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Performance of advanced Extremum Seeking Control algorithm to accurately locate the extremes on multimodal benchmark patterns

Marin Radut1

1)University POLITEHNICA of Bucharest,

3)Doctoral School of Electronics, Communications and Information Technology Faculty,

Bucharest, Romania, [email protected]

Abstract –The performance of an advanced Extremum Seeking Control (ESC) scheme will be evaluated in this paper based on multimodal benchmark pattern proposed in the literature to test the classical ESC schemes. The performance indicators such as searching resolution, tracking accuracy, and tracking speed are usually used in real-time optimization (RTO) strategy for Fuel Cell (FC) systems or photovoltaic (PV) systems. The advanced ESC scheme will locate and track the maximum value of the optimization function considered for the system under test. For this, two loops must be designed: the locating loop based on the classical Perturbed ESC (PESC) loop, with the tuning parameter k1, and the scanning loop having the tuning parameter k2. Besides the mentioned performance indicators, the percentage of hit count (PHC) indicator, which estimates the ratio of positive tests obtained versus all tests made, is other performance indicator that must be close to 100% to assure high efficiency for energy systems.

Keywords: Real-time optimization (RTO) strategy; Fuel Cell Hybrid Power Sources (FCHPS); photovoltaic (PV) systems; Extremum Seeking Control (ESC); multimodal benchmark pattern; Percentage of hit count (PHC)


The multimodal enhancement algorithms can locate the extremes in a specific searching range. The extremes can be Global Maximum Points (GMPs) and Local Maximum Points (LMPs). If the GMP is not in the searching range defined, then the higher LMP must be identified. Thus, the knowledge of the LMPs is important for the control system in order to convert from current solution to a LMP solution [1]. Consequently, the system performance remains over the imposed level [2].

Tracking of the extremes (GMPs and LMPs) on dynamic multimodal function based on Real-Time Optimization (RTO) algorithms is not always possible because of the high costs necessary for implementation and because of some physical constraints [3], but Extremum Seeking Control (ESC) schemes can be used for RTO algorithms [4].

The ESC-based RTO algorithms will quickly track the optimum point in the searching range maintaining the performance under dynamical changes of the system under test [4-6]. Dynamic multimodal patterns exhibit nonlinear systems such as Hybrid Power Sources (HPSs) in different real operation-modes [5-8]. The energy efficiency of Fuel Cell Vehicle (FCV) is a multimodal function that relies upon [5,6]: (1) the kESS ratio, which defines the charge-sustaining (kESS = 0), charging (kESS < 0), and discharging (kESS > 0) operation modes for the Energy Storage System (ESS); (2) the regenerative braking factor, kL [7]; (3) the kFC ratio, which defines the ratio from FC power flow that will charge the ESS; (4) the energy efficiency of multi-port converters used [8]; and (5) the designed Energy Management Strategy (EMS) to operate the FC System [9,10]. Note that all constraints must be compliant with safety rules of the ESS (for example, the State-Of-Charge (SOC) level for battery must be in the admissible range) and FC system (for example, the slopes of the fuelling rates (oxygen and hydrogen) for the FC system must be limited), but safe operation of the FC system means much more requirements [11].

Consequently, the RTO techniques will locate the optimal solution considering the input variables and appropriate constrains [12-14].

The ESC scheme proposed in [15] was analyzed by the author for different multimodal benchmark patterns, but here will be tested to locate the extremes of a multimodal benchmark pattern proposed in the literature [16].

The ESC scheme will be briefly explained in section II, but note that advanced ESC scheme has two loops [15] instead of one as in classical ESC scheme [13]. Besides the locating loop determined by the classical Perturbed ESC (PESC) loop with the tuning parameter k1, the second loop will scan the searching range to locate the LMP or the GMP by using the tuning parameter k2. The main contribution of this study is the design of the tuning parameters of advanced ESC scheme for special multimodal benchmark pattern used in the literature to test the classical PESC schemes [16]. The appropriate design assures 100% percentage of hit count (PHC) [17] as was obtained in tests performed for FC systems and photovoltaic (PV) systems in [18-21] and [22,23]. Note that PHC indicator is the ratio of positive tests obtained versus all tests made to find one LMP or GMP [17,22,23]. Besides PHC indicator, the performance of MPP tracking algorithms is evaluated based on searching resolution, tracking accuracy, and tracking speed [24-29]. Also, besides PHC indicator, the success rate (which is defined as the percentage of successfully found of extremes desired in all runs performed [30]) is used for firmware-based algorithms used in multimodal optimization based on Evolutionary Algorithms (EAs) [31].

Some design rules have been shown in [32-34] but using multimodal benchmark patterns specific to PV system under partially shading conditions [15]. Thus, the paper is planned as following. The ESC scheme will be briefly shown in Section 2. Some multimodal benchmark patterns used in the literature will be shown in Section 3. The design of the tuning parameters will be detailed in Section 4. The results presented in Section 4 will validate the achievement of the advanced ESC scheme proposed in [15]. The last Section presents the conclusions drew from the study.


The ESC scheme is shown in Fig. 1 and the main relations of model are given below:



 , ,  












where equations (1a) to (1i) represent the static map, normalization, band pass filter (BPF) filter, demodulation, integration, computing of the dither’s gain (Gd), and p1, p2, and p3 components of the searching signal (p).

Usually, any multimodal pattern can be approximated by the model given by (2):


considering the small signals for perturbation and response are   and  , where pLMP define the LMP position and the derivatives of the multimodal function are noted with  .

