# Impact Study of PMSG-Based Wind Power Penetration on Power System Transient Stability Using EEAC Theory

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## Abstract

**:**

## 1. Introduction

## 2. PMSG Modeling and Transient Behavior Analysis

#### 2.1. PMSG Simplified Model for Transient Stability Study

**Figure 1.**Configuration of a wind turbine with direct-driven permanent magnet synchronous generator (PMSG).

_{gd}, I

_{gq}are the active and reactive components of the grid side current, respectively.

_{g}, P

_{gref}are the actual and reference values of the active power output of a PMSG respectively. Q

_{g}, Q

_{gref}are the actual and reference values of the reactive power output of a PMSG respectively. I

_{gdcmd}, I

_{gqcmd}are the reference values of I

_{gd}and I

_{gq}respectively. Before obtaining I

_{gd}and I

_{gq}, the amplitudes of I

_{gdcmd}and I

_{gqcmd}need to be limited according to the converter’s maximum current rating, which is denoted by I

_{max}. The validity of using the PMSG controlled current source model in transient stability studies has been proved in [23] through the comparison with the practical measurement data.

#### 2.2. PMSG Transient Behavior Analysis

_{gdO}+ j0. I

_{gdO}is the pre-fault value of I

_{gd}. A PMSG with the reactive power support control mode has the same output current since the voltage deviation of the PMSG’s grid side terminal is within the dead band during the pre-fault period.

_{max}, in order to promote the active power output. The output current of a PMSG with the unity power factor control mode thus becomes I

_{max}+ j0. Comparatively, a PMSG with the reactive power support control mode outputs more reactive current. Its output current becomes I

_{gdD}+ jI

_{gqD}. I

_{gdD}, I

_{gqD}are the during fault values of I

_{gd}and I

_{gq}respectively. I

_{gqD}is controlled to comply with grid codes. Under the reactive power support control mode, the converter capacity is preferentially used to carry out the reactive power control [25]. I

_{gdD}is limited by I

_{max}as follows:

_{gdO}+ j0, no matter what kind of power control mode is adopted [26].

## 3. EEAC Theory

_{i}, ω

_{i}, M

_{i}, P

_{mi}, P

_{ei}are the rotor angle, rotor speed, inertia coefficient, mechanical input power and the electrical output power of the i-th synchronous generator respectively.

_{m}, P

_{e}are the equivalent inertia coefficient, equivalent rotor angle, equivalent mechanical input power and the equivalent electrical output power of the whole system respectively. Their specific expressions are described by:

_{S}, δ

_{S}are the equivalent inertia coefficient and the equivalent rotor angle of the specific cluster respectively. M

_{A}, δ

_{A}are the equivalent inertia coefficient and the equivalent rotor angle of the remaining cluster respectively. M

_{j}, δ

_{j}, P

_{mj}, P

_{ej}are the inertia coefficient, rotor angle, mechanical input power and the electrical output power of the j-th synchronous generator respectively. P

_{C}, P

_{max}, γ are the coefficients of the sinusoidal expression of P

_{e}, which are described by:

_{i}, E

_{j}, E

_{k}, E

_{l}are the internal voltages of the i-th, j-th, k-th and l-th synchronous generators respectively. G

_{ij}, G

_{ik}, G

_{jl}are the conductances between the internal generator nodes of the i-th and j-th synchronous generators, the i-th and k-th synchronous generators, the j-th and l-th synchronous generators respectively. B

_{ij}is the susceptance between the internal generator nodes of the i-th and j-th synchronous generators. C and D are the intermediate variables. G

_{ij}+ jB

_{ij}is an element of the admittance matrix reduced at the internal generator nodes.

