Dynamic Stability Performance of Autonomous Microgrid Involving High Penetration Level of Constant Power Loads
Abstract
:1. Introduction
2. Autonomous Microgrid Model
3. Constant Power Load (CPL) Model
4. Problem Formulation
4.1. Objective Function and Problem Constraints
4.2. Particle Swarm Optimization
- Initially, n particles and their velocities are randomly created. The time starts counting. The initial related objective functions are determined. The lowest one is nominated as a global best function Jbest. Its associated particle is chosen as the global best particle xbest. The inertia weight is carefully initiated to control the current velocity.
- The time counter is updated.
- The new inertia weight is calculated asw(t) = α*w(t − 1); α is close to 1.
- At each time step, each particle velocity is adapted depending on the distance between the particle and its personal position, the distance between the particle and the global best position, and the current velocity.At time n, gbest is the best position in the swarm; pbest is the best position for particle i.It is worth mentioning that based on its own thinking and memory, the particle changes its velocity. The second term is the PSO cognitive part, while the third term is the PSO social part. It is based on the social–psychological adaptation of knowledge.
- At iteration n + 1, the new particle position is determined based on the updated velocities as follows,
- From all global best values Jj*, the minimum one will be chosen as given.
- If Jmin > J**, then update the global best as X** = Xmin and J** = Jmin
- PSO stops searching when the number of iterations exceeds the pre-specified number or maximum allowable iterations.
4.3. PSO Implementation
- Inertia weight factor =1
- Generation or iteration = 100
- Population size = 20
- Acceleration constants: c1 = c2 = 2
- Decrement constant (α) = 0.98.
5. Results and Discussion
5.1. Simulation Results
5.2. Impacts of the Uncertainties of Controller Parameters
5.3. Proposed Method Limitations
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
vod, voq | dq components of the inverter output voltage vo |
iod, ioq | dq components of the inverter output current io |
Pm, Qm | instantaneous active and reactive powers |
Pc, Qc | average active and reactive powers |
mp, nq | droop controller gains |
θ | phase reference |
ω | nominal frequency |
ωc | cut-off frequency of the low-pass filter |
ωn | nominal angular frequency of DG |
Vn | nominal magnitude of the DG voltage |
F | voltage controller feed-forward gain |
v*od, v*oq | dq components of the reference output voltage |
i*ld, i*lq | dq components of inductor reference current |
v*id, v*iq | dq components of the reference inverter voltage |
ild, ilq | dq components of the coupling inductor current iL |
vid, viq | dq components of the inverter voltage vi |
Cf, Lf, Rf | capacitance, inductance, and resistance of the LC filter |
Lc, Rc | inductance and resistance of the coupling inductor |
Cdc, Rdc | capacitance and resistance of the DC load of the active load |
δi | angle between the reference frame of each inverter (dq)and the common reference frame (DQ) |
δAL | angle between the reference frame of CPL (dqAL)and the common reference frame (DQ) |
ilineDQ | DQ components of the line |
iloadDQ | DQ components of the load currents |
vdc, idc | DC voltage and DC current of the active load respectively |
v*DC | DC Reference voltage of the CPL |
iconv | DC side current of the CPL |
vidqAL | dq components of the active load output voltage (viAL) |
iodqAL | dq components of the active load output current (ioAL) |
ildqAL | dq components of the input current to the bridge (ilAL) |
Kpv, Kiv | PI voltage controller parameters of the DG inverter |
Kpc, Kic | PI current controller parameters of the DG inverter |
Kpv_AL, Kiv_A | PI controller parameters of the DC voltage of the CPL |
Kpc_AL, Kic_AL | PI controller parameters of the AC current of the CPL |
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Microgrid Parameters | |||
---|---|---|---|
Parameter | Value | Parameter | Value |
fs | 8 kHz | Vn | 381 V |
Lf | 1.35 mH | Lc | 0.35 mH |
Cf | 50 × 10−6 F | Cb | 50 × 10−6 F |
rf | 0.1 Ω | rc | 0.03 Ω |
ωn | 314.16 rad/s | ωc | 31.416 rad/s |
r1 + jx1 | (0.23 + j0.1) Ω | r2 + jx2 | (0.35 + j0.58) Ω |
CPL Parameters | |||
Lf | 2.3 mH | Lc | 0.93 mH |
Cf | 8.8 × 10−6 F | rc | 0.03 Ω |
rf | 0.1 Ω | ||
Rdc | 72 Ω | Cdc | 2040 × 10−6 F |
Power Sharing Parameters of the Three DG Units | |||
mp | 3.79404 × 10−7 | nq | 9.36593 × 10−5 |
6.75934 × 10−7 | 1.86121 × 10−5 | ||
1.71857 × 10−7 | 3.21349 × 10−5 | ||
md | 0.2957 × 10−5 | nd | 0.2618 × 10−6 |
0.1572 × 10−5 | 0.2374 × 10−6 | ||
−0.0030 × 10−5 | 0.2374 × 10−6 | ||
PLL Parameters | |||
KpPLL | 1 | KIPLL | 50 |
Controller Parameters of the Three DG Units | |||
Parameter | Value | Parameter | Value |
Kpv (Amp/Watt) | 1.19585 | Kpc (Amp/Watt) | 44.1091 |
1.43531 | 31.8037 | ||
1.6380 | 40.8816 | ||
Kiv (Amp/Joule) | 4.4568 | Kic (Amp/Joule) | 35.8275 |
6.17159 | 26.904 | ||
−0.69434 | 13.4463 | ||
CPL Parameters | |||
Kpv_AL (Amp/Watt) | 0.331792 | Kpc_AL (Amp/Watt) | 33.2732 |
Kiv_AL (Amp/Joule) | 4.33114 | Kic_AL (Amp/Joule) | −4.61844 |
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Hassan, M.A.; Worku, M.Y.; Eladl, A.A.; Abido, M.A. Dynamic Stability Performance of Autonomous Microgrid Involving High Penetration Level of Constant Power Loads. Mathematics 2021, 9, 922. https://doi.org/10.3390/math9090922
Hassan MA, Worku MY, Eladl AA, Abido MA. Dynamic Stability Performance of Autonomous Microgrid Involving High Penetration Level of Constant Power Loads. Mathematics. 2021; 9(9):922. https://doi.org/10.3390/math9090922
Chicago/Turabian StyleHassan, Mohamed A., Muhammed Y. Worku, Abdelfattah A. Eladl, and Mohammed A. Abido. 2021. "Dynamic Stability Performance of Autonomous Microgrid Involving High Penetration Level of Constant Power Loads" Mathematics 9, no. 9: 922. https://doi.org/10.3390/math9090922