Application-Based Simulations for Diagnosing the Power Quality When Changes in Load’s Power on Each Bus as an Analogy of the Steady-State Stability Phenomena

Abstract

This chapter describes the system’s ability to recover frequency synchronisation across all networks after a gradual change in electric power is determined by simulating the phenomenon of steady-state stability. The simulation is done by choosing and figuring out an analogy of the structure of an electric power network with four connected buses and assuming values for a number of parameters. The objectives for this diagnostic are set on a pair of primary levels, namely, to create the three syntax structures of the programme based on MATLAB and to simulate a phenomenon of steady-state stability based on the programme. The results of the implementation of steady-state stability simulation have been obtained in the form of phenomena such as a reduction of power on bus #1, an addition of power on bus #2, an addition of power on bus #3, and a reduction of power on bus #4. Based on the simulation results, it can be concluded that the MATLAB-based programme is used for the diagnosis of steady-state stability.

1. Introduction

MATLAB-based application is an interactive system and an application with a programming language. The basic data element is a matrix that does not need to declare the size or type of data, so many calculation problems can be solved in a short time, and calculations can be written into the FORTRAN or C language [1, 2]. Workarounds for complex calculation cases, such as power flow studies, make it difficult to change one value or more variables or commands for reevaluation. How to solve complex cases easily and efficiently to problems in the MATLAB command windows, done through providing input into a text file commonly called script files or M-files [1]. For the creation of an M-file with the selection of a new M-file from the File Menu, a text editor window is displayed [3].

The term power quality in the IEEE Standard Dictionary of Electrical and Electronics Terms is defined as “the concept of powering and grounding sensitive electronic equipment in a manner that is suitable to the operation of that equipment” [4]. The quality of electrical power is a problem in the form of voltage deviations, frequencies, or phase angles that result in failures or operating errors in electrical equipment on the consumer side. The diagnosis of electrical power quality is an effort to determine the type of deviation through the examination of various forms of phenomena that occur. The quality of electrical power is characterised by a voltage waveform in accordance with established guidelines. Poor quality of electrical power is a cause of loss, both on the customer side and the electric power generation side. Various factors can cause poor power quality, which can include voltage, current, frequency, and load factors [5].

Stability in electrical power systems (EPS) has been recognised as problematic for the safe operation of power systems since the 1920s [6], after the commercial use of electricity in the late 1870s with the installation of lights for lighthouses and street lighting [7]. Stability is defined as the ability of an electrical power system to return to normal or stable conditions after being disrupted [8, 9, 10]. System stability depends primarily on the behaviour of synchronous machines after a disturbance [8, 11], since the limits of system stability are characterised by the presence of a maximum amount of power passing through a point in the system without resulting in a loss of stability [11, 12]. The loss of stability in the electrical power system is strongly influenced by the presence of a number of disturbances [13], so the continuity of the electrical power system must be maintained in synchronisation conditions as perfectly as possible in a steady state [6, 12]. Disturbances in the system can be of various types [13, 14, 15], such as sudden load changes, sudden short circuits between channels and the ground [16], inter-channel disturbances, line-to-ground symmetrical disturbances, switching phenomena, and others.

Steady-state stability is essential for planning and designing power systems, developing specialised automatic control devices, placing new elements into system operation, or modifying operating conditions [17]. Stability selection is made based on the requirements of stability limits or the quality of electrical power in a steady state or during transients. The steady-state limit is guided by the maximum power flow through a given point without resulting in a loss of stability, when the power is gradually increased or decreased [18, 19]. Steady-state limit estimation is important for electrical power system analysis, with coverage on checking the electric power system in stable conditions, determining stability limits, and qualitative estimation of transient conditions [11, 17]. The existence of stability and limit estimation at steady-state is necessary for the estimation of the choice of excitation and controller system types, control modes, excitation control system parameters, and automation, including power breaker (circuit breaker, CB) performance [20, 21]. It has to do with the presence of an electrical power system made up of numerous synchronous machines that operate in unison [11, 22]. In conditions in which the system experiences interference, a force to return to a normal or stable state develops [8, 12, 23].

By simulating steady-state stability, the problem is put into words based on the system’s ability to get all networks back in sync with the same frequency after a gradual change in power. The simulation is carried out through the selection and determination of an analogy of the structure of an electric power network [24]. An analogy of the electric power structure consists of four interconnected buses with assumed values against a number of accompanying parameters. An analogy of the electrical power system structure for steady-state stability simulation [25] is shown in Figure 1.

Figure 1.

Analogy of electrical power system structure for steady-state stability simulation.

