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A Neuro-fuzzy Adaptive Power System Stabilizer Using Genetic Algorithms M. A. Awadallah a; H. M. Soliman b a Department of Electrical Power and Machines, University of Zagazig, Zagazig, Egypt b Electrical Engineering Department, Cairo University, Giza, Egypt Online Publication Date: 01 February 2009
To cite this Article Awadallah, M. A. and Soliman, H. M.(2009)'A Neuro-fuzzy Adaptive Power System Stabilizer Using Genetic
Algorithms',Electric Power Components and Systems,37:2,158 — 173 To link to this Article: DOI: 10.1080/15325000802388740 URL: http://dx.doi.org/10.1080/15325000802388740
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Electric Power Components and Systems, 37:158–173, 2009 Copyright © Taylor & Francis Group, LLC ISSN: 1532-5008 print/1532-5016 online DOI: 10.1080/15325000802388740
A Neuro-fuzzy Adaptive Power System Stabilizer Using Genetic Algorithms M. A. AWADALLAH1 and H. M. SOLIMAN2 1
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Department of Electrical Power and Machines, University of Zagazig, Zagazig, Egypt 2 Electrical Engineering Department, Cairo University, Giza, Egypt Abstract This article presents the design technique of an adaptive power system stabilizer using adaptive neuro-fuzzy inference systems trained via data obtained from genetic algorithms. The parameters of a standard power system stabilizer are tuned using adaptive neuro-fuzzy inference systems to achieve a certain damping ratio and settling time at all load points within a wide region of operation. The overall transfer function of the system is derived in terms of the power system stabilizer parameters. A genetic algorithm is used to minimize a multi-objective optimization function that forces the damping ratio and settling time of the system to desired values. The optimization process is separately conducted at selected operating points to yield power system stabilizer parameters that change with load variations. Results of genetic algorithm optimization are used to form a training dataset of an adaptive neuro-fuzzy inference systems agent, which could give the power system stabilizer parameters at any load within the specified region of operation. Results of power system stabilizer testing show that the desired performance indices could be fulfilled from light load to over load under both lagging and leading power factor conditions. System performance shows a remarkable improvement of dynamic stability by obtaining a well-damped time response. Keywords power system stabilizers, adaptive control, neuro-fuzzy systems, genetic algorithms
1. Introduction Steady-state stability, which is also known in control literature as small-signal stability, represents one vital challenge for electrical power system engineers. Transient stability concerns severe changes of the system such as short-circuit faults and substation outage. In contrast, studies of steady-state stability involve small changes such as load variations and adjustment of generation schedules. Oscillations of small magnitude and low frequency—linked with the electromechanical modes in power systems—often persist for long periods of time, and in some cases, present limitations on the power transfer capability. In addition, such oscillations may sustain and grow to cause system separation if adequate damping is not available. Therefore, the generators are equipped with power system stabilizers (PSSs) that provide supplementary feedback stabilizing signals in the excitation systems [1]. A conventional PSS can be considered as a single-input singleReceived 6 April 2008; accepted 29 July 2008. Address correspondence to Dr. Mohamed A. Awadallah, Department of Electrical Power and Machines, University of Zagazig, Zagazig, Egypt. E-mail:
[email protected]
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output feedback controller installed on the generator, while linearized models can be used for such study [2]. The input to a conventional PSS can be machine shaft speed, AC bus frequency, or accelerating power. The most commonly used input is machine shaft speed [1, 2]. On the other hand, the output of the PSS, as a supplementary control signal, is applied to the machine voltage regulator. The structure of a PSS is usually considered fixed, while its parameters are adjusted based on rated operating conditions [1, 2]. However, power systems constantly experience changes in operating conditions due to variations in generation and load patterns. Accordingly, a large variation in the small-signal dynamic behavior of a power system exists. A parametric uncertainty in the small-signal linearized model of the system is a common attribute of power systems. Therefore, a fixed-parameter optimum PSS for a nominal point would provide sub-optimum performance under variation of system parameters and operating conditions. Control strategies based on adaptive control, robust control, and variable structure control (VSC) have been reported in literature, aiming at developing PSS configurations that can provide adequate damping over wide operating conditions. Recently, intelligent control and evolutionary techniques, such as fuzzy logic systems (FLSs), artificial neural networks (ANNs), genetic algorithms (GAs), tabu