AGC of Interconnected Multi-source Power System with Considering GDB and GRC Nonlinearity Effects Javad Morsali Elect. & Comp. Eng. Faculty Tabriz University Tabriz, Iran
[email protected]
Kazem Zare Elect. & Comp. Eng. Faculty Tabriz University Tabriz, Iran
[email protected]
Abstract—In this paper, automatic generation control (AGC) problem of an interconnected two-area multi-source power system is investigated. Each control area includes generations from reheat thermal, gas, and hydro units. Appropriate dynamic models are used for simulation of physical constraints of governor dead band (GDB) effect in the reheat thermal and generation rate constraint (GRC) in the reheat thermal and hydro generating units. This paper demonstrates that evaluating the dynamic performance of AGC without regarding these issues does not show precise and realistic results. A simple integral (I) controller is considered as secondary load frequency controller (LFC) loop. The LFC design is formulated as an optimization problem in which integral of time multiplied squared error (ITSE) performance index is minimized by an improved particle swarm optimization (IPSO) algorithm to adjust I controller. The LFC performance is evaluated under step, sinusoidal and random load perturbation patterns. To show the robustness of the proposed approach, sensitivity analyses are performed under various uncertainty scenarios. All simulations are performed in MATLAB/SIMULINK environment. The results are compared with the case of without considering the GDB and GRC nonlinearity effects. The results demonstrate that with considering the GRC and GDB, the oscillations are not damped effectively and even they are increasing under uncertainty conditions. Keywords—AGC; GRC; GDB; multi-source power system; load perturbation pattern; frequency stability
I. INTRODUCTION Automatic generation control (AGC) is an important issue in frequency stability studies of interconnected power Systems. In order to obtain an accurate realization and valid insight of the AGC problem, it is important to take into consideration the realistic constraints and natural requirements such as physical constraint of generation rate constraint (GRC) [1] in thermal and hydro power units. The GRC has negative impact on the power system performance due to its non-linearity nature. The GRC means practical constraint on the rate of the change in the generating power due to physical limitations [2]. Practically, the rate of active power change that can be acquirable by both reheat and non-reheat thermal units and also in hydro units have a maximum limit [3]. Hence, the arranged AGC for the unconstrained generation rate condition may no longer be precise and realistic. Since the GRC has great influence on the dynamic performance of the AGC, appropriate modeling of this physical constraint in practical power systems is vital [4]. After load disturbance in a power system, the attainable
M. Tarafdar Hagh Elect. & Comp. Eng. Faculty Tabriz University Tabriz, Iran
[email protected]
power to keep the balance between the generation and load is limited due to the GRC. The deteriorating effect of the GRC will be more considerable if the power system experiences larger load disturbance. In such a condition, the power system tries to deliver greater amount of the power in a short time horizon to guaranty the frequency stability of the interconnected system, but the GRC restricts the output of generators by limiting the ramp rate of the generating power which is required for satisfying the load disturbance. The GRC of 10%/min for the thermal unit is considered for both raising and falling rates [4-6]. It has been demonstrated that if the GRC of the reheat thermal unit is considered smaller than the 10%/min, the system dynamic performance is deteriorated severely [5]. For the hydro unit, typical GRC of 270%/min for raising generation and 360%/min for falling generation is considered [1, 5]. The typical value of permissible GRC for hydro plant is relatively much larger than that for reheat thermal unit which has GRC. Another physical constraint of thermal units is the governor dead band (GDB) effect. The GDB is defined as the total magnitude of a sustained speed change within which there is no change in valve position of the turbine [1]. A literature survey reveals that in the presence of the system inherent constraints and nonlinearities like the GRC and GDB, the dynamic characteristics of the power system show larger overshoots and longer settling times of frequency and tieline power oscillations, in comparison with the case of without considering GRC and GDB effects [6, 7]. Also, considering these constraints affect the values of optimal gains of the LFC so that, the system frequency stability may be very challenging. Maybe, the under frequency protections are activated and the system encounters the frequency instability. The GDB and GRC restrict the instantaneous reaction of the generators to modify disturbances. Hence, to realistic evaluate the AGC performance; these constraints should be taken into account. In the presence of GRC and GDB effects, the system becomes seriously nonlinear even for small load disturbance. In [4], the authors evaluate the various modeling methods for dynamic simulation of the GRC in MATLAB/SIMULINK environment to choose an appropriate approach. In [7-9], LFC of interconnected single-source reheat thermal power system considering GDB and GRC nonlinearities has been evaluated. In [1012], LFC of two-area multi-source power system which has various generations from thermal, gas, and hydro units have been presented. However, the nonlinearity effect of GRC for thermal and hydro units and the deteriorating effect of GDB for thermal unit have not been taken into account.