Based on Figure 1, it is shown and can be explained that the load and connecting transmission line have impedance values, namely R + jX values (ohms). A circle with the notation “number” is a synchronous generator node in a rotating generation system. System structure with four buses, namely bus #1, bus #2, bus #3, and bus #4. Bus #1 is noted as a swing bus (slack bus, reference bus, $V-\phi$ bus), which serves as a bearer of the lack of generation power. Buses #2 and #3 are noted as generation buses (voltage control buses, $P-V$ buses), while bus #4 is noted as a load bus ($P-Q$ bus). The next phasing is in the form of making algorithms and compiling MATLAB-based syntax, a number of simulations of the steady-state stability phenomenon with initial assumption values, and making conditions for the steady-state stability phenomenon through changes in power on the load on each bus [26, 27].

The objectives for this diagnostic are set on a pair of primary levels, referring to the formulation of the problem. First, the acquisition of the three syntax structures of the programme, namely the programme structure for inputting data on initial assumption values, the programme structure for simulating the phenomenon of power flow, and the programme structure for simulating the phenomenon of steady-state stability with data on assumption values; The second is the implementation of steady-state stability simulation with (i) data on initial assumption values and (ii) assumption data values when there are changes in load on each bus under conditions, namely a reduction of power on bus #1, an addition of power on bus #2, an addition of power on bus #3, and a reduction of power on bus #4.

2. Materials and methods of simulation

2.1 Materials of simulation

The bus types for simulation consist of (a) swing buses (slack buses, reference buses, $V-\phi$ buses), (b) generation buses (voltage control buses, $P-V$ buses), and (c) load buses ($P-Q$ buses). The swing bus is notated against bus #1, which serves as a bearer of the lack of generation power after the power flow solution is obtained. The generation bus is denoted against bus #2 and bus #3, which function as buses with active power injection parameters and the amount of bus voltage known. Load the bus for bus #4. Serves as a bus with known active power injection and reactive power bus parameters [8]. The parameters in the mathematical model are given assumed values for simulating steady-state stability parameters in the analogy of a four-bus, three-centre power system with variable changes in the value of the load (demand) power and its effect on each engine [28, 29, 30].

Guided by Figure 1, the initial assumption values in the model of a four-bus electric power system with three generation centres, in the form of transmission line values with a base of 100 MVA, so that units per unit are obtained. The values of the connecting transmission line components as shown in Table 1.

Buses		R (p.u.)	X (p.u.)
From	To
1	2	9.23	79.98
1	3	12.93	71.69
1	4	17.82	79.98
2	3	6.28	47.45
2	4	6.66	73.57
3	4	9.26	65.03

Table 1.

The values of the connecting transmission line components.

The value of the load power (demand) on each bus is shown in Table 2.

Bus	Pd (MW)	Qd (MVAr)	Vmax. (p.u.)	Vmin. (p.u.)
1	42.50	203.80	1.1	0.9
2	11.21	122.60	1.1	0.9
3	26.63	171.40	1.1	0.9
4	42.18	147.50	1.1	0.9

Table 2.

The value of the load power (demand) on each bus.

The value of the initial state of each generation system generator before the power flow calculation process is shown in Table 3.

Bus	Pm (MW)	Qm (MVAr)	V (p.u.)	$\delta$ (°)	H; (p.u. inertia constant; sec.)	f0 (Hz)
1	0	0	1.030	0	2.32	60
2	45	0	1.025	0	3.40	60
3	50	0	1.030	0	4.63	60

Table 3.

The value of the initial state of each generation system generator before the power flow calculation process.

Based on Table 3, it is shown and has been determined that the first bus is a swing bus, and then the generation value is adjusted after the system is operational.

Theoretically and mathematically, expressions are described in the following description: A set of rotating machines is modelled with a generator, both mechanically and electrically. Modelling the mechanical and electrical relationships of a generator, as shown in Figure 2.

Figure 2.

Modelling the mechanical and electrical relationships of a generator.

Based on Figure 2, it is shown and can be explained that model block diagram electric generator on mechanical model expression is shown in Eq. (1) [8, 10, 12]:

$$T_m=J\bullet {\alpha }_m+T_D+T_e\left(\delta \right),$$E1

where:

$T_m$ = a mechanical input torque [N m];

$J$ = a moment of inertia of the turbine and rotor [s];

${\alpha }_m$ = an angular acceleration of the turbine and rotor [rad/s²];

$T_D$ = a damping torque [N m];

$T_e\left(\delta \right)$ = an equivalent electrical torque [N m].

