Structure-Preserving Model Order Reduction for Nonlinear DAE Models of Power Networks

Abstract

This paper deals with the joint reduction of the number of dynamic and algebraic states of a nonlinear differential-algebraic equation (NDAE) model of a power network. The dynamic states depict the internal states of generators, loads, renewables, whereas the algebraic ones define network states such as voltages and phase angles. In the current literature of power system model order reduction (MOR), the algebraic constraints are usually neglected and the power network is commonly modeled via a set of ordinary differential equations (ODEs) instead of NDAEs. Thus, reduction is usually carried out for the dynamic states only and the algebraic variables are kept intact. This leaves a significant part of the system's size and complexity unreduced. This paper addresses this aforementioned limitation by jointly reducing both dynamic and algebraic variables. As compared to the literature the proposed MOR techniques are endowed with the following features: (i) no system linearization is required, (ii) require no transformation to an equivalent or approximate ODE representation, (iii) guarantee that the reduced order model to be NDAE-structured and thus preserves the differential-algebraic structure of original power system model, and (iv) can seamlessly reduce both dynamic and algebraic variables while maintaining high accuracy. Case studies performed on a 2000-bus power system reveal that the proposed MOR techniques are able to reduce system order while maintaining accuracy.

Figures (6)

• Figure 1: HSVs and their cumulative sum contained in $\boldsymbol\Sigma_d$ (above) while below is for $\boldsymbol\Sigma_a$; 39-bus system. The first $r_d= 7$ HSVs in $\boldsymbol\Sigma_d$ contain $99\%$ of the cumulative sum, similarly for $\boldsymbol\Sigma_a$ the first $r_a= 3$ HSVs contain $97\%$ of the cumulative sum. Thus, the size of ROM is selected to be $r = 10$.
• Figure 2: Comparison of FOM and ROM for 39-bus system; rotor angle of Gen. at Bus $35$ (top-left), frequency of Gen. at Bus 35 (top-right), relative angle of solar plant (bottom left), and overall error norm (bottom right).
• Figure 3: Comparison of algebraic variables between FOM and ROM for the 39-bus system; Bus 5 real and imaginary current (above), and Bus 5 real and imaginary voltage (below).
• Figure 4: Comparison of FOM and ROM for 39-bus system under fault; solar power plant voltage (top-left), relative angle (top-right), reactive power output (bottom left), and rotor angle of Gen. at Bus 30 (bottom right).
• Figure 5: HSVs and their cumulative sum contained in $\boldsymbol\Sigma_d$ (above) while below is for $\boldsymbol\Sigma_a$, 2000-bus Texas system. The first $r_d= 10$ HSVs in $\boldsymbol\Sigma_d$ contain $99.99\%$ of the cumulative sum, similarly for $\boldsymbol\Sigma_a$ the first $r_a= 30$ HSVs contain $99.99\%$ of the cumulative sum. Thus, the size of ROM is selected to be $r = 40$.