Locational Marginal Prices

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 in a DC-OPF (Direct Current Optimal Power Flow) formulation.

When to Use

• When computing nodal electricity prices for a power system solved via DC-OPF.
• When calculating reserve clearing prices (MCP) from a system-wide reserve requirement constraint.
• When performing price-impact or counterfactual analysis to see how transmission upgrades affect costs and congestion.
• When identifying congestion points (binding lines) in a DC-OPF solution.

Do Not Use

• AC-OPF: Do not use for AC-OPF calculations where voltage magnitude and reactive power (Q) are critical; LMPs will differ.
• Non-optimal solves: Do not use if the optimization problem is not solved to optimality; dual values may be unstable or incorrect.
• Deprecated solvers: Avoid legacy solvers such as ECOS for large-scale DC-OPF; they may produce inaccurate duals. Prefer CLARABEL (v0.7+) or OSQP (v0.6+) with dual extraction support.
• Insecure data handling: Never expose raw dual values or model data through unauthenticated APIs; they can reveal system operating points. Apply proper access controls and sanitise outputs.

Prerequisites

• Python 3.9+ with cvxpy>=1.4.0, numpy.
• A solver that returns reliable duals: CLARABEL v0.7+ (recommended default) or OSQP v0.6+.
• A solved or solvable DC-OPF problem with nodal power-balance constraints stored as references.
• If using OSQP, enable eps_dual_inf and verify dual feasibility explicitly.

Procedure

Step 1 — Store references to balance constraints

You must keep a handle to each nodal balance constraint so duals can be read after solving.

import cvxpy as cp
import numpy as np

# --- Problem data (example placeholders) ---
n_bus, n_gen, n_branch = 5, 8, 7
baseMVA = 100.0
B = np.random.randn(n_bus, n_bus)          # susceptance matrix
theta = cp.Variable(n_bus)                 # voltage angles
Pg = cp.Variable(n_gen)                    # generator outputs
Rg = cp.Variable(n_gen)                    # reserve provision
gen_bus = np.random.randint(0, n_bus, n_gen)
buses = np.random.rand(n_bus, 3)           # [bus_id, ?, load_MW]
branches = np.random.rand(n_branch, 6)     # [from, to, ?, x, ?, rate_MW]

# --- Build constraints ---
constraints = []
balance_constraints = []   # keep a handle for dual extraction

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

    bal = pg_at_bus - pd == B[i, :] @ theta
    balance_constraints.append(bal)
    constraints.append(bal)

# Example system-wide reserve requirement
reserve_requirement = 50.0
reserve_con = cp.sum(Rg) >= reserve_requirement
constraints.append(reserve_con)

# Objective (simple quadratic cost placeholder)
cost = cp.sum_squares(Pg) + 0.01 * cp.sum_squares(theta)

Step 2 — Solve with a dual-enabled solver

# Solve with Clarabel (default dual-enabled solver)
prob = cp.Problem(cp.Minimize(cost), constraints)
prob.solve(solver=cp.CLARABEL, verbose=False)   # set verbose=True for debugging

Note: CLARABEL (v0.7+) provides high-precision duals and is the recommended default for DC-OPF. If you must use OSQP, enable eps_dual_inf and verify dual feasibility.

Step 3 — Extract LMPs from dual values

Read dual values only after prob.solve() completes without errors. Convert from per-unit to $/MWh by multiplying by baseMVA.

# --- Extract LMPs ---
lmp_by_bus = []
for i, bal in enumerate(balance_constraints):
    bus_num = int(buses[i, 0])
    dual_val = bal.dual_value

    # Convert from per-unit to $/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)
    })

Step 4 — Extract reserve clearing price (MCP)

The reserve MCP is the dual of the system-wide reserve requirement constraint.

# After solving (see above)
reserve_mcp = float(reserve_con.dual_value) if reserve_con.dual_value is not None else 0.0
print(f"Reserve MCP: ${reserve_mcp:.2f}/MWh")

Step 5 — Identify binding lines

Lines operating at or above a configurable loading threshold are considered binding.

BINDING_THRESHOLD = 99.0  # percent loading

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

    if x == 0 or rate <= 0:
        continue

    b = 1.0 / x
    flow_MW = b * (theta.value[f_idx] - theta.value[t_idx]) * 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),
            "loading_pct": round(loading_pct, 1)
        })

Step 6 — Counterfactual analysis (transmission upgrade)

Assess the impact of relaxing a transmission constraint by increasing a line's thermal limit and re-solving.

def relax_line(branches, target_from, target_to, factor=1.20):
    """Increase thermal limit of a specific line by `factor`."""
    for k in range(branches.shape[0]):
        f, t = int(branches[k, 0]), int(branches[k, 1])
        if (f == target_from and t == target_to) or (f == target_to and t == target_from):
            branches[k, 5] *= factor
            break
    return branches