In general, power is equal to torque multiplied by angular velocity or $P_m=T_m\bullet {\omega }_{\mathit{sm}}$. Hence, when a generator is spinning at a velocity of ${\omega }_s$, Eq. (1) becomes Eq. (2).

$$P_m=T_m\bullet {\omega }_{\mathit{sm}}=\left[J\bullet {\alpha }_m+T_D+T_e\left(\delta \right)\right]\bullet {\omega }_{\mathit{sm}},$$E2

which is simplified into an Eq. (3) or Eq. (4).

$$P_m=J\bullet {\alpha }_m\bullet {\omega }_{\mathit{sm}}+T_D\bullet {\omega }_{\mathit{sm}}+T_e\left(\delta \right)\bullet {\omega }_{\mathit{sm},}$$E3

$$P_m=J{\alpha }_m\bullet {\omega }_s+T_D\bullet {\omega }_{\mathit{sm}}+P_e\left(\delta \right),$$E4

where, $T_e\left(\delta \right)\bullet {\omega }_{\mathit{sm}}=P_e\left(\delta \right)$.

Initially, no damping is assumed, i.e., $T_D$ = 0, then Eq. (4) becomes Eq. (5) [8, 10, 12].

$$P_m-P_e\left(\delta \right)=J\bullet {\alpha }_m\bullet {\omega }_{\mathit{sm}},$$E5

$P_m$ is the mechanical power input, which is assumed to be constant throughout the study time period.

Regarding the rotor’s angle, it is described by Eq. (6).

$${\theta }_m={\omega }_{\mathit{sm}}\bullet t+{\delta }_m.$$E6

where, ${\theta }_m$ = the rotor angle. The change in the angle of the rotor to the attitude, therefore Eq. (6), becomes Eq. (7).

$${\omega }_m=\frac{d}{\mathit{dt}}{\theta }_m=\dot{{\theta }_m}=\frac{d}{\mathit{dt}}\left({\omega }_{\mathit{sm}}\bullet t+{\delta }_m\right)$$

or

$$\dot{{\theta }_m}={\omega }_{\mathit{sm}}+\dot{{\delta }_m}.$$E7

While the angular acceleration of the turbine and rotor changed with time, as explained by Eq. (8).

$${\alpha }_m=\frac{d}{\mathit{dt}}{\omega }_m=\dot{{\omega }_m}=\ddot{{\delta }_m}.$$E8

The substitution of Eq. (8) to Eq. (5) is obtained like Eq. (9).

$$P_m-P_e\left(\delta \right)=J\bullet {\omega }_s\bullet \ddot{{\delta }_m.}$$E9

where $J\bullet {\omega }_s$ is an inertia of the machine at synchronous velocity.

Convert to per unit by dividing by the MVA rating [8, 12, 10], $S_B$, explained by Eq. (10).

$$\frac{P_m}{S_B}-\frac{P_e\left(\delta \right)}{S_B}=\frac{J\bullet {\omega }_{\mathit{sm}}\bullet \ddot{{\delta }_m}}{S_B}\bullet \frac{2{\omega }_{\mathit{sm}}}{2{\omega }_{\mathit{sm}}}.$$E10

Simplification of Eq. (10) results in Eq. (11) since ${\omega }_{\mathit{sm}}=2\bullet \pi \bullet f_{\mathit{sm}}$.

$$\frac{P_m-P_e\left(\delta \right)}{S_B}=\frac{J\bullet {{\omega }_{\mathit{sm}}}^2}{2S_B}\bullet \frac{1}{\pi \bullet f_{\mathit{sm}}}\bullet \ddot{{\delta }_m}.$$E11

Define

$$\frac{J\bullet {{\omega }_{\mathit{sm}}}^2}{2S_B}=H.$$E12

where $H$ is a per unit inertia constant [s].

The substitution of Eq. (12) to Eq. (11) is obtained like Eq. (13).

$$\frac{P_m-P_e\left(\delta \right)}{S_B}=H\bullet \frac{1}{\pi \bullet f_{\mathit{sm}}}\bullet {\delta }_m.$$E13

All values are converted to per unit, Eq. (13) results in Eq. (14) since $M=\raisebox{1ex}{$H$}\!\left/ \!\raisebox{-1ex}{$\pi \bullet f_{\mathit{sm}}$}\right.$ [8, 10, 12].

$$P_m-P_e\left(\delta \right)=M\bullet \ddot{{\delta }_m}.$$E14

Eq. (14) is known as the generator swing eq. [8, 10, 12]. Adding a damping results in Eq. (15).

$$P_m-P_e\left(\delta \right)=M\bullet \ddot{{\delta }_m}+D\bullet \dot{{\delta }_m.}$$E15

The simplest generator model, known as the classical model, treats the generator as a voltage source behind the direct-axis transient reactance (${{\mathit{X}}_{\mathit{d}}}^{\prime }$); the voltage magnitude is fixed, but its angle changes according to the mechanical dynamics. The explanation in the form of an equation is obtained like Eq. (16) [8, 10, 12].

$$P_e\left(\delta \right)=\frac{\left|V_t\right|\bullet \left|E_a\right|}{{X_d}^{\prime }}sin\delta .$$E16

2.2 Methods of simulation

In order to do simulations, we can use the MATLAB application to create program structures, set up assumptions, algorithms, and syntax for the simulation process, and then run the programs to simulate the steady-state stability in the electrical power system. The flow diagram of the simulation implementation of the existence of the steady-state stability phenomenon in the electric power system simulated with the MATLAB application is shown in Figure 3.

