Custom Cost Distributions

This page explains how BICEP models cost uncertainty through probability distributions and how to work with custom distributions.

Overview

BICEP uses probability distributions to model infrastructure upgrade costs because:

  1. Real costs vary - Building characteristics, regional factors, and market conditions all affect costs
  2. Uncertainty exists - We don’t know exact future prices or site-specific requirements
  3. Risk assessment - Distributions help quantify best-case, typical-case, and worst-case scenarios

Cost Uncertainty Sources

Source Impact Example
Building size/type Large variation Small single-family vs. large multi-unit
Regional labor costs 30-50% variation Bay Area vs. rural areas
Upgrade complexity 2-3x variation Simple service panel vs. transformer
Supply chain Temporal variation 2020 vs. 2030 pricing
Material availability 20-40% variation Standard vs. specialty components

BICEP’s Built-in Distributions

BICEP uses three main probability distributions:

1. Panel Utilization Distribution

Models how fully existing electrical panels are currently utilized.

Characteristics:

  • Range: 0.1 - 0.95 (10% to 95% utilization)
  • Based on empirical data from building surveys
  • Affects how much spare capacity exists

Usage:

  • Determines which buildings need upgrades
  • Directly influences total number of upgrades needed

2. PV Sizing Distribution

Represents the relationship between building loads and solar PV system sizes.

Characteristics:

  • Range: 0.5 - 3.0 times building load
  • Log-normal distribution (right-skewed)
  • Reflects installer and owner preferences

Usage:

  • Determines electrical capacity needed for PV
  • Affects timing of upgrades

3. Panel Upgrade Cost Distribution

Models the cost variation for electrical panel upgrades.

Characteristics:

  • Residential: $2,000 - $50,000
    • Log-normal distribution
    • Mean approximately $8,000-$12,000
  • Commercial: $5,000 - $100,000+
    • Higher variability than residential
    • Mean approximately $15,000-$25,000

Includes:

  • Equipment costs
  • Labor costs
  • Permits and inspections
  • Site-specific modifications

Usage:

from utils.sampling import PanelUpgradeCostDistribution

# Residential costs
res_dist = PanelUpgradeCostDistribution(residential=True)
res_costs = res_dist.constrained_samples(n=1000, min_value=0, max_value=50000)

# Commercial costs
com_dist = PanelUpgradeCostDistribution(residential=False)
com_costs = com_dist.constrained_samples(n=1000, min_value=0, max_value=100000)

Working with Distributions

Interactive Notebook

The Custom Distributions notebook demonstrates:

  • Sampling from built-in distributions
  • Visualizing cost uncertainty
  • Analyzing cost variation by technology
  • Creating custom distributions
  • Assessing distribution impact on total costs

Open the Custom Distributions Notebook

Code Examples

Sampling from Distributions 

from utils.sampling import PanelUpgradeCostDistribution

# Create distribution
cost_dist = PanelUpgradeCostDistribution(residential=True)

# Sample 5000 costs
samples = cost_dist.constrained_samples(
    n=5000,
    min_value=1000,
    max_value=40000
)

# Analyze
print(f"Mean: ${samples.mean():,.0f}")
print(f"Median: ${np.median(samples):,.0f}")
print(f"95th percentile: ${np.percentile(samples, 95):,.0f}")

Understanding the Distribution 

import numpy as np
import plotly.graph_objects as go

# Sample from distribution
costs = cost_dist.constrained_samples(n=10000)

# Calculate statistics
stats = {
    'mean': np.mean(costs),
    'median': np.median(costs),
    'std': np.std(costs),
    'q5': np.percentile(costs, 5),
    'q95': np.percentile(costs, 95)
}

# Visualize
fig = go.Figure()
fig.add_trace(go.Histogram(x=costs, nbinsx=50))
fig.update_layout(
    title='Panel Upgrade Cost Distribution',
    xaxis_title='Cost ($)',
    yaxis_title='Frequency'
)
fig.show()

Creating Custom Distributions

If BICEP’s default distributions don’t match your region or situation, you can create custom distributions.

Approach 1: Empirical Distribution (from your data)

Use observed costs from your region or organization.

def create_empirical_distribution(observed_costs):
    """
    Create a distribution from observed cost data.
    
    Args:
        observed_costs: array of actual costs from your data
    
    Returns:
        Function that samples from the empirical distribution
    """
    sorted_costs = np.sort(observed_costs)
    
    def sample(n=1):
        # Random sampling with replacement from observed data
        return np.random.choice(sorted_costs, size=n, replace=True)
    
    return sample

# Example usage
my_costs = np.array([5000, 7500, 8000, 9000, 12000, 15000, ...])
my_distribution = create_empirical_distribution(my_costs)

# Sample from custom distribution
samples = my_distribution(n=1000)

Approach 2: Parametric Distribution (Log-Normal)

Use a statistical distribution fitted to your data.

from scipy import stats

def create_lognormal_distribution(observed_costs, min_val=0, max_val=np.inf):
    """
    Fit a log-normal distribution to cost data.
    
