Bus Types: Slack, PV, and PQ

Every bus in a power flow study is classified as one of three types: Slack, PV, or PQ. This classification determines which quantities you specify as inputs and which the solver computes.

Understanding bus types is essential for setting up power flow cases correctly and interpreting results.

The Core Idea

At each bus, there are four electrical quantities:

• $P$ — real power injection (MW)
• $Q$ — reactive power injection (MVAR)
• $|V|$ — voltage magnitude (p.u. or kV)
• $\theta$ — voltage angle (degrees or radians)

Power flow is a system of $2n$ equations (for $n$ buses), so we need $2n$ unknowns. At each bus, we must specify two quantities and solve for the other two.

PQ Bus: The Load Bus

Most buses are PQ buses. These represent:

• Load points (substations, industrial customers)
• Buses with no generation or voltage control

What's Specified

You tell the solver:

• $P$ — real power consumed (negative injection)
• $Q$ — reactive power consumed

What's Solved

The solver finds:

• $|V|$ — resulting voltage magnitude
• $\theta$ — resulting voltage angle

Physical Interpretation

PQ buses model constant power loads — loads that draw a fixed amount of P and Q regardless of voltage (within reason). This is realistic for most loads over normal voltage ranges.

Example

A substation with 50 MW load and 20 MVAR reactive consumption:

• $P = -50$ MW (negative = consuming)
• $Q = -20$ MVAR
• $|V|$ and $\theta$ = solved by power flow

PV Bus: The Generator Bus

Generator buses with voltage control are PV buses. These represent:

• Power plants with automatic voltage regulators (AVRs)
• Synchronous condensers
• STATCOMs and other voltage-controlling devices

What's Specified

You tell the solver:

• $P$ — real power output (dispatch setpoint)
• $|V|$ — voltage magnitude setpoint (what the AVR maintains)

What's Solved

The solver finds:

• $Q$ — reactive power needed to maintain voltage
• $\theta$ — voltage angle

Physical Interpretation

Generators have governors that control $P$ (real power output) and exciters/AVRs that control $|V|$ (terminal voltage). The reactive power $Q$ adjusts automatically to maintain the voltage setpoint.

Key insight: At a PV bus, reactive power is a result, not an input. The generator produces whatever $Q$ is needed to hold voltage at the setpoint.

Reactive Limits

Real generators have reactive capability limits:

$$Q_{\min} \leq Q \leq Q_{\max}$$

If the solved $Q$ exceeds these limits, the bus converts to PQ at the violated limit:

• If $Q > Q_{\max}$: Fix $Q = Q_{\max}$, let $|V|$ drop
• If $Q < Q_{\min}$: Fix $Q = Q_{\min}$, let $|V|$ rise

This is called PV-PQ switching and is handled automatically by power flow solvers.

Example

A 200 MW generator with voltage setpoint 1.02 p.u. and reactive limits ±100 MVAR:

• $P = 200$ MW
• $|V| = 1.02$ p.u.
• $Q$ = solved (say, 45 MVAR needed to maintain voltage)
• If $Q$ would need to be 150 MVAR, the bus switches to PQ with $Q = 100$ MVAR

Slack Bus: The Reference

Every power flow needs exactly one slack bus (per synchronous island). This special bus:

• Sets the angle reference ($\theta = 0$)
• Absorbs power mismatch (losses + any imbalance)

What's Specified

• $|V|$ — voltage magnitude (like a PV bus)
• $\theta$ — voltage angle (typically set to 0°)

What's Solved

• $P$ — real power injection (whatever's needed to balance the system)
• $Q$ — reactive power injection

Why Is It Necessary?

Two fundamental reasons:

1. Angle Reference

Voltage angles are only meaningful relative to a reference. Without fixing one angle, the solution is not unique (all angles could shift together).

2. Loss Absorption

The total generation must equal total load plus losses. But we don't know losses until we solve the power flow! The slack bus provides the "swing" generation to cover this unknown quantity.

$$P_{\text{slack}} = \sum P_{\text{load}} + P_{\text{losses}} - \sum P_{\text{other gen}}$$

Choosing the Slack Bus

Typically the slack bus is:

• The largest generator (provides most "swing" capacity)
• The system's major interconnection point
• A bus near the electrical center of the network

Practical tip: Slack bus choice affects the solution distribution of losses but not the physics. If the slack $P$ comes out unreasonably large, your input data may be unbalanced.

Example

A large coal plant serving as slack bus:

• $|V| = 1.0$ p.u.
• $\theta = 0°$ (by definition)
• $P$ = solved (e.g., 450 MW to balance the system)
• $Q$ = solved (e.g., 120 MVAR)

Summary Comparison

The Mathematical Picture

Power flow solves $2n$ nonlinear equations. For an $n$-bus system with:

• 1 slack bus
• $n_G$ PV buses
• $n - 1 - n_G$ PQ buses

We have:

• $(n-1)$ unknown angles (slack angle fixed)
• $(n - 1 - n_G)$ unknown voltage magnitudes (slack and PV magnitudes fixed)

Total unknowns: $(n-1) + (n - 1 - n_G) = 2n - 2 - n_G$

The power balance equations provide:

• $(n-1)$ real power equations (slack P not constrained)
• $(n - 1 - n_G)$ reactive power equations (slack and PV bus Q not constrained)

The system is square: same number of equations and unknowns. ✓

GAT Bus Types

In GAT's Arrow format, bus type is stored in the type column of buses.arrow:

When you run gat pf, GAT:

1. Reads bus types from the network
2. Sets up equations based on types
3. Handles PV→PQ switching if generators hit limits
4. Reports slack bus P and Q in the solution

Common Mistakes

Forgetting the Slack Bus

Power flow will fail with "no slack bus" or produce nonsense. Every synchronous island needs exactly one.

Multiple Slack Buses

Having two slack buses over-constrains the problem. Only one bus can set the angle reference and absorb mismatch.

Ignoring Reactive Limits

If a PV bus solution shows $Q$ outside limits, the voltage setpoint cannot be maintained. Enable PV-PQ switching or adjust limits.

Wrong Sign Convention

Remember: positive injection means generation, negative means consumption.

• Load bus: $P < 0$, $Q < 0$ typically
• Generator bus: $P > 0$, $Q$ = whatever's needed

Key Takeaways

1. PQ buses specify $P$ and $Q$; solve for $|V|$ and $\theta$
2. PV buses specify $P$ and $|V|$; solve for $Q$ and $\theta$
3. Slack bus specifies $|V|$ and $\theta$; solves for $P$ and $Q$
4. Every synchronous island needs exactly one slack bus
5. PV buses can convert to PQ when reactive limits bind

See Also

• Power Flow Theory — The equations solved for each bus type
• Newton-Raphson Method — How the solver handles bus types
• Y-Bus Matrix — Network representation used in power flow
• Glossary — Quick definitions