Above literature survey reveals that no effort has been made to evaluate the impacts of the GDB and GRC nonlinearity on dynamic performance of the LFC loop in an interconnected multi-source power system having diverse generations from gas, reheat thermal and hydro units in each control area. It has been clarified that ignoring these lead to nonrealistic results in evaluation of the LFC performance [1]. In this paper, a two-area multi-source power system which has reheat thermal, gas, and hydro plants in each area is proposed with considering nonlinearity and deteriorating effects of the GRC and GDB. For this study, a simple integral (I) controller is chosen as LFC loop. The integral (I) and proportional integral (PI) controllers remain still as industrially preferred structures due to easy modeling, straightforward tuning, reliability, and cost-effective choice. The integral gains are adjusted by an improved particle swarm optimization (IPSO) algorithm. II.
POWER SYSTEM STUDIES
A. Realistic interconnected multi-source power system Fig. 1 shows the transfer function model of proposed interconnected multi-source power system including GDB and GRC constraints. The block descriptions are shown on Fig. 1. The parameters are given in [1]. In this work, the GRC for reheat thermal unit is considered as 10%/min. The hydro unit has GRC of 270%/min and 360%/min for raising and falling generations, respectively. According to [1, 4], N1 and N2 in the GDB transfer function model are N1=0.8 and N2=-0.2/π, respectively. Droop of thermal units is set to 4%.
ITSE
Tsim
t f12 f 2 2 P122 dt
(1) where Tsim depicts the simulation time. The ITSE index uses squared error and time multiplication to penalize large deviation and long settling time. Hence, the ITSE profits strong points of both integral of squared error (ISE) and integral of time multiplied absolute error (ITAE) indices. The ITSE has been used lately in [1, 4] as objective function to design AGC. In this paper, an optimization problem is solved by improved particle swarm optimization (IPSO) algorithm [1, 4] to minimize the ITSE index to obtain the optimal parameters of the I controllers, subject to the gains constraints: 0
0 KI1, K I 2 2
III.
(2)
SIMULATION RESULTS AND DISCUSSES
A. Simulation results for step load perturbation In this case, the dynamic simulations are carried out for 0.01pu step load perturbation (SLP) in the area 1 with and without considering the GRC and GDB effects. Several near-optimal set of adjusted parameters obtained after numerous runs are listed in TABLE I. The parameters corresponding to the minimum value of the ITSE index is selected as final solution which is highlighted in TABLE I. It is obvious from TABLE I that with considering the GRC and GDB, the minimally calculated ITSE index (0.0621) is approximately four times higher than the next case (0.0158) which means lowering in damping performance. Also, the converging profiles as shown in the Fig. 2 demonstrate that the IPSO algorithm surpasses the PSO in minimizing the ITSE index. The damping criteria such as the system oscillatory modes and corresponding damping ratios (ζ), settling time (TS) with 5% criterion, maximum peak (Mp), and peak time (Tp) of the area frequencies and tie-line power oscillations are listed in TABLE II. Mentioned damping criteria are determinant in assessing the AGC dynamic performance. It is clear from TABLE II that considering the physical constraints of the GRC and GDB leads to obtaining remarkably larger Mp and TP, and longer TS than those of obtained from the next case. Furthermore, with considering the GRC and GDB effects, the system minimum damping ratio (0.0837) is two times smaller than the case of without the GRC and GDB (0.1656). Briefly, the system damping measures decline remarkably after taking into account the physical constraints of the GRC and GDB. TABLE I.