_{i}, P

_{mi}, E

_{i}are assumed to be constant throughout the transient period. Thereby, the P-δ curves provided by Equations (5) and (6) can be plotted as in Figure 4. P

_{eO}, P

_{eD}, P

_{eP}are the pre-fault, during fault and post-fault values of P

_{e}respectively. δ

_{O}, δ

_{τ}are the pre-fault value and the fault clearing moment value of δ respectively. N is the system’s pre-fault operating point. A

_{acc}, A

_{decmax}are the equivalent accelerating area and the equivalent maximum decelerating area of the whole system respectively.

_{ts}, the better the system transient stability. V

_{ts}> 0 and V

_{ts}≤ 0 correspond to the stable and unstable conditions, respectively. After integrating PMSGs into a multi-machine power system, the changes in the system’s equivalent power curves mentioned above can reflect the impact of integrating PMSGs on the system transient stability. Therefore, based on the theory of EEAC, an analysis of these changes is carried out in the following section to study the influence mechanisms and find out some influence rules of integrating PMSGs on the transient stability of multi-machine power systems.

## 4. PMSG Impact Analysis

Scenarios | Machines Used to Balance Wind Power | PMSG Control Mode |
---|---|---|

Scenario one | Remaining synchronous generators | Unity power factor |

Scenario two | Remaining synchronous generators | Reactive power support |

Scenario three | Critical synchronous generators | Unity power factor |

Scenario four | Critical synchronous generators | Reactive power support |

**Figure 5.**Western Systems Coordinating Council (WSCC) 3-machine-9-bus system integrated with PMSG-based wind farms.

#### 4.1. Scenario One

_{mi}

_{1}, P

_{mi}

_{0}are the mechanical input power of the i-th synchronous generator of the wind case operating in scenario one and of the no-wind power case respectively. P

_{mj}

_{1}, P

_{mj}

_{0}are the mechanical input power of the j-th synchronous generator of the wind case operating in scenario one and of the no-wind power case respectively. P

_{ePMSG}is the pre-fault value of the sum of all the integrated PMSGs’ active power outputs.

_{m}of the wind case operating in scenario one, which is represented by P

_{m}

_{1}, is bigger than that of the no-wind power case, as described by:

_{m}

_{0}is the value of P

_{m}of the no-wind power case.

_{m}

_{1}and P

_{m}

_{0}are also constant like the P

_{m}plotted in Figure 4. From Equation (10), it can be known that the incremental amount of P

_{m}

_{1}compared with P

_{m}

_{0}depends on the pre-fault value of the active power outputted by PMSGs. Considering the system’s most stressed pre-fault operating point, it is assumed that all the PMSGs operate at their rated active power before grid faults happen [9], so P

_{m}

_{1}can be determined by the wind power penetration level, which is the ratio between the rated values of the PMSGs’ active power output and the total load [14]. The higher the wind power penetration level, the bigger the P

_{m}

_{1}.

_{e}curves of the wind case and the no-wind power case will be compared with those of the CCS case respectively, because these comparisons are easier to get results. Afterwards, the obtained comparison results will be analyzed together to find out the differences between the P

_{e}curves of the wind and no-wind power cases.

Scenarios | Machines Used to Balance the CCS Power | Corresponding Operating Scenarios of the Wind Case |
---|---|---|

Scenario I | Remaining synchronous generators | Scenario one, Scenario two |

Scenario II | Critical synchronous generators | Scenario three, Scenario four |

_{e}curves of the CCS case remain the same as the ones of the no-wind power case, no matter which scenario the CCS case is operating in. The corresponding equations are described by:

_{eO}

_{I}, P

_{eD}

_{I}, P

_{eP}

_{I}are the values of the P

_{eO}, P

_{eD}and P

_{eP}curves of the CCS case I respectively. P

_{eO}

_{II}, P

_{eD}

_{II}, P

_{eP}

_{II}are the values of the P

_{eO}, P

_{eD}and P

_{eP}curves of the CCS case II respectively. P

_{eO}

_{0}, P

_{eD}

_{0}, P

_{eP}

_{0}are the values of the P

_{eO}, P

_{eD}and P

_{eP}curves of the no-wind power case respectively.

