Multi-Objective Optimal Sizing of Shunt Braking Resistor for Transient State Improvement

Power system transient stability can be effectively improved by applying shunt braking resistors. This paper proposes multi-objective heuristics-based optimization for shunt braking resistor sizing. The proposed approach addresses three objectives: transient angle stability, transient voltage response, and mechanical stress of the turbine-generator shaft. The optimization problem was solved using a Python implementation of the multi-objective evolutionary NSGA-II algorithm. Based on the optimization model, comprehensive tests for multimachine IEEE 39-bus power system including single- and multi-objective simulations were performed. Pareto sets for various sets of objectives are obtained and discussed. The results show the complexity of the shunt braking resistor optimization process. Optimization studies were complemented by simulation tests performed for assessing of the impact of optimal shunt braking resistor on power system dynamic response. Detailed discussion of test results has confirmed the correctness of the proposed approach.

SECTION I. Introduction

A. Motivation

The power system is developed in such a way as to ensure its effective and secure operation. However, in some areas large amounts of electric energy are transmitted from power plants over long distances [1] or there are delays in the transmission network development process for reasons over which the investor has no control of [2]. In addition, more electricity generated from renewable energy sources exposes conventional power plants to increased stress during disturbances [3]. In such conditions, a severe disturbance like a fault near the power plant may lead to power system instability.

The loss of transient stability in modern power systems leads to extremely high costs. Since the security of energy supply over time is increasingly important, stability will play a key role in the future. Consequently, utility engineers often introduce various corrective measures, also called emergency controls, to avoid the problem [4].

Dynamic braking using a resistor is considered to be one of the promising solutions and an effective and an economically feasible way to enhance transient stability [5], [6]. An adequate braking resistor application to enhance transient stability of a power system requires selecting: (i) the optimal location, (ii) the parameter of the braking resistor, (iii) the algorithm to control the braking resistor. The location of the (series or shunt) braking resistor on the generator feeder is considered to be the most effective measure to improve transient stability [7]. Regarding the selection of braking resistance, power system operators must take into consideration multiple objectives due to various security limits (i.e. angle and voltage stability limits) and mechanical limits (i.e. turbine-generator shaft fatigue). The selection of braking resistance is followed by the control algorithm determination of the braking resistor.

This paper aims at the comprehensive sizing of the shunt braking resistor (ShBR) for transient state improvement. The problem of shunt braking resistor selection is formulated as a multi-objective optimization problem with such objectives as transient stability, transient voltage response, and mechanical stress of the turbine-generator shaft. A two-stage optimization process has been proposed, which includes: (1) finding a set of Pareto front solutions, (2) determining the best acceptable solution using a fuzzy-set approach.

B. Literature Review

Severe disturbances, such as three-phase short-circuits near the power plant pose a high risk, as the loss of stability can lead to generators tripping and loss of power balance in the electric power system [7]. To reduce the risk of power system instability, various corrective measures are used, including dynamic braking [8], [9]. This paper focuses on using dynamic braking to improve transient stability taking also into account the effect of braking on various aspects of power system operation.

In general, series or shunt elements can be used for dynamic braking [10]. Due to the cost and technical advantages, resistive braking has found a variety of applications [6]. Recently, the application of series fault current limiters has been proposed for power system stability improvement. However, at the current stage of technology development, series elements are mostly used for transient stability enhancement of generators with low and medium power ratings or they need complex connection circuit [6], [11].

As an alternative to series elements, shunt resistors may be used to provide the braking effect. The effect of connecting an ShBR on the power system transient stability is described in [7]. To switch on an ShBR, a mechanical circuit-breaker or power electronics devices can be used. A survey of early application of a mechanically switched ShBR is reported in [12]. It is proposed to apply shunt resistors for braking hydro [13] or thermal [14] units. Current studies on ShBRs mostly focus on brake controller design [15], [16], and coordinated braking with fast valving [9].

Although selecting the parameter of the braking resistor is a very important step when an ShBR is used for transient stability improvement, only a limited research has been conducted on this topic. In [15], a brake controller is designed for one arbitrarily defined value of braking resistor’s resistance. In [9], studies are conducted for a set of the ShBR resistances using an approach similar to the trial and error method. In [16], one objective function, i.e. the maximum power transfer theorem, is used to find the best size of the brake to provide the maximum power transfer which helps to increase the stability margin of the system. If the ShBR is applied to thermal units, the effect on mechanical stress or shaft fatigue must be examined [10]. Therefore, [14] proposes damping of transient torques as a criterion for choosing braking resistor values.

Severe disturbance in a bulk power system simultaneously affects rotor angles, voltage magnitudes, and shaft mechanical stress. Therefore, various security indicators [17], [18] should be considered to assess power system response and to select ShBR resistance. In this case, the selection of the ShBR resistance can be analyzed as a multi-objective task. Out of many multi-objective optimization algorithms [19] available, Non-dominated Sorting Genetic Algorithm-II, a fast and elitist multi-objective genetic algorithm, has been recently applied as an acceptable tool [18], [20].