    Log-normal distributions are ideal for costs because:
    - Always positive (no negative costs)
    - Right-skewed (most common, some expensive outliers)
    - Common in engineering cost estimates
    
    Args:
        observed_costs: array of observed costs
        min_val, max_val: constraints on sampled values
    
    Returns:
        Function that samples from the distribution
    """
    # Fit log-normal parameters to data
    shape, loc, scale = stats.lognorm.fit(observed_costs)
    
    def sample(n=1):
        samples = stats.lognorm.rvs(shape, loc, scale, size=n)
        # Constrain to range
        samples = np.clip(samples, min_val, max_val)
        return samples
    
    return sample

# Example usage
my_distribution = create_lognormal_distribution(
    observed_costs=my_costs,
    min_val=1000,
    max_val=50000
)

samples = my_distribution(n=1000)

Approach 3: Regional Adjustment Factors

Adjust BICEP’s default distribution based on regional factors.

def create_regional_distribution(base_distribution, regional_factor):
    """
    Scale costs by regional factor.
    
    Example factors:
    - 0.8: Low cost region (rural, lower labor costs)
    - 1.0: National average (BICEP default)
    - 1.3: High cost region (urban, high labor costs)
    
    Args:
        base_distribution: BICEP's distribution function
        regional_factor: Multiplier for regional variation
    """
    def sample(n=1):
        base_samples = base_distribution.constrained_samples(n=n)
        return base_samples * regional_factor
    
    return sample

# Example: California costs are ~20% higher than national average
from utils.sampling import PanelUpgradeCostDistribution
base_dist = PanelUpgradeCostDistribution(residential=True)
ca_distribution = create_regional_distribution(base_dist, regional_factor=1.2)

ca_costs = ca_distribution(n=1000)

Impact on Results

Cost Estimate Uncertainty

Different distributions lead to different total cost estimates.

Example ranges:

  • Conservative (lower distribution): $X
  • Mid-point (BICEP default): $Y
  • High-end (upper distribution): $Z

Sensitivity Analysis

Test how distribution choice affects conclusions:

from bicep.upgrades import UpgradeEstimator

scenarios = {
    'conservative': {'distribution_multiplier': 0.8},
    'mid_point': {'distribution_multiplier': 1.0},
    'high': {'distribution_multiplier': 1.2}
}

for scenario_name, params in scenarios.items():
    estimator = UpgradeEstimator(scenario='high')
    
    # Apply custom distribution adjustment
    # (pseudocode - actual implementation depends on BICEP architecture)
    estimator.cost_multiplier = params['distribution_multiplier']
    estimator.calculate_costs()
    
    total = estimator.total_cost
    print(f"{scenario_name}: ${total:,.0f}")

Best Practices

When Choosing Distributions:

  1. Use empirical data when available - Real observed costs are more credible
  2. Validate assumptions - Check that distribution parameters match your data
  3. Document sources - Record where distribution parameters come from
  4. Perform sensitivity analysis - Test high and low scenarios
  5. Regional adjustments - Account for local cost differences

Questions to Ask:

  • What cost data do I have access to?
  • How applicable are BICEP’s national distributions to my region?
  • What’s the range of uncertainty I should plan for?
  • How do costs vary by building type?
  • Are there temporal variations I should account for?

Advanced Topics

Monte Carlo Simulation

Run uncertainty analysis with many sample iterations:

import numpy as np
from bicep.analysis import BicepResults

# Run multiple analyses with different random seeds
n_iterations = 100
total_costs = []

for i in range(n_iterations):
    np.random.seed(i)  # Different distribution sample each iteration
    results = BicepResults(scenario='high')
    total_costs.append(results.total_cost)

# Analyze uncertainty
print(f"Mean: ${np.mean(total_costs):,.0f}")
print(f"5th percentile: ${np.percentile(total_costs, 5):,.0f}")
print(f"95th percentile: ${np.percentile(total_costs, 95):,.0f}")

Distribution Comparison

Formally compare different distributions using statistical tests:

from scipy import stats

# Test if two distributions are significantly different
statistic, p_value = stats.ks_2samp(default_samples, custom_samples)

if p_value < 0.05:
    print("Distributions are significantly different")
else:
    print("Distributions are not significantly different")

Next Steps

Notice

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