Fig. 1. Block diagram of the proposed interconnected multi-source power system with GDB and GRC effects
B. Objective function for LFC design In order to mitigate the oscillations as much as possible, regarding an appropriate objective function is very essential to obtain the optimal controller parameters. In this paper, the integral of time multiplied squared error (ITSE) performance index is used as:
With GRC and GDB
SEVERAL NEAR-OPTIMAL SOLUTIONS
IPSO
PSO Without GRC and GDB
IPSO
PSO
No. 1 2 3 4 5 1 1 2 3 4 5 1
KI1
KI2
ITSE
0.0811 0.0851 0.0850 0.0791 0.0851 0.0751
0.0417 0.0427 0.0424 0.0524 0.0457 0.0351
0.0622 0.0622 0.0622 0.0622 0.0621 0.0626
0.3258 0.3207 0.3647 0.3007 0.3380 0.2278
0.0132 0.0416 0.0163 0.0142 0.0600 0.0369
0.0159 0.0162 0.0158 0.0160 0.0164 0.0170
0.2
-3
PSO without GRC and GDB IPSO without GRC and GDB IPSO with GRC and GDB PSO with GRC and GDB
0.18 0.16
5
x 10
without GRC and GDB with GRC and GDB
0.14 0 Delta P12 (pu)
ITSE
0.12 0.1 0.08 0.06
-5
0.04 0.02 0 0
5
10
15 Iteration
20
25
-10 0
30
15
20
25 Time(s)
30
35
40
45
50
SYSTEM DAMPING CRITERIA
MP
TP
0.0448
3.7452
0.0837
Δf1
0.7333
Δf2
>50
0.0585
2.4691
0.9407
ΔP12
38.2690
0.0088
1.6309
0.1656
Δf1
32.5047
0.0381
1.0328
0.5097
Δf2
33.7985
0.0419
2.0541
0.9837
ΔP12
21.2241
0.0077
1.5461
48.4881
Fig. 3 illustrate the frequencies and tie-line power oscillation responses under the SLP for the cases of with and without considering the GRC and GDB. It can be seen from Fig. 3 that with considering the GRC and GDB, the area frequencies and tie-line power deviations are damped to zero with difficulty. As it is obvious, with the optimal I controllers, the oscillations persist for a longer time with larger amplitudes than the in the case of without considering the GRC and GDB. In this case, the AGC system can no longer suppress the frequency and tie-line power oscillations and drive back them to zero, effectively. It is notable that the precise performance of the LFC is realized with regarding of the GRC and GDB effects.
B. Simulation results for sinusoidal load perturbation In order to evaluate the effectiveness of the controllers in stabilization of area frequencies and tie-line power oscillations under continuous load pattern, the sinusoidal load perturbation is applied in the area 1 as following [13]: Pd 1 0.03sin(4.36t ) 0.05sin(5.3t ) 0.1sin(6t ) (3) Fig. 4 illustrates the frequencies and tie-line power oscillations under the sinusoidal load perturbation for the cases of with and without considering GRC and GDB. The illustrations reveal that with regarding the physical constraints of the GRC and GDB, the oscillations are not restricted effectively and the LFC is incapable to stabilize the oscillations especially in the Fig. 4(b) and Fig. 4(c). 0.25 without GRC and GDB with GRC and GDB
0.2 0.15 0.1 Delta f1 (Hz)
TS
With GRC and GDB
ζ
Without GRC and GDB
oscillatory modes -0.1711 ± 2.0357i -0.6231 ± 0.5778i -0.0695 ± 0.0251i -0.1846 ± 1.0992i -0.4993 ± 0.8427i -0.1530 ± 0.0280i
10
(c) Fig. 3. Dynamic responses to the SLP: (a) Δf1 , (b) Δf2, and (c) ΔP12
Fig. 2. Converging profiles of PSO and IPSO algorithms TABLE II.
5
0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 0
5
10
15 Time(s)
20
25
30
(a) 0.02 without GRC and GDB with GRC and GDB
0.015 0.01
0.02 without GRC and GDB with GRC and GDB
0.005 Delta f2 (Hz)
0.01
Delta f1 (Hz)
0
0 -0.005 -0.01
-0.01 -0.015
-0.02
-0.02 -0.025
-0.03 -0.03 0
5
10
-0.04
-0.05 0
20
25
30
(b) 5
10
15
20
25 Time(s)
30
35
40
45
50
with GRC and GDB without GRC and GDB
(a)
0.01
without GRC and GDB with GRC and GDB Delta P12 (pu)
0.01 0 -0.01 Delta f2 (Hz)
15 Time(s)
-0.02
0
-0.01
-0.03 -0.04
-0.02 0
5
10
15 Time(s)
20
25
30
-0.05 -0.06 0
5
10
15
20
25 Time(s)
(b)
30
35
40
45
50
(c) Fig. 4. Dynamic responses to the sinusoidal load perturbation: (a) Δf 1, (b) Δf2, and (c) ΔP12
C. Simulation results for random load perturbation In this case, a random step load perturbation as depicted in Fig. 5 is applied in the area 1 [14]. The steps are random both in magnitude and duration in accordance with the nature of load changes in a realistic power system. 0.045 0.04
Randomload (pu)
0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 -0.005 0
20
40
60
80
100 Time(s)
120
140
160
180
D. Sensitivity analysis against uncertainty scenarios In order to to show the robustness under large changes in the system loading condition and parameters, the sensitivity analysis is done. The uncertainty scenarios are examined on loading condition, steam turbine time constant Tt, and the synchronizing torque T12 which are changed in the range of ±50% from nominal values, separately. The aforementioned scenarios are examined for 0.01pu SLP in area 1. The results are listed in TABLE III. It can be seen that with considering ±50% uncertainty scenarios, the ITSE index and minimum damping ratios deviate remarkably from their nominal values. The Increase/decrease in the ITSE index can be interpreted as decline/improvement of system damping performance. Also, the smaller damping ratios mean the deteriorative damping behavior.