search, simulated annealing, particle swarm optimization (PSO), and ant colony optimization (ACO), have been applied to solve many complex problems in emerging control fields. The noticeable advantage of modern optimization tools is that the objective functions need not be explicit or differentiable. Moreover, non-linearity or non-convexity is not an issue of concern in obtaining optimal solutions. Constraints on controller parameters, performance, or structure can be simply applied. A gain scheduling control for PSSs was presented in [3]; the controller parameters were tuned based on minimizing the distance between the present operating point and a desired one. The self-tuning control scheme of PSSs was reported in [4], where the amount of pole shifting was adjusted depending upon the system conditions. A model reference adaptive control (MRAC) strategy was applied in [5], where the error between the power system response and the reference model output was used to modify the controller parameters. Consequently, the system behavior is driven to match that of the reference model. Another adaptive algorithm is the multiple-model adaptive control (MMAC) [6], in which the actual system was represented by a finite number of linearized models. Separate controllers are designed to ensure satisfactory performance for each of the models. Theoretically, it cannot be claimed that a convex combination of stabilizing controllers necessarily produce a stable closed-loop response. Adaptive controllers generally have poor performance during the learning phase unless they are properly initialized. Successful operation of adaptive controllers requires continuous measurement of plant variables to fulfill strict persistent tuning conditions; otherwise, the adjustment of controller parameters fails. VSC has been proposed in the literature as an alternative to adaptive PSSs in order to counteract the problem of variation of system parameters and operating conditions. VSC is insensitive to system parameter variations and can be simply realized using microcomputers. However, VSC applied to a PSS results in high control activity (chattering) [7]. The emerging tools of modern artificial intelligence (AI) paradigms have been successfully applied to the topic of PSS design. AI techniques are characterized by the ability of reasoning, learning, decision making, and non-linear modeling; such features would make AI tools suitable to solve many control problems. ANNs are based on the concept of parallel processing and possess great ability in realizing complicated nonlinear mapping from the input space to the output space. Therefore, they provide an
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extremely fast processing facility for complicated non-linear problems [8]. The flexibility and subjectivity of knowledge representation is a distinct feature of fuzzy logic, which allows the technique to be a notable candidate for PSS design, as reported in [9]. Heuristic search algorithms, such as GAs [10], tabu search [11], and simulated annealing [12], have been applied to the problem of PSS design. The performance characteristics of PSSs, such as damping and speed of response, can be expressed in terms of an admissible pole region for the linearized, small-signal model. An approach based on linear matrix inequalities (LMIs) was presented in [13]. Placing the closed-loop poles inside a desired domain is termed D-stability. However, the PSS is usually obtained with state feedback that requires measuring all the states. Such a problem is alleviated in the present work by using a single-input single-output standard PSS with simple lead control. The present work presents the design procedures of an adaptive PSS based on GA optimization and adaptive neuro-fuzzy inference systems (ANFISs). The task of PSS design is formulated as an optimization problem with relaxed constraints and two different eigenvalue-based objective functions. The closed-loop transfer function of the system is evaluated at a certain load point in terms of the controller parameters; the eigenvalues are then computed. The maximum real part of all eigenvalues and the minimum damping ratio (of the dominant eigenvalues only) are compared with certain desired values. The GA routine is then employed to solve such an optimization problem at a grid of load points covering a wide range of operation. The obtained list of PSS parameters is used to train an ANFIS agent that tunes the parameters of the PSS to achieve the desired performance at different operating conditions. A single-machine infinite-bus system has been considered to investigate the potential of the proposed approach; the infinite bus could be the equivalent of a large interconnected power system. Testing of the presented PSS at many operating points different from training data shows its ability to maintain the required performance of the power system.
2. Problem Formulation 2.1. System Model and PSS Structure The system under study consists of a single machine connected to an infinite bus through a tie-line, as shown in the block diagram of Figure 1. It should be emphasized that the infinite bus could be representing the Thévenin equivalent of a large interconnected
Figure 1. Block diagram representation of the power system.