# 1. Base case solved above -> store results
base_cost = prob.value
base_lmp_map = {item["bus"]: item["lmp_dollars_per_MWh"] for item in lmp_by_bus}
base_binding = binding_lines.copy()

# 2. Modify constraint
branches_cf = relax_line(branches.copy(), target_from=2, target_to=5, factor=1.20)

# 3. Re-solve with modified data
cf_constraints = []
cf_balance_constraints = []

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

    bal = pg_at_bus - pd == B[i, :] @ theta
    cf_balance_constraints.append(bal)
    cf_constraints.append(bal)

cf_constraints.append(cp.sum(Rg) >= reserve_requirement)

# Re-solve
cf_prob = cp.Problem(cp.Minimize(cost), cf_constraints)
cf_prob.solve(solver=cp.CLARABEL, verbose=False)

# Extract counterfactual LMPs
cf_lmp_by_bus = []
for i, bal in enumerate(cf_balance_constraints):
    bus_num = int(buses[i, 0])
    dual_val = bal.dual_value
    lmp = float(dual_val) * baseMVA if dual_val is not None else 0.0
    cf_lmp_by_bus.append({
        "bus": bus_num,
        "lmp_dollars_per_MWh": round(lmp, 2)
    })

# 4. Compare
cost_reduction = base_cost - cf_prob.value
cf_lmp_map = {item["bus"]: item["lmp_dollars_per_MWh"] for item in cf_lmp_by_bus}
lmp_deltas = {bus: cf_lmp_map[bus] - base_lmp_map[bus] for bus in base_lmp_map}

# Find new binding lines for counterfactual
cf_binding_lines = []
for k, br in enumerate(branches_cf):
    f_idx = int(br[0])
    t_idx = int(br[1])
    x, rate = br[3], br[5]
    if x == 0 or rate <= 0:
        continue
    b = 1.0 / x
    flow_MW = b * (theta.value[f_idx] - theta.value[t_idx]) * baseMVA
    loading_pct = abs(flow_MW) / rate * 100
    if loading_pct >= BINDING_THRESHOLD:
        cf_binding_lines.append({
            "from": int(br[0]),
            "to": int(br[1]),
            "flow_MW": round(float(flow_MW), 2),
            "limit_MW": round(float(rate), 2),
            "loading_pct": round(loading_pct, 1)
        })

target_from = 2
target_to = 5
congestion_relieved = any(
    b["from"] == target_from and b["to"] == target_to for b in base_binding
) and not any(
    b["from"] == target_from and b["to"] == target_to for b in cf_binding_lines
)

print(f"Cost reduction: ${cost_reduction:.2f}")
print(f"LMP changes: {lmp_deltas}")
print(f"Congestion relieved: {congestion_relieved}")

Economic Intuition

• Relaxing a binding constraint cannot increase total cost; it either reduces cost or leaves it unchanged.
• The shadow price of the original constraint equals the cost reduction per MW of capacity added.
• Convergence of LMPs after relief indicates diminished congestion and a more uniform price field.

LMP Sign Convention

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

• Positive LMP: Increasing load at that bus raises total system cost (typical).
• Negative LMP: Increasing load decreases total cost, often because cheap generation is trapped behind a congested line.

Negative LMPs are physically valid in congested networks and indicate that additional demand can relieve congestion.

Pitfalls

• Accessing duals before solve completes: dual_value will be None if read before prob.solve() finishes. Always check prob.status first.
• Forgetting baseMVA conversion: Duals are in per-unit; without multiplying by baseMVA, LMPs will be off by a factor of 100 (or whatever baseMVA is).
• Using ECOS for large DC-OPF: Legacy solvers may produce inaccurate or unstable duals. Use CLARABEL v0.7+ or OSQP v0.6+.
• Misinterpreting negative LMPs: Negative LMPs are physically valid in congested networks; do not treat them as errors.
• Counterfactual cost increase: Relaxing a binding constraint must never increase total cost. If cost_reduction < 0, the solve likely failed or the wrong constraint was relaxed.
• Exposing raw dual values via API: Dual values reveal system operating points. Sanitise outputs and apply access controls before returning LMP data to any client.
• Sign mismatch with constraint formulation: If the balance constraint is written as load - generation == net_import, the LMP sign flips. Ensure the sign convention matches your formulation.

Verification

• Verify that dual_value is accessed only after prob.solve() completes without errors.
• Confirm that baseMVA is applied to the dual value to convert per-unit prices to $/MWh.
• Check that the sign of each LMP matches the constraint formulation (generation - load).
• Validate that cost reduction is non-negative when relaxing a binding constraint.
• Ensure the selected solver reports dual_feasibility status (e.g., prob.solver_stats.dual_feasibility).
• Run a regression test: compare LMPs against a known benchmark case (e.g., IEEE 14-bus) with tolerance ±1 $/MWh.
• Perform a security audit: confirm no raw model data is logged or returned in API responses.

Related Skills

• dc-power-flow: Used for calculating line flows and theta values.
• optimal-power-flow: The overarching optimization framework for LMPs.