Figure 3.

The flow diagram of the simulation implementation of the existence of the steady-state stability phenomenon in the electric power system simulated with the MATLAB application.

3. Results and discussion

3.1 Acquisition of the three syntax structures of the programme

Acquisition of three programme syntax structures for this simulation activity, including the structure of programme syntax for data input purposes on initial assumption values, the programme syntax structure for simulating the phenomena of power flows, and the structure of programme syntax for simulating the phenomena of steady-state stabilities with data on assumption values.

3.1.1 Structuring the programme syntax for inputting data on initial assumption values

For data input of assumed values, the existence of a MATLAB-based application syntax structure of the programme is required. The programme syntax structure for the input of assumed values is shown in Figure 4.

Figure 4.

The programme syntax structure for the input of assumed values.

Based on Figure 4, it is shown and can be explained that the programme syntax structure consists of (i) bus data, which includes bus number, bus type, and load power data (demand), (ii) generator data, which includes generation power, voltage, and base data, and (iii) inter-bus connextion impedance data.

3.1.2 Structuring the programme syntax for simulating the power flows phenomena

After the programme syntax structure for data entry of initial assumed values is created, it proceeds with the creation of a programme syntax structure for simulating the phenomenon of power flow. The structure of the programme syntax for simulating power flow phenomena is shown in Figure 5.

Figure 5.

The structure of the programme syntax for simulating power flow phenomena.

Based on Figure 5, it is shown that the programme syntax structure for simulating power flow phenomena is used to determine the mechanical power of each plant due to load changes, with operation through data input in the programme based on initial assumption values.

After obtaining data on the mechanical power value of each power generation machine, it is necessary to calculate the power flow with data on the values of the initial assumption. The initial mechanical power value is calculated with the programme syntax structure for the power flow phenomenon shown in Table 4.

Bus	Voltages		Power from synchronous generator		Power to Loads
	Mag. (p.u.)	Angle (°)	Pe (MW)	Qe (MVAr)	Pd (MW)	Qd (MVAr)
1	1.030	0.00	35.43	258.06	42.50	203.8
2	1.025	5.464	45.00	176.20	11.21	122.6
3	1.030	3.912	50.00	239.05	26.63	171.4
4	0.514	−0.723	—	—	42.18	147.5
		Total:	130.43	673.31	122.52	645,.0

Table 4.

The initial mechanical power value is calculated with the programme syntax structure for the power flow phenomenon.

3.1.3 Programme syntax structure for simulating the phenomena of steady-state stabilities

The final stage of creating a programme syntax structure that will be used to simulate steady-state stability through changes in load values on each bus. The structure of the programme syntax for simulating the phenomena of steady-state stability is shown in Figure 6.

Figure 6.

The structure of the programme syntax for simulating the phenomena of steady-state stability.

Based on Figure 6, it can be seen that the programme syntax structure for the steady-state stability phenomenon is made up of (i) mechanical power values as input data from power flow programmes; (ii) data on each machine in the form of voltage value, moment of inertia, dumping ratio, frequency, and resistance in the machine; and (iii) equations for finding the current, voltage, frequency, rotor angle, rotor angle delta, and angular velocity.

3.2 Simulation results

3.2.1 The result of the simulation on the phenomenon of steady-state stability uses the data from the initial assumption values

Some curves of the result of the simulation on the phenomenon of steady-state stability using the data from the initial assumption values are shown in Figure 7.

Figure 7.

Some curves of the result of the simulation on the phenomenon of steady-state stability using the data from the initial assumption values. (A) rotor angle curves (B) rotor angle difference curves (C) rotor angular velocity difference curves (D) frequency curves.

Based on Figure 7, it is shown and can be explained that the time for achieving stable conditions is preceded by machine #1, then machine #2, and finally machine #3 for all parameters. The values of a number of parameters are simulated with data assumption values, as shown in Table 5.