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power system. The machine is equipped with a static exciter. The non-linear model of the system is given through the following differential equations: ıP D !o !; .Tm Te / ; M xd C xd0 1 xd C xe 0 0 P Eq D 0 Efd E C 0 V cos ı ; Td 0 xd0 C xe q xd C xe !P D
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1 .KE Eref EP fd D TE
KE Vt
(1)
Efd /:
For PSS design purposes, the linearized incremental model around an equilibrium point is usually employed [1, 2]; see Figure 2. The parameters of the model have to be computed at each operating point since they are load dependent. Analytical expressions for the parameters (k1 through k6 ) are listed in the Appendix as derived in [14, 15]. The parameters are functions of the loading condition (P and Q). Typical data for the system under consideration, which are used in the present work, are given as follows. Synchronous machine parameters: xd D 1:6 p.u., xd0 D 0:32 p.u., xq D 1:55 p.u., !0 D 2 50 rad/sec, Td0 0 D 6 sec, and M D 10 sec. Transmission line reactance: xe D 0:4 p.u. The state equation of the system under study is given by [1] as xP D Ax C Bu; (2) y D Cx;
Figure 2. Linearized model of the power system in terms of the k-parameters.
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Figure 3. Block diagram of the closed-loop system.
where
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x D Œı 2
Eq0
! 0
Efd ;
!0
6 6 k1 6 6 M 6 AD6 6 k4 6 6 Td0 o 6 4 k5 kE TE BD 0 0 0
0 0
kE TE
0
k2 M 1 T
7 7 0 7 7 7 1 7 7; 0 7 Td o 7 7 1 5
k6 kE TE
0 T
3
0
(3)
TE
;
C D Œ0 1 0 0; T D k3 Td0 o: In order to cover the wide operating conditions of the machine, the following loadings are selected: P D Œ0:2; 1:25 p.u. and Q D Œ 0:4; 0:4 p.u. A widely used conventional lead PSS [1, 2] is considered in the present work; it can be described as U DK
1 C T1 s !; 1 C T2 s
(4)
where U is the PSS output signal, and ! is the machine speed deviation from the synchronous speed. Stabilizer gain K and time constants T1 and T2 represent the parameters to be tuned. The block diagram model of the system is shown in Figure 3. 2.2. Objective Functions The ultimate goal of the present work is to design an adaptive PSS that meets specified closed-loop performance criteria under all loading conditions. The desired performance indicates a closed-loop damping ratio of 0.25, while the system oscillations should settle within 10–15 sec ,as typically followed by power system utilities [16]. Therefore, the
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desired damping ratio (d ) and desired relative stability (d ) are around 0.25 and 0.3, respectively. The multi-objective optimization function, then, is formulated as follows: min J D J1 C J2 :
(5)
K;T1 ;T2
The individual objective functions are expressed as J1 D .max
d /2 ;
(6)
J2 D .min
d /2 ;
(7)
where
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max D maxfi g for all eigenvalues .i D i C j!i /, and
min D min
8 ˆ < ˆ :
k q
9 > =
> k2 C !k2 ;
for dominant eigenvalues .k D k C j!k /. In the optimization process, the goal is to minimize J1 in order to shift the poorly damped eigenvalues to the left in the complex plane until the desired relative stability, d D 0:3, is achieved. On the other hand, it is also a goal to minimize J2 in order to increase the damping of dominant modes until the desired damping ratio, d D 0:25, is fulfilled. The resulting controller parameters at each load compromise the two objectives by yielding a minimum for the multi-objective function, J . It should be emphasized that a mild constraint has been adopted by maximizing the damping of dominant eigenvalues only. The reason is that non-dominant eigenvalues result in a fast component of the response, which would not affect the overall behavior, even if it is poorly damped. Such a relaxed constraint helped in realizing a feasible solution of the above-mentioned problem.
3. Intelligent Tools 3.1.
GAs
The GA is a probabilistic, random-guided search technique inspired by the Darwinian theory of evolution, which employs the “survival of the fittest” concept of natural biology [17–20]. The most distinct feature of the GA is that the routine starts searching from a population of points—not a single point—with no need for information about the derivatives of the objective function. The algorithm codes prospective solutions of the problem as a population of individual chromosomes of different genes. The population is randomly initialized, and the individuals are evaluated based on the corresponding values of an objective fitness function. Fit individuals are probabilistically copied to the “mating pool,” while weak individuals are likely to die, as their probability of selection is small due to the poor fitness. The natural genetic processes of crossover and mutation are then imitated in order to mate parents of the current generation and produce the offspring of the new generation.