_{eO}and P

_{eP}curves of the wind case are equal to those of the CCS case. According to Equation (11), they are also equal to those of the no-wind power case. As a result, after integrating PMSGs, the system’s P

_{eO}and P

_{eP}curves will keep their original forms. In scenario one, the corresponding equations are described by:

_{eO}

_{1}, P

_{eP}

_{1}are the values of the P

_{eO}and P

_{eP}curves of the wind case operating in scenario one respectively.

_{gdO}, while the fault period output current of a CCS in the CCS case I is still equal to I

_{gdO}, so in the wind case, the bigger active current injection from PMSGs makes the transmission lines near PMSGs possess larger current magnitudes compared with the same lines in the CCS case I. These transmission lines in the wind case thus have more active and reactive power losses, which can cause the network near PMSGs to have a lower voltage. Therefore, the critical synchronous generators near PMSGs may output less active power during the fault period, compared with the same critical machines in the CCS case I, but the remaining synchronous generators in the wind case can barely be influenced due to their long electrical distances from the PMSGs [14]. Their fault period active power generation may be equal to that of the remaining machines in the CCS case I, with considering the same settings of the wind case and the CCS case previously introduced. The corresponding equations are described by:

_{eDi}

_{1}, P

_{eDj}

_{1}are the during fault values of the electrical output power of the i-th and j-th synchronous generators in the wind case operating in scenario one respectively. P

_{eDi}

_{I}, P

_{eDj}

_{I}are the during fault values of the electrical output power of the i-th and j-th synchronous generators in the CCS case I respectively.

_{eD}curve value of the wind case operating in scenario one is smaller than that of the CCS case I, as described by:

_{eD}

_{1}is the value of the P

_{eD}curve of the wind case operating in scenario one.

_{eD}

_{1}is also smaller than P

_{eD}

_{0}. Therefore, the system’s P

_{eD}curve can be brought down after integrating PMSGs with the unity power factor control mode, as described by:

_{0}, δ

_{O}

_{0}, δ

_{τ}

_{0}, A

_{acc}

_{0}, A

_{decmax}

_{0}are the variables of the no-wind power case. They represent the system’s pre-fault operating point, the pre-fault value of δ, the fault clearing moment value of δ, the equivalent accelerating area and the equivalent maximum decelerating area of the whole system, respectively. N

_{1}, δ

_{O}

_{1}, δ

_{τ}

_{1}, A

_{acc}

_{1}, A

_{decmax}

_{1}are the corresponding variables of the wind case operating in scenario one.

**Figure 7.**The comparison of the equivalent P-δ curves between the wind case operating in scenario one and the no-wind power case.

_{m}

_{1}curve rise, so δ

_{O}

_{1}> δ

_{O}

_{0}. The accelerating power of the wind case, (P

_{m}

_{1}− P

_{eD}

_{1}), is larger than that of the no-wind power case due to the P

_{m}

_{1}curve rise and the P

_{eD}

_{1}curve drop. Therefore, according to Equation (3), it can be deduced that the equivalent rotor angle variation of the wind case is larger too during the fault period, with considering the same fault duration in these two cases. The corresponding equation is shown below:

_{O}

_{1}> δ

_{O}

_{0}and Equation (16), it can be known that δ

_{τ}

_{1}is bigger than δ

_{τ}

_{0}. As a result, compared with the no-wind power case, it can be seen from Figure 7 that the equivalent accelerating area of the wind case is increased due to the P

_{m}

_{1}curve rise, P

_{eD}

_{1}curve drop and the increment of δ

_{τ}

_{1}− δ

_{O}

_{1}. The equivalent maximum decelerating area of the wind case is decreased due to the P

_{m}

_{1}curve rise and the increment of δ

_{τ}

_{1}. The corresponding equation is shown below:

_{ts}

_{0}, V

_{ts}

_{1}are the transient stability indexes of the no-wind power case and the wind case operating in scenario one respectively.

