The Measurement Model: z = h(x) + e
- Part 1 — What Is State Estimation? SCADA, EMS and the State Vector
- Part 2 — The Measurement Model: z = h(x) + e ← you are here
- Part 3 — Weighted Least Squares (WLS) State Estimation
- Part 4 — Building the Jacobian Matrix
- Part 5 — Observability Analysis and Measurement Placement
- Part 6 — Bad Data Detection and Identification
- Part 7 — Topology Error Processing
- Part 8 — Linear and DC State Estimation
- Part 9 — PMU-Based and Hybrid State Estimation
- Part 10 — PowerFactory + Python Implementation (IEEE 14-Bus)
Part 1 introduced why state estimation exists and what the state vector represents. This article derives the mathematical model that connects the state vector to the physical measurements a SCADA system collects. This model — z = h(x) + e — is the foundation on which every SE algorithm, bad-data test, and observability analysis is built.
1. The Vector Measurement Model
Each measurement zk (k = 1, …, m) is related to the true system state x by a known nonlinear function hk(x), corrupted by additive noise ek:
Stacking all m measurements into vectors gives the compact form:
where z ∈ ℜm is the measurement vector, x ∈ ℜ2n−1 is the state vector, h: ℜ2n−1 → ℜm is the vector of measurement functions encoding the network physics, and e ∈ ℜm is the error vector.
The function h(x) answers: if the true state were x, what would each meter read in the absence of noise? SE finds the x that makes h(x) match the actual readings z as closely as possible in a weighted sense.
2. The State Vector
For an n-bus system with bus 1 as reference (θ1 = 0):
For the IEEE 14-bus system (n = 14): 2(14)−1 = 27 state variables — 13 voltage angles and 14 voltage magnitudes. For a 500-bus transmission system the state vector has 999 elements.
3. Measurement Functions h(x)
The explicit form of hk(x) depends on which physical quantity is being measured. All functions derive from the standard AC power flow equations using the Ybus admittance matrix.
3.1 Active Power Flow — Line i→j
where θij = θi − θj, and gij + jbij is the series admittance of line i–j.
3.2 Reactive Power Flow — Line i→j
where bsh/2 is the half-line charging susceptance of the π-equivalent circuit.
3.3 Active Power Injection — Bus i
where Gij + jBij = Yij is the (i,j) element of Ybus, and Ni is the set of all buses connected to bus i (including i itself via the diagonal element Yii).
3.4 Reactive Power Injection — Bus i
3.5 Voltage Magnitude — Bus i
The simplest measurement function. Its row in the Jacobian H has exactly one non-zero entry (= 1) in the column for |Vi|.
3.6 PMU Voltage Phasor — Rectangular Form
A PMU at bus i provides voltage magnitude and angle. In rectangular coordinates (ei = real part, fi = imaginary part):
These are nearly linear in x, which is why PMU-only SE can be formulated as a purely linear problem (Part 9).
4. Measurement Noise Assumptions
The statistical properties of e determine which estimator is optimal. Standard SE assumes:
- Zero mean: E[e] = 0 — meters are unbiased
- Gaussian: ek ~ N(0, σk2)
- Uncorrelated: E[ekej] = 0 for k ≠ j
Under these assumptions the noise covariance matrix R is diagonal:
The weight matrix is W = R−1 = diag(1/σ12, …, 1/σm2). Measurements with lower noise receive higher weight. A PMU with σ = 0.001 receives ~100× higher weight than a SCADA meter with σ = 0.01.
Bad data (faulty meters, communication errors) produces non-Gaussian, non-zero-mean errors — violations of assumption 1. The chi-squared bad data test in Part 6 detects these violations by testing whether the weighted residuals at the SE solution are statistically consistent with the noise model.
5. Worked Example: 3-Bus System
Consider a 3-bus system (n = 3) with bus 1 as reference. State vector:
Suppose 7 measurements are available (redundancy ρ = 7/5 = 1.4):
| Index k | Measurement zk | Type | σk |
|---|---|---|---|
| 1 | P12 = 0.888 pu | Active power flow, bus 1→2 | 0.010 |
| 2 | P13 = 0.568 pu | Active power flow, bus 1→3 | 0.010 |
| 3 | P2 = −0.501 pu | Active power injection, bus 2 | 0.010 |
| 4 | Q12 = 0.201 pu | Reactive power flow, bus 1→2 | 0.010 |
| 5 | Q2 = −0.101 pu | Reactive power injection, bus 2 | 0.010 |
| 6 | |V1| = 1.006 pu | Voltage magnitude, bus 1 | 0.004 |
| 7 | |V2| = 0.968 pu | Voltage magnitude, bus 2 | 0.004 |
Equations (4)–(8) give h1(x) through h7(x) using the network Ybus data. The WLS algorithm (Part 3) minimises:
6. Redundancy and Its Value
Having m > 2n−1 means the system of equations z = h(x) is overdetermined — no exact solution exists because of noise. The least-squares criterion selects the x that comes closest in a weighted sense. The key benefits of redundancy:
- Noise averaging: random meter errors cancel across redundant measurements
- Bad data detection: a faulty meter creates large inconsistent residuals visible in the chi-squared test (Part 6)
- Resilience: losing one measurement does not necessarily make the network unobservable
References
- A. Abur and A. G. Exposito, Power System State Estimation: Theory and Implementation, Marcel Dekker, 2004 — Chapters 2–3.
- A. Monticelli, State Estimation in Electric Power Systems, Kluwer Academic, 1999 — Chapter 3.
- A. J. Wood, B. F. Wollenberg, and G. B. Sheblé, Power Generation, Operation, and Control, 3rd ed., Wiley, 2014 — Chapter 9.
- F. C. Schweppe et al., “Power system static-state estimation,” IEEE Trans. Power Apparatus and Systems, vol. PAS-89, 1970.