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Crossover means probabilistic gene exchange between randomly selected individuals. The process is usually carried out between two individuals of the current generation (parents) to yield two different individuals of the next generation (offspring). On the other hand, mutation is the random gene alteration, which is a typical phenomenon in natural biology. Mutation means that one offspring might have a genetic characteristic that is not directly found in either of the parents. Mutation is performed in the GA at a very small probability; yet, it helps greatly in making the routine escape from local minima. The routine continues until the termination criterion is satisfied, where the best individual of the last generation is taken as the solution of the problem. The termination condition could be stagnation of fitness value over a certain number of generations, reaching runtime limit, obtaining desired fitness value, executing a preset number of generations, or a combination of such criteria. Based on Darwin’s theory, the population evolves from one generation to the next as the best fitness improves. In spite of the noteworthy robustness of the GA in finding global optima, the slowness of operation could be a significant obstacle in some applications. However, the GA has proven notable effectiveness in many types of optimization problems. Coding of individual solutions has granted the GA one more apparent plus, which is its adaptability to certain optimization problems that could not be solved by classical or even some other evolutionary techniques. The sequentially applied genetic operations of selection, crossover, and mutation attributed the GA with the ability to escape local optima, besides the robustness of attaining global ones. 3.2. ANFIS The ANFIS refers, in general, to an adaptive network that performs the function of a fuzzy inference system [21–23]. The most commonly used fuzzy system in ANFIS architectures is the Sugeno model, since it is less computationally exhaustive and more transparent than other models. A consequent membership function (MF) of the Sugeno model could be any arbitrary parameterized function of the crisp inputs, most likely a polynomial. Zero- and first-order polynomials are used as consequent MF in constant and linear Sugeno models, respectively. In addition, the defuzzification process in Sugeno fuzzy models is a simple weighted average calculation. The fuzzy space is divided via grid partitioning according to the number of antecedent MFs, and each fuzzy region is covered with a fuzzy rule. On the other hand, each fixed and adaptive node of the network performs one function or sub-function of the Sugeno model, as shown in Figure 4, such that the overall performance of the network is functionally the same as that of the fuzzy model. The adaptive network employs an optimization algorithm in order to modify the parameters of the fuzzy inference system. The adaptation process aims at obtaining a set of parameters at which an error measure between the actual performance of the fuzzy inference system and a targeted set of training data is minimized. Classical optimization techniques, such as back propagation, could be used as well as hybrid algorithms. The total number of ANFIS-modifiable parameters is a crucial factor of the computational effort required before the adaptation process is completed. Therefore, the antecedent Gaussian MF, which is defined through two parameters only, is more preferable than other forms of MFs, which require three or more parameters. The ANFIS combines the advantages of fuzzy systems and adaptive networks in one hybrid intelligent paradigm. The flexibility and subjectivity of fuzzy inference systems, when added to the optimization
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Figure 4. (a) Two-input, one-output, two-rule Sugeno model and (b) equivalent ANFIS structure.
strength of adaptive networks, give the ANFIS its remarkable power of modeling, learning, non-linear mapping, and pattern recognition. ANFISs have numerous applications in power systems such as fault diagnosis in electric motor drives [24–26].