The authors of this paper note the fact that designing dynamic braking is a multi-objective optimization task that must consider the effect of dynamic braking on various aspects of power system operation, including transient voltage response.

C. Contribution and Organization

The main contributions of this paper are as follows:

1. This paper proposes a multi-objective heuristics-based optimization method for shunt braking resistor sizing.

2. A set of objectives and constraints for the comprehensive sizing of shunt braking resistor is proposed.

3. Single- and multi-objective optimization approaches are compared.

4. A methodology based on Python programming language is also proposed for solving the multi-objective optimization problem.

This paper is organized in the following way: Section II states the problem, describing selected objectives and constraints which should be considered when selecting the size of an ShBR. Section III provides a brief description of the proposed implementation of the optimization procedure, including multi-objective optimization procedure, NSGA-II algorithm and the best compromise solution method. Section IV describes the results of simulation tests for single-machine and multi-machine test systems. Section V provides conclusions.

SECTION II. Problem Formulation

A. Shunt Braking Resistor

This paper solves the problem of the ShBR sizing for transient state improvement in bulk power systems. The transient state is caused by severe three-phase faults in transmission lines near power plant substation. It is assumed that an ShBR with mechanical circuit breakers for connecting and disconnecting the braking resistor from the circuit is used. The connection of the ShBR to the generator unit’s circuit is shown in Figure 1.

FIGURE 1.

Power system with ShBR device.

During a system contingency, the ShBR must be connected as fast as possible. In addition, to reduce mechanical stress of the generator shaft, it is assumed that during the transient state the ShBR is connected to the circuit only once to improve first swing stability.

B. Objectives

This paper proposes a multi-objective optimization method for the ShBR resistance sizing. A ShBR connected to the circuit of a generating unit not only affects rotor speed during and after a fault, but also affects nodal voltages and mechanical quantities. On this basis, the proposed approach tries to simultaneously optimize the following three objectives.

1) Transient Angle Stability

Fault clearing time is a major factor in determining the generator stability [7], [17]. The relative difference between the longest clearing time for which the generator will remain in synchronism and the actual clearing time can be provided by a measure of the transient stability margin:

$$\begin{equation*} k_{\textrm {t}} =\frac {t_{\textrm {CCT}} -t_{\textrm {SHC}}}{t_{\textrm {SHC}} }\tag{1}\end{equation*}$$ where $t_{\textrm {CCT}}$ is critical clearing time and $t_{\textrm {SHC}}$ is actual clearing time resulting from the operation of the protection.

In this paper, to assess how the ShBR resistance $R_{\textrm {sh}}$ enhances transient angle stability, the following function based on the transient stability index is used:

$$\begin{equation*} f_{1} \left ({{R_{\textrm {sh}}} }\right)=-k_{\textrm {t}}\tag{2}\end{equation*}$$

2) Transient Voltage Response

Dynamic braking based on connecting a large artificial load can significantly affect voltage response of a synchronous generator during a fault and after its clearance, which may lead to further instability. Hence, the optimization approach takes into account a quantitative assessment of transient voltage deviation using Voltage Deviation Security Margin Index (VSDMI) introduced in [21]:

$$\begin{equation*} \eta =\frac {s_{\textrm {d}}}{k_{\eta } \left ({{v_{\textrm {n}} -c_{\textrm {cr}}} }\right)t_{\textrm {cr}}}\tag{3}\end{equation*}$$ where $s_{\textrm {d}}$ is defined as the violation area limited by the curves of voltage $v\left ({t }\right)$ and security limit $v_{\textrm {cr}}$ , denominator $k_{\eta } \left ({{v_{\textrm {n}} -v_{\textrm {cr}}} }\right)t_{\textrm {cr}}$ describes the area limited by rated voltage $v_{\textrm {n}}$ and voltage security limit within the analysis time window $t_{\textrm {cr}}$ (Figure 2). Index $k_{\eta } =1$ is introduced in the formula for the analysis of the low voltage deviation security margin, $k_{\eta } =-1$ occurs in the formula for high voltage deviation security margin.

FIGURE 2.

Voltage response curve for the definition of the voltage deviation security margin index.

Depending on whether the voltage curve passes the defined voltage security limit $v_{\textrm {cr}}$ , $s_{\textrm {d}}$ area can be formulated as follows:

$$\begin{align*}&s_{\textrm {d}} \\&\quad \!\!=\,\,\begin{cases} {k_{\eta } \left \{{{\left ({{v_{\textrm {n}} -v_{\textrm {cr}}} }\right)t_{\textrm {cr}} -\displaystyle \int \limits _{t_{\textrm {s}}}^{t_{\textrm {s}} \!+\!t_{\textrm {cr}}} {\left [{ {v_{\textrm {n}} \!-\!v\left ({t }\right)} }\right]\textrm {d}t}} }\right \}} \\ \qquad \qquad \qquad \qquad \qquad \quad ~\quad \qquad \qquad \;\textrm {if}\;\textrm {t}_{\textrm {b}} \!=\!0 \\ {\displaystyle \sum \nolimits _{\textrm {i=1}}^{\textrm {N}} {\displaystyle \int _{t_{\textrm {si}} }^{t_{\textrm {si}} +t_{\textrm {bi}}} {k_{\eta } \left [{ {v\left ({t }\right)-v_{\textrm {cr}}} }\right]\textrm {d}t,\;\;\textrm {if}\;\;\textrm {t}_{\textrm {b}} \ne 0}}} \end{cases}\tag{4}\end{align*}$$ where $t_{\textrm {si}}$ and $t_{\textrm {bi}}$ denote the beginning time and the duration of the $i$ -th violation of the voltage security level, respectively, $N$ is the number of the events in which $v_{\textrm {cr}}$ voltage is exceeded.

The presented Voltage Deviation Security Margin Index allows us to assess transient voltage response regardless of whether security voltage level has been passed or not. Therefore, weighted average of VDSMI indicators is used in this paper to formulate the objective function, which determines the impact of the ShBR resistance on transient voltage response, according to the following equation:

$$\begin{equation*} f_{2} \left ({{R_{\textrm {sh}}} }\right)=-\textrm {max}\left \{{ {\frac {w_{\textrm {h}} \eta _{\textrm {h}g} +w_{1} \eta _{1g}}{w_{\textrm {h}} +w_{1}}} }\right \}_{g=1,2,\ldots,N_{\textrm {G}}}\tag{5}\end{equation*}$$ where indicators $\eta _{\textrm {h}g}$ and $\eta _{\textrm {l}g}$ refer to voltage deviation security margin for high and low voltage limit crossing for in the $g$ -th synchronous generator, $w_{\textrm {h}}$ and $w_{\textrm {l}}$ are weight factors for each of the analyzed voltage index, respectively, and $N_{\textrm {G}}$ is the number of considered synchronous generators.

3) Mechanical Stress of the Turbine-Generator Shaft

In general, the ShBR can be used for braking hydro or thermal generating units. The application of the ShBR for thermal generating units in particular requires the analysis of the effect it has on the mechanical stress of the turbine-generator shaft. Step changes in power that arise during the ShBR switching can cause extreme mechanical stress in a multi-mass shaft system, with potentially adverse effects on torsional oscillations of the shaft.

As shown in Figure 3, a three-phase short circuit on transmission lines near a power plant substation and the ShBR operation produce transient electrical power in which three step-changes $\Delta P_{1}$ , $\Delta P_{2}$ and $\Delta P_{3}$ can be distinguished. The first step-change $\Delta P_{1}$ is due to the short circuit fault event. The second step-change $\Delta P_{2}$ is due to fault clearing, and the third one $\Delta P_{3}$ is due to switching off the ShBR. The time after which the ShBR is switched off is determined by the control algorithm of the ShBR. Detailed analysis of mechanical fatigue of the turbine- generator multi-mass shaft system is very complex [22]. This paper uses a simplified approach based on integral torsional torque index to minimize mechanical effects due to multiple switching during the fault and its clearance:

$$\begin{equation*} f_{3} \left ({{R_{\textrm {sh}}} }\right)=\max \limits _{g=1,2,3,\ldots,N_{\textrm {G}}} \left \{{{\int _{t=0}^{t_{\textrm {k}}} {\left ({{\left |{ {\tau _{g} -\tau _{0}} }\right |} }\right)\textrm {d}t}} }\right \}\tag{6}\end{equation*}$$ where $\tau _{g}$ is the torque between the low-pressure turbine and synchronous generator in the $g$ -th turbine-generator shaft system, and $\tau _{0} \tau _{0}$ is the initial value of torque before the considered fault event occurs.

FIGURE 3.

Step changes of generator’s active power output during a fault event with dynamic braking.

C. Constraints

1) Transient Angle Stability Constraints

Transient stability constraints are implemented by using generator rotor angles with respect to the centre of inertia of all the generators. In general, angle transient stability constraints can be expressed as follows:

$$\begin{equation*} \delta ^{\min }\le \delta _{g} \left ({t }\right)-\delta _{\textrm {COI}} \left ({t }\right)\le \delta ^{\max }\tag{7}\end{equation*}$$ where: $\delta _{g} \left ({t }\right)$ - rotor angle of $g$ -th generator $(g=1,2,3,\ldots, N_{\textrm {G}})$ , $N_{\textrm {G}}$ is the number of generators, $\delta ^{\min }$ , $\delta ^{\max }$ are the minimum and maximum rotor angle differences limits, respectively, $\delta _{\textrm {COI}} \left ({t }\right)$ is the centre of inertia defined as follows: $$\begin{equation*} \delta _{\textrm {COI}} \left ({t }\right)=\frac {1}{M_{\textrm {T}} }\sum \limits _{g=\textrm {1}}^{N_{\textrm {G}}} {\left [{ {M_{g} \Delta \delta _{g} \left ({t }\right)} }\right]},\quad M_{\textrm {T}} =\sum \limits _{g=\textrm {1}}^{N_{\textrm {G}}} {M_{g}}\tag{8}\end{equation*}$$

2) Transient Voltage Response Constraints

Transient voltage response of the generating unit terminals, during the fault and after its clearance, is important for maintaining a secure operation of the generating unit auxiliary service. Continuous voltage deviation for a short period may cause serious damage to the auxiliary services motor drives of power units, therefore voltage fault ride-through (FRT) requirements specified in grid codes need to be met [23].

