Active Distribution Networks Implications on Transmission System Stability

Abstract

Renewable power generation sources are imposing new operational challenges for power systems around the world. The traditional passive operating conditions may change since the power flow can reverse its direction under certain conditions. When the penetration of distributed generation is low, the effects comprise only distribution system. On the other hand, high distributed generation penetration may harm the whole system. In this sense, the connection of these sources can cause a considerable impact on operation and planning of transmission and distribution systems. This paper focuses on the implications on transmission systems when active distribution networks are considered. Dynamic studies are performed to assess the effects in transmission systems of renewable power generation connected to the distribution systems, considering different models and several operating scenarios. The consequences of applying complete and equivalent models are also investigated. The analysis and approaches presented here demonstrate the active distribution networks may influence transmission system behavior and also support that choosing the adequate model to represent the system is essential according to the kind of investigation someone is performing.

Appendix

1.1 Synchronous Generator Model

The synchronous machine model represents the transient and subtransient effects in d and q axes and considers the variation of the machine parameters and voltages with frequency and the stator dynamics. It is described in Eqs. (5)–(10) (CEPEL 2016).

$$ \frac{{{\text{d}}E_{d}^{{\prime \prime }} }}{{{\text{d}}t}} = \frac{1}{{T_{q0}^{''} }}\left[ { - E_{d}^{{\prime \prime }} + E_{d}^{{\prime }} + \left( {X_{q}^{{\prime }} - X_{q}^{{\prime \prime }} } \right)I_{q} + \frac{{X_{q}^{{\prime \prime }} - X_{l} }}{{X_{q}^{{\prime }} - X_{l} }}\frac{{{\text{d}}E_{d}^{{\prime }} }}{{{\text{d}}t}}} \right] $$

(5)

$$ \frac{{{\text{d}}E_{q}^{{\prime \prime }} }}{{{\text{d}}t}} = \frac{1}{{T_{d0}^{{\prime \prime }} }}\left[ { - E_{q}^{{\prime \prime }} + E_{q}^{{\prime }} + \left( {X_{d}^{{\prime }} - X_{d}^{{\prime \prime }} } \right)I_{d} + \frac{{\left( {X_{d}^{{\prime \prime }} - X_{l} } \right)}}{{X_{d}^{{\prime }} - X_{l} }}\frac{{{\text{d}}E_{q}^{{\prime }} }}{{{\text{d}}t}}} \right] $$

(6)

$$ \frac{{{\text{d}}E_{d}^{{\prime }} }}{{{\text{d}}t}} = \frac{1}{{T_{q0}^{{\prime }} }}\left[ {\frac{{X_{q} - X_{q}^{{\prime }} }}{{X_{q}^{{\prime }} - X_{l} }}E_{d}^{{\prime \prime }} - \frac{{X_{q} - X_{l} }}{{X_{q}^{'} - X_{l} }}E_{d}^{{\prime }} + \frac{{\left( {X_{q} - X_{q}^{{\prime }} } \right)\left( {X_{q}^{{\prime \prime }} - X_{l} } \right)}}{{X_{q}^{{\prime }} - X_{l} }}I_{q} + SAT_{q} } \right] $$

(7)

$$ \frac{{{\text{d}}E_{q}^{{\prime }} }}{{{\text{d}}t}} = \frac{1}{{T_{d0}^{{\prime }} }}\left[ {E_{fd} + \frac{{X_{d} - X_{d}^{{\prime }} }}{{X_{d}^{{\prime }} - X_{l} }}E_{q}^{{\prime \prime }} - \frac{{X_{d} - X_{l} }}{{X_{d}^{{\prime }} - X_{l} }}E_{q}^{{\prime }} + \frac{{\left( {X_{d} - X_{d}^{{\prime }} } \right)\left( {X_{d}^{{\prime \prime }} - X_{l} } \right)}}{{X_{d}^{{\prime }} - X_{l} }}I_{q} + SAT_{d} } \right] $$

(8)

$$ V_{d} = E_{d}^{{\prime \prime }} + X_{q}^{{\prime \prime }} I_{q} - R_{a} I_{d} $$

(9)

$$ V_{q} = E_{q}^{{\prime \prime }} + X_{d}^{{\prime \prime }} I_{d} - R_{a} I_{q} $$

(10)

1.2 Excitation System

The excitation system model is described in Eqs. (11)–(13) (Ajjarapu 2006).