Parameters	Machines
	#1	#2	#3
(A) Rotor angle
The initial value unstable (°)	45.5198	36.7641	33.9222
Stable value (°)	35.5185	26.7676	23.9201
Stable time achievement (s)	0.55	0.72	0.91
(B) Rotor angle difference
The initial value unstable (°)	0.1745	0.1745	0.1745
Stable value (°)	0.0000	0.0000	0.0000
Stable time achievement (s)	0.48	0.64	0.68
(C) Rotor angular velocity difference
The initial value unstable (rad/s)	−0.4400	−0.3119	−0.359
Stable value (rad/s)	0.0000	0.0000	0.0000
Stable time achievement (s)	0.56	0.68	0.76
(D) Frequency
The initial value unstable (Hz)	59.9300	59.9504	59.9428
Stable value (Hz)	60.0000	60.0000	60.0000
Stable time achievement (s)	0.54	0.71	0.76

Table 5.

The values of a number of parameters are simulated with data assumption values.

Based on Table 5, it is shown and can be explained that there are a number of phenomena, namely (a) changes in rotor angle values when achieving stable, (b) achieving time when delta rotor reaches 0°, achieving stable, (c) achieving time when delta omega values reach 0 rad/s, achieving stable, and (d) achieving time when frequency values reach 60 hertz, achieving stable. The generating engine with the largest power generation (in this case, machine #3) takes longer to achieve a state of stability again after a change in load.

3.2.2 The result of the simulation on the phenomenon of steady-state stability uses the data from the assumption values for load changes on each bus

Depending on the needs of the user, today’s linked power networks may supply a range of loads. Naturally, these demands change continually, which causes the system’s loading to vary and has a variety of effects [26, 27]. Variation in loading has certain negative impacts, with the following listed as the most notable, namely (i) costlier power generation, (ii) complex system control, (iii) a need for more equipment, and (iv) higher losses are just a few of the consequences [27, 28, 29]. The reduction or addition of power due to the release or inclusion of a number of type and range of loads [25, 28, 29, 30]. In the simulation of the steady-state stability phenomenon, it is made with four conditions: (a) power reduction on bus #1, (b) power addition on bus #2, (c) power addition on bus #3, and (d) power reduction on bus #4.

3.2.2.1 The simulation on the phenomenon of steady-state stability uses the data of power reduction on bus #1

Changes in load power values, in the form of reducing active and reactive power from P_d = 42.50 MW and Q_d = 203.8 MVAr to P_d = 25.00 MW and Q_d = 125.5 MVAr. The value of the load power of each bus after the reduction of power on bus #1 is shown in Table 6.

Bus	Pd (MW)	Qd (MVAr)	Vmax. (p.u.)	Vmin. (p.u.)
1	25.00	125.5	1.1	0.9
2	11.21	122.6	1.1	0.9
3	26.63	171.4	1.1	0.9
4	42.18	147.5	1.1	0.9

Table 6.

The value of the load power of each bus after the reduction of power on bus #1.

The values in Table 6 are used for mechanical power gain after a change in power values on bus #1. The mechanical power values simulated with the power flow programme with the case of power change on bus #1 are shown in Table 7.

Bus	Voltages		Power from the synchronous generator		Power to loads
	Mag. (p.u.)	Ang. (°)	Pe (MW)	Qe (MVAr)	Pd (MW)	Qd (MVAr)
1	1.030	0.000	17.93	179.76	25.50	125.50
2	1.025	5.464	45.00	176.20	11.21	122.60
3	1.030	3.912	50.00	239.05	26.63	171.40
4	0.514	−0.723	—	—	42.18	147.50
Total value:			112.93	595.01	105.02	567.00

Table 7.

The mechanical power values simulated with the power flow programme with the case of power change on bus #1.

Some curves from the simulation of steady-state stability with power changes on bus #1 are shown in Figure 8.

Figure 8.

Some curves from the simulation of steady-state stability with power changes on bus #1. (A) Rotor angle curves, (B) rotor angle difference curves, (C) rotor angular velocity difference curves and (D) frequency curves.

Based on Figure 8, it is shown and can be explained that the time for achieving stable conditions is preceded by machine #1, then machine #2, and finally machine #3 for all parameters.

When the active power and reactive power at the connected load on bus #1 went from P_d = 42.50 MW and Q_d = 203.8 MVAr to P_d = 25.00 MW and Q_d = 125.5 MVAr, a number of things started to happen. A number of phenomena due to power reduction on bus #1 are shown in Table 8.