200
Fig. 5. Random step load perturbation
TABLE III.
0.2 0.15 Delta f1 (Hz)
0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 0
20
40
60
80
100 Time(s)
120
140
160
180
200
(a) without GRC and GDB with GRC and GDB
+50% in Tt
0.25 0.2 0.15 0.1 Delta f2 (Hz)
-50% in loading +50% in loading
% with GRC and GDB without GRC and GDB
0.25
0.05 0 -0.05
Constraint With GRC and GDB Without GRC and GDB With GRC and GDB Without GRC and GDB With GRC and GDB Without GRC and GDB
-0.1 -0.15
-0.25 0
20
40
60
80
100 Time(s)
120
140
160
180
200
(b) without GRC and GDB with GRC and GDB
0.05
-50% in Tt
-0.2
0.04
Without GRC and GDB
0.02 0.01
+50% in T12
0 -0.01 -0.02 -0.03 -0.04 -0.05 0
20
40
60
80
100 Time(s)
120
140
160
180
200
(c) Fig. 6. Dynamic responses to the random step load perturbation: (a) Δf1, (b) Δf2, and (c) ΔP12
The dynamic responses of the multi-source power system under the random perturbation are shown in Fig. 6. It can be observed that with considering the GRC and GDB effects, the amplitude of the oscillations grows continuously which may lead to system instability, in final.
-50% in T12
Delta P12 (pu)
0.03
With GRC and GDB
With GRC and GDB Without GRC and GDB With GRC and GDB Without GRC and GDB
SENSITIVITY ANALYSIS AGAINST UNCERTAINTIES
oscillatory Modes -0.1953 ± 2.0363i -0.6589 ± 0.5899i -0.0697 ± 0.0252i -0.1932 ± 1.0112i -0.5355 ± 0.8548i -0.1581 ± 0.0201i -0.1469 ± 2.0350i -0.5865 ± 0.5646i -0.0693 ± 0.0249i -0.1532 ± 1.1056i -0.4629 ± 0.8312i -0.1489 ± 0.0327i -0.1319 ± 2.0296i -0.5943 ± 0.6238i -0.0692 ± 0.0253i -0.1332 ± 1.0239i -0.4402 ± 0.8743i -0.1534 ± 0.0291i -5.7878 ± 0.4882i -5.7896 ± 0.5471i -0.2202 ± 2.0169i -0.6305 ± 0.5358i -0.0698 ± 0.0248i -5.9771 ± 0.6337i -5.9624 ± 0.5728i -0.2982 ± 1.0198i -0.5364 ± 0.7915i -0.1526 ± 0.0268i -0.1317 ± 2.4318i -0.6231 ± 0.5778i -0.0698 ± 0.0254i -0.1569 ± 1.2402i -0.4976 ± 0.8395i -0.1543 ± 0.0299i -0.2386 ± 1.5156i -0.6232 ± 0.5779i -0.0688 ± 0.0240i -0.4024 ± 1.1457i -0.5044 ± 0.8524i -0.1493 ± 0.0210i
ζ 0.0955 0.7451 0.9403 0.1877 0.5309 0.9920 0.0720 0.7204 0.9411 0.1373 0.4865 0.9767 0.0649 0.6898 0.9391 0.1290 0.4497 0.9824 0.9965 0.9956 0.1085 0.7620 0.9422 0.9944 0.9954 0.2807 0.5610 0.9849 0.0541 0.7333 0.9396 0.1255 0.5099 0.9818 0.1555 0.7332 0.9440 0.3314 0.5093 0.9903
ITSE 0.0424
0.0130
0.5793
0.0207
0.0949
0.0273
0.0483
0.0119
0.6585
0.0201
0.0442
0.0140
For better insight to the results of sensitivity analyses, Figs. 7-9 depict the dynamic responses for the considered uncertainty scenarios with considering the GRC and GDB effects. It can be inferred from these illustrations that
0.03 0.02 0.01 0 Delta f2 (Hz)
considering ±50% variations in the system loading condition and parameters can bring about increasing oscillations in the frequencies and tie-line power responses as seen in Figs. 7 and 8. The performed sensitivity analyses demonstrate that the power system is not robust to the considered uncertainties. I.e., the optimized LFC once adjusted for the nominal condition and nominal parameters; can no longer provide satisfactory results for the considered changes in the system parameters and loading condition.
