State Estimation in Power Distribution Network Operation Ravindra Singh Dept. of Electrical & Electronic Engg. Imperial College London 1
Outline: • Introduction • Research Objectives – Identification of DSSE solver – Modelling of pseudo measurements – Measurement placement • Conclusion and future Work 2
Introduction Distribution networks of today: – Old design and ageing equipments – Passive operation – Based on “Fit and Forget” concept This concept can not work considering – The deregulation of power system – Emphasis towards low carbon technology – Introduction of DGs at distribution level 3
Distribution Management System (DMS) • Introduction of DGs at distribution level requires monitoring and control of distribution networks through Distribution Management System (DMS) • DMS has two functional block • State Estimation • Control Scheduling • SE will be the core function of the DMS which will provide the input to control scheduling block for various control functionality 4
Challenges in DSSE • State estimation is routine task in transmission systems • Several SE techniques exist • But these techniques can not be duplicated in distribution networks mainly due to – lack of available measurements – lack of methodologies and tools that can be applied to restricted measurements 6
Objectives 1. Development of DSSE Algorithm • Choice of DSSE Solver 2. Modelling of Pseudo Measurements • To ensure network observability • To handle non Gaussian load distribution 3. Measurement placement • Location • Type • Number 4. SE with network topology changes 7
Objective 1: Choice of DSSE Estimator Review of state estimation in transmission system • State estimation is a data fitting problem: ∈ ℜ n x – State Vector: z i ∈ ℜ , i = 1, K , m – Measurements: z i = h i ( x ) + e i , i = 1, K , m – Model: m > n , e i = measurement error (Gaussian) • The solution of the state estimation problem requires x finding that minimises a (weighted) norm of the residuals: = − = ( ) ( ), 1 , , r x z h x i m i i i 8
L 2 and L 1 norms • L1 norm: • L2 norm: (Weighted) Least (Weighted) Least Absolute Value Squares LS LAV approach approach. – More robust than LS in the – Performs well presence of bad in the presence data of noise – Ineffective for – Fails to reject noise filtering bad data m ∑ 12 r i ( x ) m min w i 1 ∑ where w i = 1 w i r i ( x ) 2 min x σ i i = 1 2 2 x i = 1 9
Compromise: Huber M-estimator • Huber function 2 t 2 , if t ≤ γ 1 ρ ( t ) = γ t − 1 2 γ 2 , if t > γ • State estimation problem γ → ∞ ⇒ m ∑ LS estimator ρ min ( ( )) r x i x = γ → 0 ⇒ 1 i LAV estimator 10
Estimators Summary ϒ= 1.5 H = ∂ h ( x ) and R z is measurement error covarince matrix ∂ x 11
Simulation study • Various estimators like WLS , WLAV, and SHGM were tested on the 95 bus UKGDS model (Fig) with • errors in real measurement 1%-3% • errors in pseudo measurements 20%-50% • DSSE relies on a large number of pseudo measurements which are statistical in nature • Thus the statistical criterion were adopted to identify the suitable solver for the DSSE problem 12
Statistical Criteria • Bias: For the estimator to be unbiased, the mean error in the estimates should be zero x − x t ) = 0 E ( ˆ • Consistency : For consistency the normalized mean square estimation error should be within confidence bound determined by chi-square test (under Gaussian assumption of error) x − x t ) T P − 1 ( ˆ ε = ( ˆ x − x t ) • Quality : Expressed as the inverse of the trace of the error covariance matrix 1 Q = log( trace ( P )) 13
Performance Summary Quality Bias Consistency 20 10 8,86 6,7 0 WLS WLAV SHGM -10 -20 -55,65 -30 -40 -50 -60 WLS is the suitable solver for the DSSE problem 14
Objective 2: Modelling of Pseudo Measurements Loads of various customer class Load profile computations 1.5 1.5 Domestic Unrestricted Domestic Economy 1 1 Load,pu Load,pu 0.5 0.5 0 0 0 5000 10000 15000 0 5000 10000 15000 time time 1.5 1.5 Commercial Industrial 1 1 Load,pu Load,pu 0.5 0.5 0 0 0 5000 10000 15000 0 5000 10000 15000 time time 15