During the fault and after it its clearance, the voltage curve must remain between two limitation curves: the upper one, HVRT (high voltage ride through), and the lower one, LVRT (low voltage ride through). To solve the optimization problem, voltage limitation curves for LVRT and HVRT are established according to documents [23], [24]. Figure 4 shows example limitations of voltage determined by two broken lines: the upper HVRT and the lower LVRT, and also shows an example of dynamic voltage response during a fault in the transmission line and after its clearance (bold line).

FIGURE 4.

Example of voltage ride through limits, based on [23], [24].

The FRT transient voltage response constraints can be expressed by the following integral indicators:

$$\begin{align*} h_{1} \left ({{R_{\textrm {sh}}} }\right)=&\int \limits _{0}^{t_{\textrm {k}}} {\left |{ {\varepsilon _{\textrm {HVRT}} \left ({t }\right)} }\right |} \textrm {d}t= \textrm {0} \tag{9}\\ h_{2} \left ({{R_{\textrm {sh}}} }\right)=&\int \limits _{0}^{t_{\textrm {k}}} {\left |{ {\varepsilon _{\textrm {LVRT}} \left ({t }\right)} }\right |} \textrm {d}t= \textrm {0}\tag{10}\end{align*}$$ where $\varepsilon _{\textrm {HVRT}}$ and $\varepsilon _{\textrm {LVRT}}$ are the deviations of the analyzed quantities, respectively and $t_{\mathrm {k}}=10$ s is the length of analysis time window for the voltage signal. For simulation tests, the following deviations are proposed: $$\begin{align*} \varepsilon _{\textrm {HVRT}} \left ({t }\right)=&V\left ({t }\right)-HVRT\quad \textrm {for}\;V\left ({t }\right) > HVRT \tag{11}\\ \varepsilon _{\textrm {HVRT}} \left ({t }\right)=&0\quad \textrm {for}\;V\left ({t }\right)\le HVRT \tag{12}\\ \varepsilon _{\textrm {LVRT}} \left ({t }\right)=&V\left ({t }\right)-LVRT\;\textrm {for}\;V\left ({t }\right) < LVRT \tag{13}\\ \varepsilon _{\textrm {LVRT}} \left ({t }\right)=&0\quad \textrm {for}\;V\left ({t }\right)\ge LVRT\tag{14}\end{align*}$$

3) Mechanical Stress of Turbine-Generator Shaft Constraints

For a switching operation in which the ShBR is switched off, the following constraint can be applied, according to guidance in [25]:

$$\begin{equation*} \Delta P_{3} \le 0.5P_{\textrm {n}}\tag{15}\end{equation*}$$

From (15) it results that ShBR switching off should result in step changes in real power of generating units less than a half of their rated power $P_{\textrm {n}}$ .

D. Optimization Problem

The multi-objective optimization model which minimizes three objectives (2), (5) and (6) for optimal resistance sizing of the ShBR can be expressed as follows:

$$\begin{align*} \min \limits _{R_{\textrm {sh}}} F\!=\!\left [{ {f_{1} \left ({{R_{\textrm {sh}},{\mathbf { x}},{\mathbf { u}},{\mathbf { y}}} }\right),f_{2} \left ({{R_{\textrm {sh}},{\mathbf { x}},{\mathbf { u}},{\mathbf { y}}} }\right),f_{3} \left ({{R_{\textrm {sh}},{\mathbf { x}},{\mathbf { u}},{\mathbf { y}}} }\right)} }\right] \\\tag{16}\end{align*}$$ subject to differential-algebraic equations: $$\begin{align*} \begin{cases} {{\dot {\mathbf {u}}}={\mathbf { h}}\left ({{{\mathbf { x}},{\mathbf { u}},{\mathbf { y}}} }\right)} \\ {0={\mathbf { g}}\left ({{{\mathbf { x}},{\mathbf { u}},{\mathbf { y}}} }\right)} \end{cases}\tag{17}\end{align*}$$ where $R_{\textrm {sh}}$ is the rated resistance of the ShBR, ${\mathbf { x}}$ is the system states vector, ${\mathbf { u}}$ is the control signals vector, ${\mathbf { y}}$ is the output signals vector. And this is also subject to technical constraints given by (7)–​(15).