-0.01 -0.02 -0.03 -0.04 nominal T12 +50% of T12 -50% of T12
-0.05 -0.06 -0.07 0
5
10
15
20
25 Time(s)
30
35
40
35
40
45
50
(b) 0.03
0.01
nominal loading -50% of loading +50% of loading
0.02
0.005
0.01
Delta P12 (pu)
Delta f1 (Hz)
0 -0.01 -0.02
0
-0.005
-0.03 -0.04
-0.01 nominal T12 +50% of T12 -50% of T12
-0.05 -0.06 0
5
10
15
20
25 Time(s)
30
35
40
45
-0.015 0
50
5
10
15
20
25 Time(s)
30
45
50
(c) Fig. 8. Sensitivity analysis against uncertainty in T12, (a) Δf1, (b) Δf2, (c) ΔP12
(a) 0.03 0.02 0.01
0.02 nominal Tt -50% of Tt +50% of Tt
0.01
-0.02
0 -0.03
Delta f1 (Hz)
Delta f2 (Hz)
0 -0.01
-0.04 -0.05
nominal loading -50% of loading +50% of loading
-0.06 -0.07 0
5
10
15
20
25 Time(s)
30
35
40
45
-0.02
-0.03
50
-0.04
(b) x 10
-0.01
-3
-0.05 0
5
10
15
20
6
25 Time(s)
30
35
40
45
50
(a)
4
0.02 0.01 0
0 -2
-0.01 Delta f2 (Hz)
Delta P12 (pu)
2
-4 -6 nominal loading +50% of loading -50% of loading
-8 -10 0
5
10
15
20
25 Time(s)
30
35
40
45
-0.02 -0.03 -0.04
50
-0.05 nominal Tt +50% of Tt -50% of Tt
-0.06
(c) Fig. 7. Sensitivity analysis against uncertainty in loading condition, (a) Δf 1, (b) Δf2 , (c) ΔP12
-0.07 0
5
10
15
20
25 Time(s)
30
35
40
45
50
(b) 5
x 10
-3
nominal Tt +50% of Tt -50% of Tt
0.03 0.02 0 Delta P12 (pu)
Delta f1 (Hz)
0.01 0 -0.01 -0.02
-5
-0.03 -0.04
nominal T12 -50% of T12 +50% of T12
-0.05 -0.06 0
5
10
15
20
25 Time(s)
(a)
30
35
40
45
-10 0
50
5
10
15
20
25 Time(s)
30
35
40
45
50
(c) Fig. 9. Sensitivity analysis against uncertainty in Tt, (a) Δf1, (b) Δf2, (c) ΔP12
IV.
CONCLUSION
In current paper, an attempt has been made to evaluate the impacts of physical constraints of the GRC and GDB on AGC performance of the interconnected multi-source power system including gas, reheat thermal, and hydro units in each control area. The results have been compared with the case of without considering the GRC and GDB effects. The simulation results under step, sinusoidal, and random load perturbation patterns reveal that with considering the GDB and GRC effects, the dynamic performance of the AGC in damping of the area frequencies and tie-line power oscillations is declined significantly due to the nonlinearity nature of the GRC and GDB. Further investigations known as the sensitivity analyses have been performed under uncertainty scenarios for the system loading condition and parameters. The simulation results demonstrate that the interconnected multi-source system with the optimized I controllers as the LFC loop is not capable to withstand the uncertainties and can not act satisfactorily under the uncertainty conditions. Maybe a more advanced LFC can resolve this problem. Clearly, this paper concludes that to obtain precise realization of the AGC problem and to realistic understanding of power system dynamics, it is inevitable to take into account the physical constraint of the GRC and GDB, in addition to the variety of the power generating units such as reheat thermal, hydro, and gas in each control area. REFERENCES
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