Causal Cost Attribution

Customers are modeled as players in a transferable-utility cooperative game where the characteristic function is $\Delta D$5 (the probabilistic Turvey LRMC). The Shapley value $\Delta D$6 uniquely allocates costs according to efficiency, symmetry, additivity, and the null-player principle. It is evaluated either as

$\Delta D$7

or, grouping by coalition size,

$\Delta D$8

Each customer $\Delta D$9 receives a charge reflecting their expected marginal contribution to future LRMC investment, averaged over all orders in which customers could join the system. The result is a cost-causal, forward-looking allocation (Azuatalam et al., 2019).

4. Clustering-Based Approximation for Scalability

Direct computation of Shapley values requires $PV(\cdot)$0 evaluations of $PV(\cdot)$1, becoming intractable for $PV(\cdot)$2. The Turvey–Shapley method addresses this via hierarchical customer clustering:

1. Partition the set of $PV(\cdot)$3 customers into $PV(\cdot)$4 clusters $PV(\cdot)$5 using k-means on half-hourly annual load profiles (typical $PV(\cdot)$6).
2. Treat clusters as “super-players” and compute exact Shapley values over the $PV(\cdot)$7 cluster coalitions.
3. For each coalition $PV(\cdot)$8, record the vector $PV(\cdot)$9 of individual customer contributions to $S$0's 50 POE peak.
4. Distribute each cluster’s Shapley value among its members in proportion to their averaged peak-load share across contributing coalitions.

This reduces the computational burden from $S$1 to $S$2 evaluations of $S$3 plus $S$4 bookkeeping, enabling practical application to hundreds of customers (Azuatalam et al., 2019).

5. Turvey–Shapley Algorithmic Workflow

The high-level procedure is as follows:

1. Forecast individual customers’ small (e.g., 1%) 50 POE annual-peak increments $S$5.
2. Partition the full set into $S$6 clusters via k-means.
3. Compute network line limit, typically $S$7 annual peak of the grand coalition.
4. For each cluster coalition $S$8:
   • Sum $S$9
   • Fit Weibull parameters with mean $x$0 and $x$1
   • Compute $x$2, set $x$3
5. Compute exact cluster Shapley allocations $x$4 as above.
6. Record for each cluster and coalition, the member's peak-load shares $x$5
7. Allocate $x$6 to members $x$7 proportional to their average $x$8
8. The resulting $x$9 is the LRMC tariff for customer $X \sim \text{Weibull}(\alpha, \beta)$0 (Azuatalam et al., 2019).

6. Comparative Performance with Established Methods

A numerical case study uses 125 residential customers from the Ausgrid Solar Home Electricity trial (half-hourly data, 2012–13), considering net loads both with and without PV. Benchmarking against alternative cost allocation approaches yields the following correlations of each method’s cost assignments with the true network peak driver (correlation $X \sim \text{Weibull}(\alpha, \beta)$1):

Method	Without PV	With PV
Shapley	0.948	0.947
Coincident	0.948	0.947
Yearly-peak	0.507	0.249
Monthly-peak	0.485	0.468
Energy-flat	0.210	0.180
Energy-ToU	0.240	0.220

Root mean squared error (RMSE) relative to the Shapley value allocation is smallest for the coincident peak, moderate for yearly/monthly peak methods, and largest for energy-based approaches. Energy-based methods are shown to result in significant cross-subsidies and poor cost-reflectivity, while Shapley allocation aligns closely with cost causality (Azuatalam et al., 2019).

7. Computational Efficiency and Practical Feasibility

Direct Shapley enumeration scales as $X \sim \text{Weibull}(\alpha, \beta)$2 in the number of customers, with runtimes approximately 10 minutes for $X \sim \text{Weibull}(\alpha, \beta)$3, over 1 hour for $X \sim \text{Weibull}(\alpha, \beta)$4, and 7 hours for $X \sim \text{Weibull}(\alpha, \beta)$5. Randomized sampling can reduce runtime by about 75% at $X \sim \text{Weibull}(\alpha, \beta)$6, but remains prohibitive as $X \sim \text{Weibull}(\alpha, \beta)$7 increases.

Cluster-based approximation reduces computational complexity dramatically, with only $X \sim \text{Weibull}(\alpha, \beta)$8 calls to $X \sim \text{Weibull}(\alpha, \beta)$9 for $\beta=1.5$0 (e.g., $\beta=1.5$1 yields 32 evaluations). Including all customer–cluster bookkeeping ($\beta=1.5$2), the end-to-end computation for $\beta=1.5$3, $\beta=1.5$4 is under 5 minutes, with performance independent of $\beta=1.5$5 beyond clustering (Azuatalam et al., 2019).

In summary, the Turvey–Shapley method adheres to economic principles of causality and fairness in LRMC allocation, incorporates robust statistical modeling of network risk, and achieves computational tractability for large customer bases via clustering, providing a practical, allocationally fair, and cost causally precise mechanism for modern distribution networks.