<｜begin▁of▁sentence｜>### Instruction:
Imagine you're working as a QA Analyst in a tech company that has developed an advanced software system for managing power grid networks. The system uses a hierarchical tree structure to represent the distribution network, with substations, transformers, and distribution lines as nodes and edges, respectively.

You've been tasked to evaluate a new algorithm designed to optimize the load distribution across the network. The network's current model is a tree with substations labeled {1, 2, 3, 4, 5}, each with an operating capacity represented by an integer weight of 1. The lines connecting them have impedance values--think of these as "resistance-to-flow" weights for current--specified as floating-point numbers. The impedance values for the lines are 0.1 between substations 1 and 2, 0.2 between 1 and 3, 0.3 between 3 and 4, and 0.4 between 3 and 5.

Your task is to conduct a test case scenario in which the algorithm attempts to partition the network into sub-networks, ensuring that the total operating capacity of the substations in each sub-network does not exceed a load limit of 3. Additionally, the objective is to verify that the algorithm minimizes the sum of the impedance values of the lines that are "cut" by the partition, thus ensuring a minimum disruption to the power flow and an optimal load distribution.

Your goal is to document the performance of the algorithm in creating these partitions and to report on whether it operates within the specified parameters. It's vital to ensure that the algorithm not only respects the load limits but also achieves this with the least possible impact on the power grid's efficiency.

Please note that in your test documentation, you should describe the conditions under which the algorithm operates rather than providing a solution or methodology for the partitioning. Consider this scenario as a practical application of the abstract problem presented earlier.
### Response:
import networkx as nx
from networkx.algorithms.tree import lukes_partitioning

# Create a weighted tree G
G = nx.Graph()
G.add_nodes_from([1, 2, 3, 4, 5], weight=1)
G.add_edges_from([(1, 2, {'weight': 0.1}), (1, 3, {'weight': 0.2}),
                  (3, 4, {'weight': 0.3}), (3, 5, {'weight': 0.4})])

# Define max size of the partition
max_size = 3

# Perform Lukes partitioning
partition = lukes_partitioning(G, max_size, edge_weight='weight')

print(partition)
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