Locational Marginal Prices (LMPs)

LMPs are the marginal cost of serving one additional MW of load at each bus. In optimization terms, they are the dual values (shadow prices) of the nodal power balance constraints.

LMP Extraction from CVXPY

To extract LMPs, you must:

1. Store references to the balance constraints
2. Solve the problem
3. Read the dual values after solving

import cvxpy as cp

# Store balance constraints separately for dual extraction
balance_constraints = []

for i in range(n_bus):
    pg_at_bus = sum(Pg[g] for g in range(n_gen) if gen_bus[g] == i)
    pd = buses[i, 2] / baseMVA

    # Create constraint and store reference
    balance_con = pg_at_bus - pd == B[i, :] @ theta
    balance_constraints.append(balance_con)
    constraints.append(balance_con)

# Solve
prob = cp.Problem(cp.Minimize(cost), constraints)
prob.solve(solver=cp.CLARABEL)

# Extract LMPs from duals
lmp_by_bus = []
for i in range(n_bus):
    bus_num = int(buses[i, 0])
    dual_val = balance_constraints[i].dual_value

    # Scale: constraint is in per-unit, multiply by baseMVA to get $/MWh
    lmp = float(dual_val) * baseMVA if dual_val is not None else 0.0
    lmp_by_bus.append({
        "bus": bus_num,
        "lmp_dollars_per_MWh": round(lmp, 2)
    })

LMP Sign Convention

For a balance constraint written as generation - load == net_export:

• Positive LMP: Increasing load at that bus increases total cost (typical case)
• Negative LMP: Increasing load at that bus decreases total cost

Negative LMPs commonly occur when:

• Cheap generation is trapped behind a congested line (can't export power)
• Adding load at that bus relieves congestion by consuming local excess generation
• The magnitude can be very large in heavily congested networks (thousands of $/MWh)

Negative LMPs are physically valid and expected in congested systems — they are not errors.

Reserve Clearing Price

The reserve MCP is the dual of the system reserve requirement constraint:

# Store reference to reserve constraint
reserve_con = cp.sum(Rg) >= reserve_requirement
constraints.append(reserve_con)

# After solving:
reserve_mcp = float(reserve_con.dual_value) if reserve_con.dual_value is not None else 0.0

The reserve MCP represents the marginal cost of providing one additional MW of reserve capacity system-wide.

Finding Binding Lines

Lines at or near thermal limits (≥99% loading) cause congestion and LMP separation. See the dc-power-flow skill for line flow calculation details.

BINDING_THRESHOLD = 99.0  # Percent loading

binding_lines = []
for k, br in enumerate(branches):
    f = bus_num_to_idx[int(br[0])]
    t = bus_num_to_idx[int(br[1])]
    x, rate = br[3], br[5]

    if x != 0 and rate > 0:
        b = 1.0 / x
        flow_MW = b * (theta.value[f] - theta.value[t]) * baseMVA
        loading_pct = abs(flow_MW) / rate * 100

        if loading_pct >= BINDING_THRESHOLD:
            binding_lines.append({
                "from": int(br[0]),
                "to": int(br[1]),
                "flow_MW": round(float(flow_MW), 2),
                "limit_MW": round(float(rate), 2)
            })

Counterfactual Analysis

To analyze the impact of relaxing a transmission constraint:

1. Solve base case — record costs, LMPs, and binding lines
2. Modify constraint — e.g., increase a line's thermal limit
3. Solve counterfactual — with the relaxed constraint
4. Compute impact — compare costs and LMPs

# Modify line limit (e.g., increase by 20%)
for k in range(n_branch):
    br_from, br_to = int(branches[k, 0]), int(branches[k, 1])
    if (br_from == target_from and br_to == target_to) or \
       (br_from == target_to and br_to == target_from):
        branches[k, 5] *= 1.20  # 20% increase
        break

# After solving both cases:
cost_reduction = base_cost - cf_cost  # Should be >= 0

# LMP changes per bus
for bus_num in base_lmp_map:
    delta = cf_lmp_map[bus_num] - base_lmp_map[bus_num]
    # Negative delta = price decreased (congestion relieved)

# Congestion relieved if line was binding in base but not in counterfactual
congestion_relieved = was_binding_in_base and not is_binding_in_cf

Economic Intuition

• Relaxing a binding constraint cannot increase cost (may decrease or stay same)
• Cost reduction quantifies the shadow price of the constraint
• LMP convergence after relieving congestion indicates reduced price separation