Besides, it is assumed that the ShBR rated resistance can be selected from a limited range of possible values specified by the user. Thus, constraints for $R_{\textrm {sh}}$ are given by:

$$\begin{equation*} R_{\textrm {sh}\min } \le R_{\textrm {sh}} \le R_{\textrm {sh}\max }\tag{18}\end{equation*}$$

SECTION III. Implementation of the Optimization Procedure

A. Multi-Objective Optimization Procedure

To obtain multi-objective sizing of the ShBR, the calculation procedure presented in Figure 5 has been formulated. The optimization process is as follows.

FIGURE 5.

Multi-objective optimization procedure diagram.

The main component of the methodology presented below is a Python-based program which includes modules responsible for conducting the respective steps of the optimization process. Transient state simulations were conducted in DIgSILENT PowerFactory power systems analysis software. One of the user-written modules allows us to run PF software in the calculation engine mode to maintain a fast and reliable data exchange of the results between simulation and optimization packages.

The user-made Python module responsible for the calculation of the objectives then processes the obtained simulation result model and the values of the constraints functions. In the next steps, prepared data sets with the results are used by a Python-based NSGA-II implementation of the optimization algorithm. The presented steps are consecutively repeated to obtain the following populations forming the set, including feasible Pareto optimal solutions. Finally, the NSGA-II algorithm results are processed by a fuzzy logic selection method to find the best compromise solution for the analyzed problem.

B. NSGA-II Algorithm

The considered optimization problem was solved using Python implementation of the multi-objective evolutionary NSGA-II algorithm (Nondominated Sorting Genetic Algorithm), which is presented in detail in [26]. The presented algorithm is a significant expansion of the original NSGA concept due to the following improvements [27]:

• reduction of the algorithmic complexity to $O(\kappa ^{2}c)$ , where $\kappa$ is the size of the population and $c$ is the number of objectives;

• replacement of the original sharing method by a niching technique without additional parameters;

• implementation of elitism in order to improve the algorithm convergence.

Due to the above features, NSGA-II method is considered to be one the most effective reference methods for multi-objective evolutionary optimization, therefore it was chosen to solve the considered problem. A brief illustration of the algorithm procedure is presented in Figure 6.

FIGURE 6.

NSGA-II algorithm procedure.

In the initial step of the algorithm, $\kappa$ individuals form a random parent population $G_{0}$ . Then, Pareto ranking using a fast non-dominated sorting approach assigns all individuals to groups. For each solution, the rank level is assigned according to its nondomination level. In the following step, the offspring population $Q_{0}$ is created using binary tournament selection, recombination and mutation genetic operators. The binary tournament selection method, specific to NSGA-II, is referred to as crowded tournament selection, which assumes to favor the selection of individuals with the equal lowest possible nondomination rank in sparsely populated areas using crowded-comparison operator $\prec _{\textrm {n}}$ [27].

Once initial parent and offspring populations are prepared, consecutive $p$ -th generations of solutions are created based on the following procedure. Offspring population $Q_{p}$ is formed from the parent population $G_{p}$ by the application of crowded tournament selection, crossover and mutation operators in the presented order. Then, the obtained combined union population $U_{p} =G_{p} \cup Q_{p}$ is sorted according to the nondomination of the individuals to form new subpopulations $H_{\textrm {r}}$ , where $r$ denotes the number of the rank level.

The next population $G_{p+1}$ is formed of the solutions in the Pareto ranked subpopulations $H_{1}$ to $H_{L}$ , where $L$ is the greatest integer such that the sum of the sizes of these subpopulations is less than or equal to $\kappa$ . To fill up the formed population to the required size, the best solutions from the following $H_{l+1}$ subpopulation are chosen using an assignment method based on crowded-comparison operator $\prec _{\textrm {n}}$ , until the number of the individuals is equal to $\kappa$ .

C. Best Compromise Solution

Once the Pareto set $G_{\textrm {opt}}$ is obtained by the optimization procedure, it is suggested to choose one solution from all feasible Pareto-optimal solutions, which would be the best compromise equally satisfying all objectives considered here. This paper contemplates a fuzzy-logic selection method based on membership functions [28].

Membership functions describe the optimization degree of a result, taking into account each objective function. The fuzzy membership of $i$ -th objective function of the $j$ -th Pareto optimal solution can be expressed as follows:

$$\begin{align*} \mu _{ij} \left ({{R_{\textrm {sh}}} }\right)=\begin{cases} {1,} & {\textrm {if}} ~{f_{ij} \le f_{i}^{\min }} \\ {\dfrac {f_{j}^{\max } -f_{ij}}{f_{i}^{\max } -f_{i}^{\min }},} & {\textrm {if}} ~{f_{i}^{\min } < f_{ij} < f_{i}^{\max }} \\ {0,} & {\textrm {if}} ~{f_{ij} \ge f_{i}^{\max }} \\ \end{cases}\tag{19}\end{align*}$$ where $f_{i}^{\min }$ and $f_{i}^{\max }$ are the minimum and the maximum values of the $i$ -th objective function from among all non-dominated solutions. The membership function $\mu _{ij}$ is a real value between 0 and 1, where $\mu _{ij} =0$ indicates the incompatibility of the solution with the set, while $\mu _{ij} =1$ means full compatibility. For each non-dominated solution $j$ , the normalized membership function is calculated as: $$\begin{equation*} \tilde {\mu }_{j} \left ({{R_{\textrm {sh}}} }\right)=\frac {\sum \limits _{i=\textrm {1}}^{N_{1}} {\mu _{ij}} }{\sum \limits _{j=1}^{N_{2}} \sum \limits _{i=\textrm {1}}^{N_{1}} {\mu _{ij}}}\tag{20}\end{equation*}$$ where $N_{2}$ is the number of non-dominated solutions in the Pareto set and $N_{1}$ is the number of objective functions. The solution with the maximum $\tilde {\mu }_{j}$ value is regarded as the best unbiased solution, which corresponds to final optimal $\tilde {R}_{\textrm {sh}} \in G_{\textrm {opt}}$ ShBR resistance value.

SECTION IV. Simulation Results and Discussion

A. Multi-Machine System Test Case

The multi-objective ShBR sizing procedure for a multi-machine system was performed on the IEEE 10-Machine 39-Bus Power System presented in [29] and shown in Figure 7. Preliminary critical clearing times for various three-phase fault locations were determined. The results show that the shortest critical clearing time is when a fault occurs at busbar B29 (near generator G9). In such a case, generator G9 is the first to lose synchronism and due to that, the ShBR location in the generator G9 feeder and the fault near G9 are selected for the analysis. The ShBR cannot replace fundamental remedial measures to be implemented in the transmission network.

FIGURE 7.

Diagram of the IEEE 10-Machine 39-Bus power system [29].

Therefore, only a single ShBR installation to enhance stability is analyzed in the test system considered here.

To illustrate the operation of the ShBR, the operating state of the test system [29] was slightly modified. For the test system under consideration with inactive ShBR, the critical clearing time of generator G9 is 64 ms, which is less than the actual clearing time $t_{\mathrm {SHC}}=\mathrm {100}$ ms. Hence, the stability margin calculated using (1) is negative and equals $k_{\mathrm {t}}=-35.1\%$ , as a result of which generator G9 losses its synchronism. It is assumed that the ShBR is connected to the circuit 40 ms after the fault event occurs and is switched off when the sign of speed deviation $\Delta \omega$ changes from positive to negative.

B. SHBR Optimization Results

Simulation tests have been performed taking into account single-and multi-objective (two- and three) optimizations. To assess the optimal resistance for the ShBR, a three-phase fault is considered, since such an event has the strongest effect on transient stability. The following fault case is considered: a three-phase fault occurs in line L34 (Figure 7) near substation busbars B29 and that fault is cleared by the line L33 being switched off permanently. If the line L34 is switched off permanently, generator G9 can operate through the line L33, if it manages to remain in synchronism. To calculate objectives $f_{2} \left ({{R_{\textrm {sh}}} }\right)$ and $f_{3} \left ({{R_{\textrm {sh}}} }\right)$ , the actual normal clearing time of a close three-phase fault $t_{\textrm {SHC}} =100\;\textrm {ms}$ is assumed.

In the case where the three-objective optimization was performed also additional following types of fault events have been taken into account:

1. two-phase to the ground fault (L-L-G),

2. two-phase fault (L-L).

Presented fault events (L-L-L, L-L-G, L-L) for the three objective optimization have been analyzed using two different approaches. The first approach includes an individual optimization procedure for each of the particular disturbances. The second approach is based on the analysis of the power system under each of the fault events in the common optimization procedure. The optimized value for each of the three considered objectives have been calculated using a weighted average of the objectives obtained for each disturbance, according to the following formula:

$$\begin{equation*} f_{i}=0.7\cdot f_{i_{L-L-L}}+0.15\cdot f_{i_{L-L-G}}+0.15\cdot f_{i_{L-L}}\tag{21}\end{equation*}$$ where $f_{i}$ is the optimized value of the considered objective, and $f_{i_{L-L-L}}$ , $f_{i_{L-L-G}}$ , $f_{i_{L-L}}$ are the values of the objective calculated for the specific fault applied.

Figure 8 and Figure 9 show the Pareto set $G_{\textrm {opt}}$ for different variants of optimization, where green dots correspond to feasible results and navy blue dots correspond to non-dominant results $G_{\textrm {non}} \in G_{\textrm {opt}}$ . Figure 8 a-c shows results for the single-objective optimization problem, while Figure 8 d-f corresponds to the two-objective optimization problems. Figure 9 shows the results for the three-objective optimization problem, considering three-phase fault as a simulated disturbance. Range of ShBR resistance values belonging to set of non-dominant results $P_{\textrm {non}}$ and optimum values $\tilde {R}_{\textrm {sh}} \in G_{\textrm {opt}}$ for the contemplated sets of objectives are presented in Table 1.

TABLE 1 Comparison of Multi-Objective Optimization Results for Different Sets of Objectives

FIGURE 8.