4. Simulation and Results The proposed control scheme of the present work aims at developing an adaptive PSS, which assures the desired relative stability and damping ratio for the power system under transient conditions. The PSS parameters given by Eq. (4) are tuned to cope with load changes. Forty equally spaced points covering the operating range, defined in Section 2.1, are selected to find the PSS parameters through an optimization process. The GA routine is run at each discrete point to find the PSS parameters that minimize the multi-objective optimization function given in Eq. (5). It should be pointed out that minimizing such a function makes the relative stability and system damping ratio get as close as possible to their desired values. It is obvious that if a minimum of zero could be reached, the required indices match the desired values exactly. Finally, the desired values of relative stability and damping ratio are taken as 0.3 and 0.25, respectively, as is well stated in the literature of power system stability and control [16]. The Genetic Algorithm and Direct Search Toolbox of MATLAB 7.1 is used to obtain the controller parameters that minimize the objective function described in Eq. (5). The optimization process is repeated at 40 equally spaced load points spanning the whole operating range defined earlier. The 40 load points step on P and Q from 0.2 to 1.25 p.u. and from 0.4 to 0.4 p.u. by steps of 0.15 and 0.2 p.u., respectively. Results of the optimization process at selected extreme loads are reported in Table 1. The PSS
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QD
0:4
PSS parameters k T1 T2 Eigenvalues
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Relative stability, max Damping ratio, min Objective function, J Q D 0:4
PSS parameters k T1 T2 Eigenvalues
Relative stability, max Damping ratio, min Objective function, J
P D 1:25
8.15 10.902 0.025 0:3 ˙ j 0:54 1:501 ˙ j 25:876 56.86 0.3 0.486 0.111
10.5 9.09 0.01 0:3114 ˙ j1:223 4:473 ˙ j 32:09 110.894 0.3114 0.2467 1:52 10 4
6.5 50 0.01 0:3465 ˙ j1:3123 4:66 ˙ j 27:465 87.374 0.3465 0.2553 2:22 10 3
7.521 21.686 0.01 0:3115 ˙ j1:207 2:853 ˙ j 37:55 114.13 0.3115 0.25 1:34 10 4
parameters, closed-loop eigenvalues, relative stability, damping ratio, and objective function value are given at four loading conditions. It should be emphasized that the PSS structure is fixed, while its parameters vary with loading. Results of Table 1 show that the optimization objectives have been met at such extreme loads. Nevertheless, it should be noted that the less the relative stability and the more the damping ratio, the better. Results of GA optimization at the 40 load points are next used to constitute the training dataset for an ANFIS agent that would be able to deduce the PSS parameters required to attain the desired performance at any load. The inputs to the ANFIS are the loading points P and Q, while the outputs signify the three parameters of the PSS. The core of the proposed ANFIS is a two-input three-output Sugeno-type fuzzy model that was designed, trained, and tested under the Fuzzy Logic Toolbox of MATLAB 7.1. The firstorder (linear) ANFIS, which has four Gaussian MFs per input, is trained using the hybrid algorithm of the back-propagation and least-square methods available through the MATLAB toolbox. The mean-square error of ANFIS has stagnated at a minimum value after training the system for 30 epochs in each case. A testing dataset, which contains 374 load points uniformly distributed in the operating region, is formed in order to test the performance of the developed ANFIS. Testing points are created by stepping on P from 0.2 to 1.25 p.u., on Q from 0.4 to 0.4 p.u. by a step of 0.05 p.u. each, and combining all P and Q values together. Consequently, the ANFIS training dataset is evidently a subset of the testing data. The testing procedures of the developed ANFIS
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agent are arranged as follows. First, the operating points P and Q are passed on to the ANFIS in order to acquire the PSS parameters. Second, the closed-loop eigenvalues are computed in order to find out the relative stability index, max , and the damping ratio of the dominant poles, min . Third, the obtained values of max and min are compared to the desired values of 0.3 and 0.25, respectively. Results of ANFIS testing are shown in Figure 5, in which the three PSS parameters and the overall objective function are separately plotted against P and Q. The testing process indicates that the developed ANFIS managed to keep max between 0.346 and 0.279, and to keep min between 0.244 and 0.486, for all testing points. The result is considered a distinct success to sustain both performance indices around the desired values. The overall objective function, J , is always less than 0.06 except for three load points corresponding to reactive power loading of 0.4 p.u. The three points occur at active power values of 0.2, 0.25, and 0.45 p.u., with objective function values of 0.111, 0.171, and 0.116, respectively. These three points appear as spikes in the plot of Figure 5(d). On the other hand, Figure 6 shows a plot of the dominant poles of the testing dataset in the complex plane, where the vertical dashed line indicates d D 0:3 and the two inclined lines denote d D 0:25. The figure authenticates the conclusion reported above that the developed adaptive PSS could preserve the designated performance indices close to their desired values. The testing phase of the neuro-fuzzy systems is the only means of assessing the performance and validating the robustness of a trained agent. Therefore, testing of the present ANFIS continued with another augmented dataset; it consists of 8586 loading
Figure 5. Results of ANFIS testing: (a) PSS gain, k; (b) PSS time constant, T1 ; (c) PSS time constant, T2 ; and (d) objective function, J .