_{m}

_{1}curve and the lower P

_{eD}

_{1}curve.

#### 4.2. Scenario Two

_{m}, P

_{eO}and P

_{eP}curves are similar to the scenario one. The analysis results are shown below:

_{m}

_{2}, P

_{eO}

_{2}, P

_{eP}

_{2}are the values of the P

_{m}, P

_{eO}and P

_{eP}curves of the wind case operating in scenario two respectively.

_{eD}curve of the wind case operating in scenario two is larger than that of the CCS case I, and is thus larger than that of the no-wind power case. The corresponding equation is shown below:

_{eD}

_{2}is the value of the P

_{eD}curve of the wind case operating in scenario two.

_{2}, δ

_{O}

_{2}, δ

_{τ}

_{2}, A

_{acc}

_{2}, A

_{decmax}

_{2}are the variables of the wind case operating in scenario two. They represent the system’s pre-fault operating point, the pre-fault value of δ, the fault clearing moment value of δ, the equivalent accelerating area and the equivalent maximum decelerating area of the whole system, respectively.

**Figure 8.**The comparison of the equivalent P-δ curves between the wind case operating in scenario two and the no-wind power case.

_{m}

_{2}and P

_{eD}

_{2}curves are increased. However, the reactive power support capability of PMSGs is restricted by the capacity of converters and the grid voltage drop [10]. The voltage improvement of the network near PMSGs is limited and so is the rise of the P

_{eD}

_{2}curve. The P

_{m}

_{2}curve thus has a larger rise amount. As a result, the accelerating power of the wind case, (P

_{m}

_{2}− P

_{eD}

_{2}), is still larger than that of the no-wind power case. During the fault period, the equivalent rotor angle of the wind case also has larger variation, as described by:

_{τ}

_{2}− δ

_{O}

_{2}and the P

_{m}

_{2}curve rise, which has greater effect than the P

_{eD}

_{2}curve rise. The equivalent maximum decelerating area of the wind case is decreased due to the P

_{m}

_{2}curve rise and the increment of δ

_{τ}

_{2}. Therefore, the transient stability of the wind case is still worse than that of the no-wind power case, as described by:

_{ts}

_{2}is the transient stability index of the wind case operating in scenario two. In this analyzed situation, the PMSG-based wind power penetration also has a detrimental impact on the system transient stability. When the wind power penetration level increases, the system transient stability will become worse too due to the higher P

_{m}

_{2}curve, which has greater effect on the system than the higher P

_{eD}

_{2}curve.

_{m}curves and the pre-fault operating points of the wind cases operating in scenario one and two will be the same when the wind power penetration level is equal in these two scenarios, but the accelerating power of the wind case operating in scenario two is smaller due to the P

_{eD}

_{2}curve rise compared with the P

_{eD}

_{1}curve drop. Therefore, the equivalent rotor angle of the wind case operating in scenario two has smaller variation during the fault period, as described by:

_{τ}

_{2}is smaller than δ

_{τ}

_{1}, so compared with scenario one, the equivalent accelerating area of the wind case operating in scenario two is decreased due to the higher P

_{eD}

_{2}curve and the smaller δ

_{τ}

_{2}− δ

_{O}

_{2}. The equivalent maximum decelerating area of the wind case operating in scenario two is increased due to the smaller δ

_{τ}

_{2}. Therefore, the wind case operating in scenario two has better transient stability.

#### 4.3. Scenario Three

_{mi}

_{3}, P

_{mj}

_{3}are the mechanical input power of the i-th and j-th synchronous generators of the wind case operating in scenario three respectively.

_{m}of the wind case operating in scenario three, which is represented by P

_{m}

_{3}, is smaller than that of the no-wind power case, as described by:

_{m}

_{3}is also constant and determined by the wind power penetration level. The higher the wind power penetration level, the smaller the P

_{m}

_{3}.