The searching gradient is given by (3) [15]:


where the two components are:



Note that the a1 amplitude of dither is very small close to LMP or GMP, so the searching gradient close to an extreme may be approximated by (19):


Thus, if the tuning parameter k1 will be designed to guarantee the stability of the locating loop, then the tuning parameter k2 will be designed to scan the searching range where the LMP is positioned, as will be shown in section IV.


The multimodal benchmark patterns nominated are available on Internet, but for mathematical descriptions the reader can use [35]. For example, the Himmelblau\'s function is defined by (7) [36]:


This function has one maximum and four identical local minima. Based on the Himmelblau\'s function can be defined other multimodal benchmark patterns [22,23]. For example, the function defined by (8):


has one minimum and four identical local maxima

Using the parameter r to set the searching resolution (RS) [15]:


other four multimodal benchmark patterns can be defined:


These function have only one GMP and three different LMPs.

An interested multimodal pattern is the 6th-order polynomial proposed as benchmark in [16]:


The GMP, LMP1, and LMP2 are located at pGMP  −0.8985, pLMP1  0.05, and pLMP2  0.8951  (see the zoom of the pattern (11) shown in top of the Figure 2, where the first and second derivative are also shown for p in searching range -1 to 1). The first and second derivatives are also shown for p in large range -3 to 3. It can be observed that first derivative have high values outside the range -1 to 1, so the components of the searching gradient (4) and (5) will have high values as well. Consequently, the searching of the extreme points starting from the searching range is possible, but the tuning parameter k2 must be carefully set in order to maintain the scanning signal in the searching range.

Figure 1. The ESC scheme

Figure 2. The multimodal pattern given by the 6th-order polynomial (5)


The searching space will be defined by setting the starting point p0=(p01,p02) and the value of the tuning parameter k2 [37]. Note that if k2 > k2(min), then GMP will be accurately positioned starting from any p0 in the searching range [32]. If k1=d2fd to improve the dither’s persistence in the ESC loop, where the fd is the dither’s frequency, then the stability condition of the ESC loop can be given considering the design parameter d and the maximum value k2(max) [37].

If k2 < k2(min) and a LMP is closed to starting point p0, then this LMP will be located instead of GMP, but the tuning parameter k2 must be appropriately designed.

So, considering the searching range -1 to 1 for the 6th-order polynomial proposed here as benchmark (see Fig. 2), the normalized gains kNy and kNp can be chosen as 20 and 2. Consequently, the tuning parameter k2 will be chosen considering (12) [32]:

pGMP = kNpk2(GMP)| pGMP - p0| (12)

where pGMP  −0.8985, and p0 is -1 and -0.3. So, k2(GMP)=0.25  max(|pGMP - p0|)/kNp is a good value for both cases. The variation of the signals y, p, and Gd are shown only for case p0 = -1 in Fig. 3 (because in case of p0 = -0.3 the shapes of signals are almost the same).

The values of the second derivate at pGMP, pLMP1, and pLMP2 are of -7, -1, and -5, so |k3(max|7.

Consequently, the tuning parameter d will be chosen considering (13) [32] and |k3(max|7:


Thus, the tuning parameter d can be chosen in range 0.5 to 1.5 (around 0.26/0.251.04). The values used in simulation are k2 =0.25 and d =1.


The searching of the GMP on the multimodal pattern (11) is shown in Fig. 3 for k2 =0.25 and d =1.

The searching and tracking of the GMP can be obtained for different starting points, but the tuning parameters must be chosen in a range more restrictive. Note that if the starting point is p0 = 1, then the GMP searching is not convergent (see Figure 4). This is happen due to high value of the components of the searching gradient (4) and (5) that will be obtained outside of the searching range.

A solution to limit these components without loses the advantages of the advanced ESC scheme is proposed below by appropriate choosing the starting point.

The LMP1 and LMP2 can be located starting, for example, from p0 = 0.6, which is positioned between pLMP1  0.05 and pLMP2  0.8951 (see Figure 5).  

The tuning parameter k2 will be chosen considering (14) [32]:

pLMP = kNpk2(LMP)| pLMP - p0| (14)

The value k2(LMP1) = 0.25 and k2(LMP2) = 0.2 will be used in simulation, and d given by (13), respectively.

Figure 3. Searching of the GMP on the multimodal pattern (11)

Figure 4. Searching of the GMP on the multimodal pattern (11) starting from p0 = 1

Figure 5. Searching of the LMP1 and LMP2 on the multimodal pattern (11)


The performance of the advanced ESC scheme proposed in [15] is performed here based on special multimodal benchmark pattern used in the literature to test the classical PESC schemes. The multimodal benchmark pattern has the GMP and LMP2 (located at pGMP  −0.8985 and pLMP2  0.8951) with about the same value (about 0.068 and 0.062). Thus, the searching resolution (RS) is about 8.8% being close to lower limit set by the tracking accuracy. Furthermore, the derivatives have high values outside of the searching range, so high value will be obtained for the components of the searching gradient (4) and (5). This can give some issues of stability if the starting point is not appropriately chosen in the searching range.

Besides the issue mentioned above, this study have shown the performance of advanced ESC scheme proposed in [15] under very difficult multimodal benchmark pattern if the design rules proposed are used. So, advanced ESC scheme could be used in RTO strategies to optimize the operation of energy systems such as FC system or PV systems.


This work was partially supported by the grant SD no #4/45. Also, I thank to Professor Nicu BIZON for the proficient advice given during the writing of this paper.


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