Multi-objective optimization results—obtained Pareto set G_opt as a function of complete set of objectives.

FIGURE 9.

Multi-objective optimization results – obtained Pareto set G_opt three-dimensional plot.

Under the single-objective approach (the results of which are presented in Figure 8.a-c, Pareto sets $G_{\textrm {opt}}$ for objective $f_{1} \left ({{R_{\textrm {sh}}} }\right)$ (transient angle stability), Figure 8.a, and for objective $f_{2} \left ({{R_{\textrm {sh}}} }\right)$ (transient voltage response), Figure 8.b, have similar shapes, but the shape of the Pareto set for objective $f_{3} \left ({{R_{\textrm {sh}}} }\right)$ (mechanical stress of the turbine-generator shaft) is different.

Under the single-objective approach, optimum values $\tilde {R}_{\textrm {sh}} =148.1\;\Omega$ for $f_{1} \left ({{R_{\textrm {sh}}} }\right)$ and $\tilde {R}_{\textrm {sh}} =161.4\;\Omega$ for $f_{2} \left ({{R_{\textrm {sh}}} }\right)$ are closer to the lower limit of resistances from set $G_{\textrm {non}}$ , while $\tilde {R}_{\textrm {sh}} =228.5\;\Omega$ for $f_{3} \left ({{R_{\textrm {sh}}} }\right)$ is closer to the upper limit of resistances from set $G_{\textrm {non}}$ . Optimum values $\tilde {R}_{\textrm {sh}}$ for each of the three functions $f_{1} \left ({{R_{\textrm {sh}}} }\right)$ , $f_{2} \left ({{R_{\textrm {sh}}} }\right)$ and $f_{3} \left ({{R_{\textrm {sh}}} }\right)$ lead to improve stability and a positive stability margin $k_{\textrm {t}}$ . The biggest contribution to stability improvement is provided by function $f_{1} \left ({{R_{\textrm {sh}}} }\right)$ ($k_{\textrm {t}} =16.8\%$ ), and the smallest – by function $f_{3} \left ({{R_{\textrm {sh}}} }\right)$ ($k_{\textrm {t}} =12.45\%$ ).

Under the two-objective approach (Table 1), the optimum value $\tilde {R}_{\textrm {sh}} =151.4\;\Omega$ for the pair of objectives $f_{1} \left ({{R_{\textrm {sh}}} }\right)$ , $f_{2} \left ({{R_{\textrm {sh}}} }\right)$ differs only slightly from $\tilde {R}_{\textrm {sh}}$ obtained under the single-objective approach for $f_{1} \left ({{R_{\textrm {sh}}} }\right)$ or for $f_{2} \left ({{R_{\textrm {sh}}} }\right)$ .

However, for the pairs of objectives $f_{1} \left ({{R_{\textrm {sh}}} }\right)$ , $f_{3} \left ({{R_{\textrm {sh}}} }\right)$ or $f_{2} \left ({{R_{\textrm {sh}} } }\right)$ , $f_{3} \left ({{R_{\textrm {sh}}} }\right)$ , optimum vales are $\tilde {R}_{\textrm {sh}} =219.4\;\Omega$ and $\tilde {R}_{\textrm {sh}} =211.6\;\Omega$ , respectively. These values significantly differ from $\tilde {R}_{\textrm {sh}}$ obtained under the single-objective approach for $f_{1} \left ({{R_{\textrm {sh}}} }\right)$ or for $f_{2} \left ({{R_{\textrm {sh}}} }\right)$ . In addition, it is notable that, under the two-objective approach, $f_{3} \left ({{R_{\textrm {sh}}} }\right)$ leads to a larger value of $\tilde {R}_{\textrm {sh}}$ than under the single-objective approach for $f_{1} \left ({{R_{\textrm {sh}}} }\right)$ or $f_{2} \left ({{R_{\textrm {sh}}} }\right)$ . Out of all sets of objective functions, the best improvement of angle stability is obtained for the pair of objectives $f_{1} \left ({{R_{\textrm {sh}}} }\right)$ , $f_{2} \left ({{R_{\textrm {sh}}} }\right)$ ($k_{\textrm {t}} =16.74\%$ ).

Under the three-objective approach considering three-phase disturbance (Table 1), $\tilde {R}_{\textrm {sh}} =152.1\;\Omega$ differs only slightly from the result for the two-objective approach taking into account $f_{1} \left ({{R_{\textrm {sh}}} }\right)$ , $f_{2} \left ({{R_{\textrm {sh}}} }\right)$ . Hence, the stability margin for the two- and three-objective approach is the same ($k_{\textrm {t}} =16.74\%$ ). This indicates that when objectives $f_{1} \left ({{R_{\textrm {sh}}} }\right)$ , $f_{2} \left ({{R_{\textrm {sh}}} }\right)$ are taken into account simultaneously, the effect of objective $f_{3} \left ({{R_{\textrm {sh}}} }\right)$ on the optimum value of $\tilde {R}_{\textrm {sh}}$ is very small.