Phenomena	Mach.	Initial value (°)	Final value (°)	Difference (°)	Information
Changes in rotor angle value when stable conditions are reached	#1	35.5198	27.3672	8.1526	Smaller
	#2	26.7676	26.7676	0	Fixed value
	#3	23.9201	23.9201	0	Fixed value
Achievement of the time when the rotor delta is stable (at a value of 0°)	#1	0.48	0.55	0.07	Slower
	#2	0.64	0.64	0	Fixed value
	#3	0.68	0.91	0.23	Slower
Achievement of time when delta omega value reaches 0 rad/s (stable)	#1	0.56	0.68	0.12	Slower
	#2	0.68	0.73	0.05	Slower
	#3	0.76	0.76	0	Fixed value
Achievement of time when the frequency value reached 60 Hz (stable)	#1	0.54	0.61	0.07	Slower
	#2	0.71	0.71	0	Slower
	#3	0.76	0.76	0	Fixed value

Table 8.

A number of phenomena due to power reduction on bus #1.

3.2.2.2 The simulation on the phenomenon of steady-state stability uses the data of power addition on bus #2

Changes in load power values, in the form of adding active and reactive power from P_d = 11.21 MW and Q_d = 126.6 MVAr to P_d = 40.00 MW and Q_d = 200 MVAr. The value of the load power of each bus after the addition of power on bus #2 is shown in Table 9.

Bus	Pd (MW)	Qd (MVAr)	Vmax. (p.u.)	Vmin. (p.u.)
1	25.00	125.50	1.1	0.9
2	40.00	200.00	1.1	0.9
3	26.63	171.40	1.1	0.9
4	42.18	147.50	1.1	0.9

Table 9.

The value of the load power of each bus after the addition of power on bus #2.

The values in Table 9 are used for mechanical power gain after a change in power values on bus #2. The mechanical power values simulated with the power flow programme with the case of power change on bus #2 are shown in Table 10.

Bus	Voltages		Power from the synchronous generator		Power to loads
	Mag. (p.u.)	Ang. (°)	Pe (MW)	Qe (MVAr)	Pd (MW)	Qd (MVAr)
1	1.030	0.00	73.60	275.64	42.50	203.80
2	1.025	−2.137	45.00	282.56	40.00	200.00
3	1.030	−1.120	50.00	269.23	26.63	171.40
4	0.325	−0.723	—	—	42.18	147.50
Total value:			168.60	827.43	151.31	722.70

Table 10.

The mechanical power values simulated with the power flow programme with the case of power change on bus #2.

Some curves from the simulation of steady-state stability with power changes on bus #2 are shown in Figure 9.

Figure 9.

Some curves from the simulation of steady-state stability with power changes on bus #2. (A) Rotor angle curves, (B) rotor angle difference curves, (C) rotor angular velocity difference curves and (D) frequency curves.

Based on Figure 9, it is shown and can be explained that the time for achieving stable conditions is preceded by machine #1, then machine #2, and finally machine #3 for all parameters.

When the active power and reactive power at the connected load on bus #2 went from P_d = 11.21 MW and Q_d = 122.6 MVAr to P_d = 40.00 MW and Q_d = 200.00 MVAr, a number of things started to happen. A number of phenomena due to power reduction on bus #2 are shown in Table 11.

Phenomena	Mach.	Initial value (°)	Final value (°)	Difference (°)	Information
Changes in rotor angle value when stable conditions are reached	#1	35.5198	35.6074	0.0876	Greater
	#2	26.7676	38.4333	11.6657	Greater
	#3	23.9201	25.2952	1.3751	Greater
Achievement of the time when the rotor delta is stable (at a value of 0°)	#1	0.48	0.57	0.09	Slower
	#2	0.64	0.64	0	Fixed value
	#3	0.68	0.88	0.20	Slower
Achievement of time when delta omega value reaches 0 rad/s (stable)	#1	0.56	0.73	0.17	Slower
	#2	0.68	0.73	0.05	Slower
	#3	0.76	0.97	0.21	Slower
Achievement of time when the frequency value reached 60 HZ (stable)	#1	0.54	0.63	0.09	Slower
	#2	0.71	0.71	0	Fixed value
	#3	0.76	0.96	0.20	Slower

Table 11.

A number of phenomena due to power reduction on bus #2.

3.2.2.3 The simulation on the phenomenon of steady-state stability uses the data of power addition on bus #3

Changes in load power values, in the form of adding active and reactive power from P_d = 26.63 MW and Q_d = 171.4 MVAr to P_d = 35.75 MW and Q_d = 195.5 MVAr. The value of the load power of each bus after the addition of power on bus #2 is shown in Table 9.

The values in Table 12 are used for mechanical power gain after a change in power values on bus #3. The mechanical power values simulated with the power flow programme with the case of power change on bus #3 are shown in Table 13.

