locational-marginal-prices

---
name: locational-marginal-prices
description: "Extract locational marginal prices (LMPs) from DC-OPF solutions using dual values. Use when computing nodal electricity prices, reserve clearing prices, or performing price impact analysis."
---

# 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

```python
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:

```python
# 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.

```python
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

```python
# 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