Under the three-objective optimization approach considering two-phase and two-phase to the ground fault events (Table 1), optimal resistance values $\tilde {R}_{\mathrm {sh}}=178.5 \Omega$ (L-L) and $\tilde {R}_{\mathrm {sh}}=192.1 \mathrm {\Omega }$ (L-L-G) tend towards higher values. This indicates that when less severe disturbances are considered, the effect of objective $f_{1}\left ({R_{\mathrm {sh}} }\right)$ on the optimum value of $\tilde {R}_{\textrm {sh}}$ is apparently smaller, as transient stability margin objective is reached with a significant reserve. The similar trend has been observed in the analysis of the power system under each of the fault event in the common optimization procedure, where optimization procedure tends to focus on higher values of resistance trying to optimize $f_{1}\left ({R_{\mathrm {sh}} }\right)$ and $f_{2}\left ({R_{\mathrm {sh}} }\right)\mathrm { }$ more, leading to the final result of $\tilde {R}_{\mathrm {sh}}=173.9 \Omega$ .

C. Impact of the Optimal SHBR on Power System Dynamic Response

The impact of the ShBR on power system dynamic response was assessed by calculating the integral squared error of the performance index given by the following formula:

$$\begin{equation*} ISE_{\textrm {y}} =\int \limits _{t=0}^{t_{\textrm {k}}} {\left [{ {e_{y} \left ({t }\right)} }\right]^{2}} \textrm {d}t\tag{22}\end{equation*}$$ where $t_{\textrm {k}} =10\;\textrm {s}$ and $e_{y} \left ({t }\right)$ is the error of the analyzed quantity. For simulation tests, the following errors are used: $e_{\textrm {p}} \left ({t }\right)=P\left ({t }\right)-P_{0}$ , $e_{\delta } \left ({t }\right)=\delta \left ({t }\right)-\delta _{0}$ , $e_{\textrm {V}} \left ({t }\right)=V\left ({t }\right)-V_{0}$ , $e_{\tau } \left ({t }\right)=\tau \left ({t }\right)-\tau _{0}$ , where “0” subscripts refer to initial values of considered transients before the fault event occurs. Simulation results in Table 2 were obtained for a three-phase fault cleared within normal time $t_{\textrm {SHC}} =100\;\textrm {ms}$ and permanent disconnection of the faulted line.

TABLE 2 Quality of Dynamic Response

In Table 2, bold numbers denote the results which correspond to minimum values of $ISE$ indicators for single- and two-objective optimization. The results presented in Table 2 indicate that for a single-objective approach, the minimum value of $ISE_{\delta }$ , $ISE_{\textrm {V}}$ , $ISE_{\tau }$ is obtained taking into account functions $f_{1} \left ({{R_{\textrm {sh}}} }\right)$ , $f_{2} \left ({{R_{\textrm {sh}}} }\right)$ and $f_{3} \left ({{R_{\textrm {sh}}} }\right)$ respectively. Under the two-objective approach, the minimum values of $ISE_{\delta }$ , $ISE_{\textrm {V}}$ are obtained for the pair of objectives $f_{1} \left ({{R_{\textrm {sh}}} }\right)$ , $f_{2} \left ({{R_{\textrm {sh}}} }\right)$ , and the minimum $ISE_{\tau }$ is obtained for $f_{1} \left ({{R_{\textrm {sh}}} }\right)$ , $f_{3} \left ({{R_{\textrm {sh}}} }\right)$ . The results shown in Table 2 are complemented by Figure 10 which presents simulation results for selected optimum values of $R_{\textrm {sh}}$ in time.

FIGURE 10.

The comparison of the dynamic performance of generator G9, in the case of t_SHC = 100 ms and different R_sh values considered.

SECTION V. Conclusion

This paper proposes and analyses a multi-objective optimization approach for ShBR sizing. The ShBR sizing problem incorporates three objectives, including transient angle stability, transient voltage response, and mechanical stress of the turbine-generator shaft. To present the complexity of the ShBR optimization process more clearly, studies were conducted for the one-, two- and three-objective approaches. The multivariate optimization analysis shows that the optimal ShBR value is mainly influenced by objectives related to transient stability and voltage dynamic response of the power system. With the above-mentioned criteria taken into account simultaneously, the criterion related to mechanical stress of the turbine-generator shaft affects the selection of the ShBR values only to a slight extent.

The presented multi-objective optimization process confirmed the high efficiency of the electric braking of the rotor of a synchronous generator which contributes to the improvement of the transient stability of the considered power systems. The NSGA-II optimization algorithm provided reliable results in the form of Pareto sets of acceptable solutions from which the final solution has been selected using the fuzzy logic-based method.

Presented research studies can be further developed, focusing on the following aspects:

• comparative analysis for different types of braking resistors, especially considering different shunt and series structures,

• comprehensive study on the impact of coordinated activation of the electric braking and other supplementary control solutions,

• comparative analysis of different multi-criteria optimization methods available for the considered shunt braking resistor dimensioning problem.