_{e}curves of the wind case operating in scenario three is similar to the scenario one, but the CCS case II is adopted as the reference case. The analysis results are shown below:

_{eO}

_{3}, P

_{eD}

_{3}, P

_{eP}

_{3}are the values of the P

_{eO}, P

_{eD}and P

_{eP}curves of the wind case operating in scenario three respectively.

_{3}, δ

_{O}

_{3}, δ

_{τ}

_{3}, A

_{acc}

_{3}, A

_{decmax}

_{3}are the variables of the wind case operating in scenario three. They represent the system’s pre-fault operating point, the pre-fault value of δ, the fault clearing moment value of δ, the equivalent accelerating area and the equivalent maximum decelerating area of the whole system, respectively.

**Figure 9.**The comparison of the equivalent P-δ curves between the wind case operating in scenario three and the no-wind power case.

_{m}

_{3}curve drop. So δ

_{O}

_{3}< δ

_{O}

_{0}. The value of the P

_{eD}

_{3}curve is also decreased compared with the P

_{eD}

_{0}curve. During the fault period, the output current increment of PMSGs with the unity power factor control mode is the main reason to cause the P

_{eD}

_{3}curve drop. But this current increment is limited especially in the high wind speed situation. So the drop of the P

_{eD}

_{3}curve may be not significant and the P

_{m}

_{3}curve has larger drop amount. As a result, the accelerating power of the wind case, (P

_{m}

_{3}− P

_{eD}

_{3}), is smaller than that of the no-wind power case. The equivalent rotor angle of the wind case thus has smaller fault period variation, as described by:

_{O}

_{3}< δ

_{O}

_{0}and Equation (27), it can be known that δ

_{τ}

_{3}is smaller than δ

_{τ}

_{0}. Compared with the no-wind power case, the equivalent accelerating area of the wind case is decreased due to the decrement of δ

_{τ}

_{3}− δ

_{O}

_{3}and the P

_{m}

_{3}curve drop, which has greater effect than the P

_{eD}

_{3}curve drop. The equivalent maximum decelerating area of the wind case is increased due to the P

_{m}

_{3}curve drop and the decrement of δ

_{τ}

_{3}. Therefore, the transient stability of the wind case is better than that of the no-wind power case, as described by:

_{ts}

_{3}is the transient stability index of the wind case operating in scenario three. In this analyzed situation, the PMSG-based wind power penetration has a beneficial impact on the system transient stability. When the wind power penetration level increases, the system transient stability will become better due to the lower P

_{m}

_{3}curve, which has greater effect on the system than the lower P

_{eD}

_{3}curve.

#### 4.4. Scenario Four

_{m}curve is similar to the scenario three and the analysis of the system’s P

_{e}curves is similar to the scenario two. The CCS case II is used as the reference case. The analysis results of scenario four are shown below:

_{m}

_{4}, P

_{eO}

_{4}, P

_{eD}

_{4}, P

_{eP}

_{4}are the values of the P

_{m}, P

_{eO}, P

_{eD}and P

_{eP}curves of the wind case operating in scenario four respectively. P

_{m}

_{4}is constant throughout the study period of transient stability.

_{4}, δ

_{O}

_{4}, δ

_{τ}

_{4}, A

_{acc}

_{4}, A

_{decmax}

_{4}are the variables of the wind case operating in scenario four. They represent the system’s pre-fault operating point, the pre-fault value of δ, the fault clearing moment value of δ, the equivalent accelerating area and the equivalent maximum decelerating area of the whole system, respectively.

**Figure 10.**The comparison of the equivalent P-δ curves between the wind case operating in scenario four and the no-wind power case.