Bus	Pd (MW)	Qd (MVAr)	Vmax. (p.u.)	Vmin. (p.u.)
1	25.00	125.50	1.1	0.9
2	40.00	200.00	1.1	0.9
3	35.75	195.50	1.1	0.9
4	42.18	147.50	1.1	0.9

Table 12.

The value of the load power of each bus after the addition of power on bus #3.

Bus	Voltages		Power from the synchronous generator		Power to loads
	Mag. (p.u.)	Ang. (°)	Pe (MW)	Qe (MVAr)	Pd (MW)	Qd (MVAr)
1	1.030	0.00	64.40	303.52	42.50	203.80
2	1.025	5.464	45.00	231.56	11.21	122.60
3	1.030	3.912	50.00	328.09	35.75	195.50
4	0.114	−0.723	—	—	42.18	147.50
Total value:			159.40	863.17	131.64	669.40

Table 13.

The mechanical power values simulated with the power flow programme with the case of power change on bus #3.

Some curves from the simulation of steady-state stability with power changes on bus #3 are shown in Figure 10.

Figure 10.

Some curves from the simulation of steady-state stability with power changes on bus #3. (A) Rotor angle curves, (B) rotor angle difference curves, (C) rotor angular velocity difference curves and (D) frequency curves.

Based on Figure 10, it is shown and can be explained that the time for achieving stable conditions is preceded by machine #1, then machine #2, and finally machine #3 for all parameters.

When the active power and reactive power at the connected load on bus #3 went from P_d = 26.63 MW and Q_d = 171.4 MVAr to P_d = 35.75 MW and Q_d = 195.5 MVAr, a number of things started to happen. A number of phenomena due to power reduction on bus #3 are shown in Table 14.

Phenomena	Mach.	Initial value (°)	Final value (°)	Difference (°)	Information
Changes in rotor angle value when stable conditions are reached	#1	35.5198	38.4580	2.9382	Greater
	#2	26.7676	33.1932	6.4256	Greater
	#3	23.9201	32.3113	8.912	Greater
Achievement of the time when the rotor delta is stable (at a value of 0°)	#1	0.48	0.57	0.09	Slower
	#2	0.64	0.64	0	Fixed value
	#3	0.68	0.71	0.03	Slower
Achievement of time when delta omega value reaches 0 rad/s (stable)	#1	0.56	0.73	0.17	Slower
	#2	0.68	0.73	0.05	Slower
	#3	0.76	0.98	0.22	Slower
Achievement of time when the frequency value reached 60 Hz (stable)	#1	0.54	0.63	0.09	Slower
	#2	0.71	0.71	0	Fixed value
	#3	0.76	0.79	0.03	Slower

Table 14.

A number of phenomena due to power reduction on bus #3.

3.2.2.4 The simulation on the phenomenon of steady-state stability uses the data of power reduction on bus #4

Changes in load power values, in the form of reducing active and reactive power from P_d = 42.18 MW and Q_d = 147.5 MVAr to P_d = 10.15 MW and Q_d = 60.75 MVAr. The value of the load power of each bus after the reduction of power on bus #1 is shown in Table 15.

Bus	Pd (MW)	Qd (MVAr)	Vmax. (p.u.)	Vmin. (p.u.)
1	25.00	125.50	1.1	0.9
2	40.00	200.00	1.1	0.9
3	35.75	195.50	1.1	0.9
4	10.15	60.75	1.1	0.9

Table 15.

The value of the load power of each bus after the reduction of power on bus #4.

The values in Table 15 are used for mechanical power gain after a change in power values on bus #4. The mechanical power values simulated with the power flow programme with the case of power change on bus #4 are shown in Table 16.

Bus	Voltages		Power from the synchronous generator		Power to loads
	Mag. (p.u.)	Ang. (°)	Pe (MW)	Qe (MVAr)	Pd (MW)	Qd (MVAr)
1	1.030	0.00	−2.18	219.65	42.50	203.80
2	1.025	8.836	45.00	124.71	11.21	122.60
3	1.030	7.622	50.00	180.21	26.63	171.40
4	0.887	5.360	—	—	10.15	60.75
Total value:			92.82	524.57	90.49	558.55

Table 16.

The mechanical power values simulated with the power flow programme with the case of power change on bus #4.

Some curves from the simulation of steady-state stability with power changes on bus #4 are shown in Figure 11.

Figure 11.

Some curves from the simulation of steady-state stability with power changes on bus #4. (A) Rotor angle curves, (B) rotor angle difference curves, (C) rotor angular velocity difference curves and (D) frequency curves.

Based on Figure 11, it is shown and can be explained that the time for achieving stable conditions is preceded by machine #1, then machine #2, and finally machine #3 for all parameters.