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Figure 6. Dominant eigenvalues of the closed-loop system at 374 testing points.
points formed by stepping on both P and Q by 0.01 p.u. within the specified operating region. Such testing data covered the whole operation range with very high resolution. The process is carried out with similar steps as given above, and the same results are obtained. Concisely, the testing procedure verifies the excellent performance of the proposed PSS in preserving the desired performance indices of the system. The last step of the present study is to compare the performance of the proposed PSS to a conventional controller. A widely used conventional PSS is given in [1], which can be described as U DK
Tw s .1 C T1 s/.1 C T3 s/ !; 1 C Tw s .1 C T2 s/.1 C T4 s/
(8)
where Tw is the time constant of a washout circuit that abandons the controller action in steady state. The design procedures given in [1] are used to obtain the parameters of the conventional PSS at nominal load. Performance of the system under three different conditions—namely, without PSS, with conventional PSS, and with proposed PSS—is compared for selected load points as shown in Figures 7, 8, and 9. A step input of 0.1 p.u. is applied in all cases. The figures demonstrate the superiority of the proposed design, especially under extreme loads. The poor stability of the open-loop system at P D 0:25 and Q D 0:35 p.u. is much better improved using the proposed PSS (Figure 7). Meanwhile, the severely unstable response of the system is stabilized using the proposed PSS with superior damping and settling time for the load points of Figures 8 and 9. It should be noted that the advantages of the
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Figure 7. Comparison of system step responses at P D 0:25 p.u. and Q D 0:35 p.u.
adaptive PSS are expected to be much more evident as the operating point is driven away from nominal load. Finally, it should be acknowledged that the single-lead adaptive PSS still outperforms the double-lead conventional PSS. System responses are further compared for a large disturbance by considering a three-phase-to-ground short circuit at the infinite bus. The fault occurs at t D 3 sec and lasts for 100 ms, while the system is loaded with P D 0:8 p.u. and Q D 0:3 p.u. It is worth mentioning that the responses are obtained through the non-linear model given by Eq. (1), and the parameters of the proposed adaptive PSS correspond to the loading just before fault. System responses with the proposed and conventional controllers are plotted in Figure 10. The plots show that the proposed PSS could damp the disturbance faster than the conventional PSS and with less oscillation.
5. Conclusions This article presents an adaptive PSS based on neuro-fuzzy systems and GAs, with which two performance objectives are to be met. The relative stability of the system—which also indicates the settling time—is to be close to 0.3, while the damping ratio is set near 0.25. A multi-objective optimization function is minimized using a GA at certain load points covering a wide range of operation. The PSS parameters at such points are used to train an ANFIS agent, which yields the controller parameters required to satisfy the performance objectives at any load condition within the operating range. Testing results show that the developed ANFIS managed to keep the relative stability index between 0.346 and 0.279 and the damping ratio between 0.244 and 0.486 for all
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Figure 8. Comparison of system step responses at P D 1:15 p.u. and Q D
0:3 p.u.
Figure 9. Comparison of system step responses at P D 0:95 p.u. and Q D
0:4 p.u.
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Figure 10. Comparison of system non-linear responses during fault at P D 0:8 p.u. and Q D 0:3 p.u.
testing points. Time-domain step responses of the system with and without the PSS are compared; results show the effectiveness of the PSS to damp the oscillations and obtain satisfactory behavior. Hence, findings of the present work validate the proposed design methodology of adaptive PSSs and solidify the suitability of the GA and ANFIS for such purpose. Finally, it should be emphasized that although the development of the proposed method has been conducted on a single-machine infinite-bus system, the bus could be representing a Thévenin equivalent of a large interconnected grid. Therefore, the implementation of the proposed method on a practical power system is evidently feasible.
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Appendix A.
k1 D C3
P2 C Q C C1 ; P 2 C .Q C C1 /2 k3 D
xd0 C xe ; xd C xe
k5 D C4 xe
Downloaded By: [Awadallah, Mohamed A.] At: 05:27 17 January 2009
k6 D C7
p
P k2 D C4 p ; 2 P C .Q C C1 /2
P ; k4 D C5 p 2 P C .Q C C1 /2
P C1 C Q C6 2 2 V C Qxe P C .C1 C Q/2
xd0 ;
C1 xq .C1 C Q/ P 2 C .C1 C Q/2 xe C 2 ; V 2 C Qxe P C .C1 C Q/2 C1 D
C3 D C1
C5 D
xd xe
V2 ; xe C xq
xq xd0 ; xe C xd0 xd0 ; xd0 C7 D
C2 D k3 ;
C4 D
C6 D C1 xe : xe C xd0
V ; xe C xd0
xq .xq xd0 / ; xe C xq