_{m}

_{4}curve drop. The accelerating power of the wind case, (P

_{m}

_{4}− P

_{eD}

_{4}), is smaller than that of the no-wind power case due to the P

_{m}

_{4}curve drop and the P

_{eD}

_{4}curve rise. As a result, during the fault period, the equivalent rotor angle of the wind case has smaller variation, as described by:

_{m}

_{4}curve drop, P

_{eD}

_{4}curve rise and the decrement of δ

_{τ}

_{4}− δ

_{O}

_{4}. The equivalent maximum decelerating area of the wind case is increased due to the P

_{m}

_{4}curve drop and the decrement of δ

_{τ}

_{4}. Therefore, the transient stability of the wind case is still better than that of the no-wind power case, as described by:

_{ts}

_{4}is the transient stability index of the wind case operating in scenario four. In this analyzed situation, the PMSG-based wind power penetration also has a beneficial impact on the system transient stability. When the wind power penetration level increases, the system transient stability will become better due to the lower P

_{m}

_{4}curve and the higher P

_{eD}

_{4}curve.

_{m}curves and the pre-fault operating points of the wind cases operating in scenario three and four will be the same when the wind power penetration level is equal in these two scenarios. The accelerating power of the wind case operating in scenario four is smaller due to the P

_{eD}

_{4}curve rise compared with the P

_{eD}

_{3}curve drop. Therefore, the equivalent rotor angle of the wind case operating in scenario four has smaller fault period variation, as described by:

_{τ}

_{4}is smaller than δ

_{τ}

_{3}. Compared with scenario three, the equivalent accelerating area of the wind case operating in scenario four is decreased due to the higher P

_{eD}

_{4}curve and the smaller δ

_{τ}

_{4}− δ

_{O}

_{4}. The equivalent maximum decelerating area of the wind case operating in scenario four is increased due to the smaller δ

_{τ}

_{4}. Therefore, the wind case operating in scenario four has better transient stability.

#### 4.5. Summary

_{m}curve and the drop of the P

_{eD}curve can make A

_{acc}increase and A

_{decmax}decrease, which are harmful to the system transient stability. But the drop of the P

_{m}curve and the rise of the P

_{eD}curve can make A

_{acc}decrease and A

_{decmax}increase, which are good for the system transient stability. The integration of PMSGs may have either detrimental or beneficial impacts on the system transient stability.

## 5. Simulation Verification

#### 5.1. WSCC 3-Machine-9-Bus System

_{0}represents the fault CCT in the original situation without wind power penetration.

**Figure 12.**Critical clearing times (CCTs) of the fault on line 7–8 in the scenarios of PMSG-based wind power balanced by G1.

#### 5.2. IEEE 10-Mahicne-39-Bus System

**Figure 14.**Institute of Electrical and Electronics Engineers (IEEE) 10-mahicne-39-bus system integrated with PMSG-based wind farms.

**Figure 16.**CCTs of the fault on line 10–13 in the scenarios of PMSG-based wind power balanced by G1.

**Figure 17.**CCTs of the fault on line 10–13 in the scenarios of PMSG-based wind power balanced by G3.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Liu, Z.; Liu, C.; Li, G.; Liu, Y.; Liu, Y.
Impact Study of PMSG-Based Wind Power Penetration on Power System Transient Stability Using EEAC Theory. *Energies* **2015**, *8*, 13419-13441.
https://doi.org/10.3390/en81212377

**AMA Style**

Liu Z, Liu C, Li G, Liu Y, Liu Y.
Impact Study of PMSG-Based Wind Power Penetration on Power System Transient Stability Using EEAC Theory. *Energies*. 2015; 8(12):13419-13441.
https://doi.org/10.3390/en81212377

**Chicago/Turabian Style**

Liu, Zhongyi, Chongru Liu, Gengyin Li, Yong Liu, and Yilu Liu.
2015. "Impact Study of PMSG-Based Wind Power Penetration on Power System Transient Stability Using EEAC Theory" *Energies* 8, no. 12: 13419-13441.
https://doi.org/10.3390/en81212377