When the active power and reactive power at the connected load on bus #4 went from P_d = 42.18 MW and Q_d = 147.5 MVAr to P_d = 10.15 MW and Q_d = 60.75 MVAr., a number of things started to happen. A number of phenomena due to power reduction on bus #4 are shown in Table 17.

Phenomena	Mach.	Initial value (°)	Final value (°)	Difference (°)	Information
Changes in rotor angle value when stable conditions are reached	#1	35.5198	33.3549	2.1649	Smaller
	#2	26.7676	20.1882	6.5794	Smaller
	#3	23.9201	18.5109	5.4092	Smaller
Achievement of the time when the rotor delta is stable (at a value of 0°)	#1	0.48	0.63	0.15	Slower
	#2	0.64	0.64	0	Fixed value
	#3	0.68	0.90	0.22	Slower
Achievement of time when delta omega value reaches 0 rad/s (stable)	#1	0.56	0.73	0.17	Slower
	#2	0.68	0.76	0.08	Slower
	#3	0.76	0.98	0.22	Slower
Achievement of time when the frequency value reached 60 hertz (stable)	#1	0.54	0.69	0.15	Slower
	#2	0.71	0.71	0	Fixed value
	#3	0.76	0.98	0.22	Slower

Table 17.

A number of phenomena due to power reduction on bus #4.

3.2.2.5 Summary of the simulation results of the steady-state stability phenomenon with data on changes in load power values on each bus

Referring to a number of tables, including Tables 8,11,14 and 17 are shown and can be explained that there are a number of phenomena, namely (a) changes in rotor angle values when achieving stable, (b) achieving time when the rotor delta reaches 0° (stable), (c) achieving time when the omega delta value reaches 0 rad/s (stable), and (d) achieving time when the frequency value reaches 60 hertz (stable). The generating engine with the largest power generation (in this case, machine #3) takes longer to achieve a state of stability again after a change in load.

4. Conclusions

Based on a number of descriptions in the discussion, conclusions were drawn according to the purpose of the study. The programme syntax structure consists of (i) bus data, which includes bus number, bus type, and load power data (demand); (ii) generator data, which includes generation power, voltage value, and base data; and (iii) inter-bus connection impedance data; The programme syntax structure for simulating power flow phenomena is used for the determination of the mechanical power of each power generation machine due to load changes, with operation through data input in the programme input data of initial assumption values. The initial mechanical power values inputted to the programme syntax structure for the steady-state stability phenomenon were 35.43 MW for machine #1, 45 MW for machine #2, and 50 MW for machine #3. The programme syntax structure for the steady-state stability phenomenon consists of (i) mechanical power values as input data obtained from the results of calculating the power flow programme; (ii) data on each machine in the form of voltage value, moment of inertia, dumping ratio, frequency, and resistance in the machine; and (iii) equations for finding the current, voltage, frequency, rotor angle, rotor angle delta, angular velocity delta, and maximum power values for each machine affecting instability.

The simulation results of the steady-state stability phenomenon with data on initial assumption values in the form of the emergence of a number of phenomena, namely (a) changes in rotor angle values when achieving stable, (b) achieving time when the rotor delta reaches 0° (stable), (c) achieving time when the delta omega value reaches 0 rad/s (stable), and (d) achieving time when the frequency value reaches 60 hertz (stable). The generating machine with the largest power generation (in this case, machine #3) takes longer to achieve a state of stability again after a change in load. The phenomena in simulating power changes in the installed load on each bus according to the research objectives, namely (a) the active power in machine #2 and machine #3 are always fixed, but the reactive power changes for voltage regulation, so that the generator voltage is equal to the system voltage; (b) the rotor angle value is directly proportional to the change in load power, if the power load rises, the angle value rises and vice versa, so that the condition affects the change in mechanical power, which is when the mechanical power stabilises according to the load power condition, where the mechanical power affects the rotor angle value; (c) the achievement of stable time of rotor angle difference (delta rotor) and stable time of machine #2 frequency is not affected by changes in load power; (d) changes in load power result in achieving a stable time difference in rotor angular velocity (delta omega) relatively slower; (e) generators with large capacities need slower stable achievement times, in which case the capacity on machine #1 is smaller than machine #2 and smaller than machine #3. The damping ratio is inversely proportional to the value of the load power; if the load power rises, the damping ratio decreases; and (f) the system returns to stability when the latter reaches a stable time, i.e. in terms of similarity of frequency, phase sequence, and phase angle. In this simulation, after the machine 3 reaches a stable time.

Completing the conclusions, recommendations are submitted, namely in the implementation of simulations for the analogy of the phenomenon of steady-state stability; it is still necessary to develop by analogy the structure of a larger and more complex electric power system, also through the provision of conditions of simultaneous change on each bus.