$$ \frac{{{\text{d}}E_{fd} }}{{{\text{d}}t}} = \frac{1}{{T_{e} }}\left[ { - \left( {K_{e} + S_{e} } \right)E_{fd} + V_{R} } \right] $$

(11)

$$ \frac{{{\text{d}}V_{R} }}{{{\text{d}}t}} = \frac{1}{{T_{a} }}\left[ { - V_{R} + K_{a} R_{f} - \frac{{K_{a} K_{f} }}{{T_{f} }}E_{fd} + K_{a} \left( {V_{ref} - V_{i} } \right)} \right] $$

(12)

$$ \frac{{{\text{d}}R_{f} }}{{{\text{d}}t}} = \frac{1}{{T_{f} }}\left[ { - R_{f} + \frac{{K_{f} }}{{T_{f} }}E_{fd} } \right] $$

(13)

1.3 Speed Governor and Hydraulic Prime Mover

The speed governor and prime mover models for hydraulic units are described in Eqs. (14)–(16) (Kundur 1993).

$$ \frac{{{\text{d}}\Delta P_{m} }}{{{\text{d}}t}} = \frac{1}{{T_{w} }}\left[ { - 2P_{m} + \left( {\frac{{2T_{w} R_{P} }}{{R_{T} T_{R} }} + 2} \right)\Delta G - \frac{{2T_{w} R_{P} }}{{R_{T} T_{R} }}\left( {1 - \frac{{T_{R} }}{{T_{g} }}} \right)\Delta y + \left( {\frac{{2T_{w} }}{{R_{T} T_{g} }}} \right)\Delta \omega - \left( {\frac{{2T_{w} R_{P} }}{{T_{g} R_{T} }}} \right)P_{c} } \right] $$

(14)

$$ \frac{{{\text{d}}\Delta y}}{{{\text{d}}t}} = \frac{1}{{T_{g} }}\left[ { - \Delta y + P_{c} - \frac{\Delta \omega }{{R_{P} }}} \right] $$

(15)

$$ \frac{{{\text{d}}\Delta G}}{{{\text{d}}t}} = \frac{1}{{T_{R} }}\left[ { - \left( {\frac{{R_{P} }}{{R_{T} }}} \right)\Delta G + \frac{{R_{P} }}{{R_{T} }}\left( {1 - \frac{{T_{R} }}{{T_{g} }}} \right)\Delta y - \left( {\frac{{T_{R} }}{{T_{g} R_{T} }}} \right)\Delta \omega + \left( {\frac{{T_{R} R_{P} }}{{T_{g} R_{T} }}} \right)P_{c} } \right] $$

(16)

1.4 Speed Governor and Steam Prime Mover

The speed governor and prime mover models for steam units are described in Eqs. (17)–(19) (Kundur 1993).

$$ \frac{{{\text{d}}\Delta P_{m} }}{{{\text{d}}t}} = \frac{1}{{T_{RH} }}\left[ { - \Delta P_{m} + \left( {1 + \frac{{F_{HP} T_{RH} }}{{T_{CH} }}} \right)\Delta P_{CH} + \left( {\frac{{F_{HP} T_{RH} }}{{T_{CH} }}} \right)\Delta y} \right] $$

(17)

$$ \frac{{{\text{d}}\Delta P_{CH} }}{{{\text{d}}t}} = \frac{1}{{T_{CH} }}\left[ {\Delta y - \Delta P_{CH} } \right] $$

(18)

$$ \frac{{{\text{d}}\Delta y}}{{{\text{d}}t}} = \frac{1}{{T_{g} }}\left[ { - \Delta y + P_{c} - \frac{1}{{R_{P} }}\Delta \omega } \right] $$

(19)

1.5 Wind Power

The complete wind turbine model is depicted in Fig. 2, where the block diagram with transfer functions is presented (ONS 2010) (Fig. 9).

Fig. 9

Wind turbine model

1.6 ZIP Load

ZIP load model is described in Eqs. (20–21) (Kundur 1993).

$$ P_{L} = P_{L0} \left[ {a\left( {\frac{V}{{V_{0} }}} \right)^{2} + b\left( {\frac{V}{{V_{0} }}} \right) + c} \right] $$

(20)

$$ Q_{L} = Q_{L0} \left[ {d\left( {\frac{V}{{V_{0} }}} \right)^{2} + e\left( {\frac{V}{{V_{0} }}} \right) + f} \right] $$

(21)