Issues with the WLS WLS algorithm assumes that the measurements are normally distributed, however: • Load pdfs do not follow normal distribution • One way could be to model them as lognormal or beta distribution • However, this is not true (pdf shown in Fig.) • This shows that the load pdfs do not follow any particular known distribution 16
How to Model the Variability in Load pdfs? Concept of Gaussian Mixture Model (GMM) • Load pdfs are represented through a mixture of Gaussian components • Resulting pdf is a weighted combination of individual Gaussian components. • Now WLS can be applied • Use of Expectation Maximization (EM) algorithm to obtain the parameters of the mixture components 17
Density Estimation Through EM Algorithm • Mixture Model M c M c ∑ ∑ γ = µ Σ = ( | ) ( | , ) a n d 1 f z w f z w i i i i = = 1 1 i i { } γ Γ = γ γ = µ Σ M c w h e r e , i s c h o s e n f r o m t h e s e t : { , , } w = 1 i i i i 1 − 1 − − µ Σ − µ 1 ( ) ( ) z z µ Σ = i i i ( | , ) 2 f z e π Σ i i / 2 1 / 2 d ( 2 ) d e t ( ) i = N o f o . m i x t u r e c o m p o n e n t s M c 18
The Algorithm The EM Algorithm works recursively to obtain the parameters of the mixture γ + γ 1 s s g i v i components. One step of recursion, yielding formulae for : Result at Bus #82 19
Comparison with other distributions •Comparison is based on both graphical and the numerical measure •For numerical measure Chi-Square goodness of fit value is used • A smaller value of goodness fit indicates a better fit •GMM gives better fit according to both measures 20
SE Results: Bus #2 • SE was performed with 1 voltage and 2 flow measurements at main substation • Loads were modelled as GMM using load profiles obtained by mapping the behaviour of various customer class at each bus • SE was run at different time steps by sampling the load profiles • Estimation errors are small close to the main substation buses (Bus #2) Bus #51 • Errors increase at the buses away from main substation (Bus #51) • This is because the buses away from main substation are predominantly influenced by pseudo measurements • Error in angle is more as compared to voltage 21
Objective 3: Measurement Placement SE Example: • 3% error in real, 50% error in pseudo • UKGDS Network • 100 Simulations • 1% voltage threshold 5% angle threshold •In this example a significant number of cases are above their respective thresholds •The reason is that the SE utilizes a large number of pseudo measurements •This problem can be eliminated by placing more number of real measurements 22
Objective To bring down the relative errors in Voltage and angle below threshold in more than 95% simulation cases. Benefit Reduction in error in estimates. Issues: To identify • Measurement location • Measurement type • Number of measurements 23
Problem Formulation Improve: (1) In order to improve above index reduce: (2) Where, 24
Geometric Interpretation of Error Covariance Matrix n-dimensional error ellipse e = error vector (n × 1) P= error covariance matrix (n × n) c=constant Using transformation , can be diagonalized and the is given by: 25
• Index in (1) can be improved by reducing the right hand side of (2) • This can be achieved by reducing the determinant of estimation error covariance matrix. • Since determinant represents the area of the error ellipse, the said probability index can be improved through reduction in area of the V- δ error ellipse. • Placement of voltage meter reduces the errors in both voltage and angle if orientation of ellipse is according to Fig. (a) (Correlated) • However if the errors are aligned with the coordinate axis (Fig. (b)). The placement of voltage measurement does not improve errors in angle (Uncorrelated) • In this situation we used flow measurements and based on the criterion of minimization of the area of P-Q_flow error ellipse 26
The Algorithm Flow Chart 27
Results Measurements Placed: V 19 ,V 20 , V 21 P 15-17 ,Q 15-17 P 34-35 ,Q 34-35 Red ellipse: Before No Placement Blue ellipse: After All Meters Placed 0.015 0.015 0.015 Bus #21 Bus #20 Bus #19 0.01 0.01 0.01 Error in Voltage 0.005 0.005 0.005 0 0 0 -0.005 -0.005 -0.005 -0.01 -0.01 -0.01 -0.015 -0.015 -0.015 -2 0 2 -2 0 2 -2 0 2 Error in Angle -3 -3 -3 x 10 x 10 x 10 Error ellipses Quality 28
Recommend
More recommend
