Online dynamic security assessment of microgrids before intentional islanding occurrence

Abstract

This paper presents a statistical learning-based method for security assessment of microgrids (MGs) in case of isolation from the main grid. Based on the stability criteria, the MG pre-islanding conditions are divided into secure and insecure regions. Critical system variables regarding the MG dynamic security are first selected via a feature selection procedure, known as minimum redundancy maximum relevance. An unsupervised learning method called pattern discovery method is then performed on the space of the critical features to extract the organization (patterns) among samples. Geometrically, the patterns are hyper-rectangles in the features space representing the system dynamic secure/insecure regions and can be effectively used for online MG security monitoring before islanding condition. Simulation results are carried out in the time domain, by using MATLAB, which demonstrate the effectiveness and accuracy of the proposed method in the MG security assessment.

Appendix: VSI, loads and network equations

1.1 VSI modeling

Figure 3 shows the block diagram of a VSI-based DG model and its controllers. This model consists of voltage controller, current controller, output filter and coupling inductance.

The following equations illustrate the relationship between the input and output of each block of a VSI-based DG connected to the network.

• (a) Voltage controller: The DAEs of voltage controller are as follows [10]:

  $$ \dot{\phi }_{d} = v_{{{\text{od}},{\text{ref}}}} - v_{\text{od}} $$

  (29)
  $$ \dot{\phi }_{q} = v_{{{\text{oq}},{\text{ref}}}} - v_{\text{oq}} $$

  (30)
  $$ i_{\text{ld}}^{*} = F \cdot i_{\text{od}} - \omega_{n} \cdot C_{f} \cdot v_{\text{oq}} + k_{\text{pv}} (v_{\text{od}}^{*} - v_{\text{od}} ) + k_{\text{iv}} \cdot \phi_{d} $$

  (31)
  $$ i_{\text{iq}}^{*} = F \cdot i_{\text{oq}} - \omega_{n} \cdot C_{f} \cdot v_{\text{od}} + k_{\text{pv}} (v_{\text{oq}}^{*} - v_{\text{oq}} ) + k_{\text{iv}} \cdot \phi_{q} $$

  (32)

• (b) Current controller: The DAEs of current controller are as follows [10]:

  $$ \dot{\gamma }_{d} = i_{\text{id}}^{*} - i_{\text{id}} $$

  (33)
  $$ \dot{\gamma }_{q} = i_{\text{iq}}^{*} - i_{\text{iq}} $$

  (34)
  $$ v_{\text{id}}^{*} = - \omega_{n} \cdot L_{f} \cdot i_{\text{lq}} + k_{\text{pc}} (i_{\text{lq}}^{*} - i_{\text{lq}} ) + k_{\text{ic}} \cdot \gamma_{d} $$

  (35)
  $$ v_{\text{iq}}^{*} = \omega_{n} \cdot L_{f} \cdot i_{\text{ld}} + k_{\text{pc}} (i_{\text{lq}}^{*} - i_{\text{lq}} ) + k_{\text{ic}} \cdot \gamma_{q} $$

  (36)

• (c) Output filter and coupling inductance: The DAEs of output filter and coupling inductance are as follows [10]:

  $$ \dot{i}_{\text{ld}} = - \frac{{r_{f} }}{{L_{f} }}i_{\text{ld}} + \omega_{\text{ref}} i_{\text{lq}} + \frac{1}{{L_{f} }}v_{\text{id}} - \frac{1}{{L_{f} }}v_{\text{od}} $$

  (37)
  $$ \dot{i}_{\text{lq}} = - \frac{{r_{f} }}{{L_{f} }}i_{\text{lq}} - \omega_{\text{ref}} i_{\text{ld}} + \frac{1}{{L_{f} }}v_{\text{iq}} - \frac{1}{{L_{f} }}v_{\text{oq}} $$

  (38)
  $$ \dot{v}_{\text{od}} = \omega_{\text{ref}} v_{\text{oq}} + \frac{1}{{C_{f} }}i_{\text{ld}} - \frac{1}{{C_{f} }}i_{\text{od}} $$

  (39)
  $$ \dot{v}_{\text{oq}} = - \omega_{\text{ref}} v_{\text{od}} + \frac{1}{{C_{f} }}i_{\text{lq}} - \frac{1}{{C_{f} }}i_{\text{oq}} $$

  (40)
  $$ \dot{i}_{\text{od}} = - \frac{{r_{c} }}{{L_{c} }}i_{\text{od}} + \omega_{\text{ref}} i_{\text{oq}} + \frac{1}{{L_{c} }}v_{\text{od}} - \frac{1}{{L_{c} }}v_{\text{bd}} $$

  (41)
  $$ \dot{i}_{\text{oq}} = - \frac{{r_{c} }}{{L_{c} }}i_{\text{oq}} - \omega_{\text{ref}} i_{\text{od}} + \frac{1}{{L_{c} }}v_{\text{oq}} - \frac{1}{{L_{c} }}v_{\text{bq}} $$

  (42)

1.2 Load modeling

A general RL load is considered in this paper. The state equations of the RL load connected at i-th node are [10]:

$$ \frac{{d(i_{{{\text{load}}\,D,i}} )}}{{{\text{d}}t}} = \frac{{ - R_{{{\text{load}}i}} }}{{L_{{{\text{load}}i}} }}i_{{{\text{load}}\,D,i}} + \omega i_{{{\text{load}}\,Q,i}} + \frac{1}{{L_{{{\text{load}}i}} }}v_{bDi} $$

(43)

$$ \frac{{d(i_{{{\text{load}}\,Q,i}} )}}{{{\text{d}}t}} = \frac{{ - R_{{{\text{load}}i}} }}{{L_{{{\text{load}}i}} }}i_{{{\text{load}}\,Q,i}} - \omega i_{{{\text{load}}\,D,i}} + \frac{1}{{L_{{{\text{load}}i}} }}v_{bQi} $$

(44)

1.3 Network modeling

On a common reference frame, the state equations of line current of i-th line connected between nodes j and k are [10]:

$$ \frac{{d(i_{{{\text{line}}\,D,i}} )}}{{{\text{d}}t}} = \frac{{ - R_{{{\text{line}}i}} }}{{L_{{{\text{line}}i}} }}i_{{{\text{line}}\,D,i}} + \omega i_{{{\text{line}}\,Q,i}} + \frac{1}{{L_{{{\text{line}}i}} }}v_{bDi} - \frac{1}{{L_{{{\text{line}}i}} }}v_{bDk} $$

(45)

$$ \frac{{d(i_{{{\text{line}}\,Q,i}} )}}{{{\text{d}}t}} = \frac{{ - R_{{{\text{line}}i}} }}{{L_{{{\text{line}}i}} }}i_{{{\text{line}}\,Q,i}} - \omega i_{{{\text{line}}\,D,i}} + \frac{1}{{L_{{{\text{line}}i}} }}v_{bQi} - \frac{1}{{L_{{{\text{line}}i}} }}v_{bQk} $$

(46)

By integrating equations of VSIs, lines and loads and using KCL for each node [10, 11], the MG general equations can be briefly written, as (17) in Sect. 3.2.