## 3 Harmonic Model in RSHPES
### The Control Aspect  
Figure 6 indicates that harmonics is introduced into the controller because \(u_{\text{o}}\) and \(u_{\text{s}}/U_{\text{s}}\) is sampled. \(P_{\text{in}}\) is the input instantaneous power, and the expression of \(P_{\text{in}}\) is shown in (6). \[P_{\text{in}}(t)=u_{\text{s}}i_{\text{s}}=[\frac{U_{\text{so}}^{2}\sin(2 \omega t)-U_{\text{AB}}U_{\text{so}}\sin(2\omega t+\varphi)}{\omega L}+\sum \limits_{k=2}^{n}\frac{U_{\text{sk}}^{2}\sin(2 kwt)}{k\omega L}]+\frac{U_{\text {AB}}U_{\text{so}}\sin(\varphi)}{\omega L} \tag{6}\]  
\(P_{\text{out}}\) is output instantaneous power, which is expressed as (7)  
\[P_{\text{out}}=u_{\text{o}}i_{\text{dc}}=u_{\text{o}}i_{\text{o}}+u_{\text{o} }i_{\text{c}}=U_{\text{o}}I_{\text{o}}+CU_{\text{o}}\frac{du_{\text{ore}}(t)} {dt} \tag{7}\]  
where \(U_{\text{o}}\) and \(I_{\text{o}}\) are the DC components, and \(u_{\text{ore}}\) represents the ripple component. When the line loss is ignored, \(P_{\text{in}}=P_{\text{out}}\), the expression of \(u_{\text{ore}}\) can be obtained as (8). \[u_{\text{ore}}(t)=\frac{U_{\text{AB}}U_{\text{so}}\cos(2\omega t+\varphi)-U_{ \text{AB}}^{2}\cos(2\omega t)}{2\omega^{2}CLU_{\text{o}}} \tag{8}\]  
As can be seen from Figure 6, \(i_{\text{sr}}\) can be obtained as (9). \[ \begin{array}{ll}i_{\text{sr}}(t)&=k_{\text{p}}[\cos(\omega t)+\sum\limits _{k=2}^{n}m\cos(k\omega t)]\{(\,U_{\text{or}}-U_{\text{o}})-\frac{U_{\text{AB} }U_{\text{so}}\cos(2\omega t+\varphi)-U_{\text{AB}}^{2}\cos(2\omega t)}{2\omega ^{2}CLU_{\text{o}}}\}\\ &+k_{\text{i}}[\cos(\omega t)+\sum\limits_{k=2}^{n}m\cos(k\omega t)]\int\{(\,U_ {\text{or}}-U_{\text{o}})-\frac{U_{\text{AB}}U_{\text{so}}\cos(2\omega t+ \varphi)-U_{\text{AB}}^{2}\cos(2\omega t)}{2\omega^{2}CLU_{\text{o}}}\}\\ &\text{where }m=U_{\text{sk}}/U_{\text{so}}\text{. When }U_{\text{dc}}\text{ reaches the reference, }U_{\text{dc}}-U_{\text{dc}}=0\text{. The simplification results of (9) is}\\ \\ i_{\text{sr}}(t)&=\frac{k_{\text{p}}}{4\omega^{2}CLU_{\text{o}}}[U_{\text{AB} }^{2}\cos(\omega t)-U_{\text{AB}}U_{\text{so}}\cos(\omega t+\varphi)+U_{\text {AB}}^{2}\cos(3\omega t)-U_{\text{AB}}U_{\text{so}}\cos(3\omega t+\varphi)]\\ &+\frac{k_{\text{p}}m}{4\omega^{2}CLU_{\text{o}}}\sum\limits_{k=2}^{n}\frac{ \{U_{\text{AB}}^{2}\cos[(k-2)\omega t]-U_{\text{AB}}U_{\text{so}}\cos[(k-2) \omega t+\varphi]+U_{\text{AB}}^{2}\cos[(k+2)\omega t]}\\ &-U_{\text{AB}}U_{\text{so}}\cos[(k+2)\omega t+\varphi]\}\\ &+\frac{k_{\text{i}}}{8\omega^{3}CLU_{\text{o}}}[U_{\text{AB}}^{2}\sin(\omega t )-U_{\text{AB}}U_{\text{so}}\sin(\omega t+\varphi)+U_{\text{AB}}^{2}\sin(3 \omega t)-U_{\text{AB}}U_{\text{so}}\sin(3\omega t+\varphi)]\\ &+\frac{k_{\text{i}}m}{8\omega^{3}CLU_{\text{o}}}\sum\limits_{k=2}^{n}\frac{ \{U_{\text{AB}}^{2}\sin[(k-2)\omega t]-U_{\text{AB}}U_{\text{so}}\sin[(k-2) \omega t+\varphi]+U_{\text{AB}}^{2}\sin[(k+2)\omega t]}\\ &-U_{\text{AB}}U_{\text{so}}\sin[(k+2)\omega t+\varphi]\}\\ \end{array}  \tag{10}\]From (10), we can deduce that the input current will contain the 3 th, \((k-2)\)th and \((k+2)\) th harmonic. When stability is achieved in the system, the expression of \(i_{\text{s}}\) is presented in (11)  
\[\begin{split} i_{\text{s}}(t)&=i_{\text{sr}}(t)+i_{ \text{st}}(t)\\ &=\frac{k_{\text{p}}}{4\omega^{2}CLU_{\text{\text{AB}}}}[U_{\text {\text{AB}}}^{2}\cos(\omega t)-U_{\text{\text{AB}}}U_{\text{\text{so}}}\cos( \omega t+\varphi)+U_{\text{\text{AB}}}^{2}\cos(3\omega t)-U_{\text{\text{AB}}}U _{\text{\text{so}}}\cos(3\omega t+\varphi)]\\ &+\frac{k_{\text{p}}m}{4\omega^{2}CLU_{\text{\text{o}}}}\sum \limits_{k=2}^{n} \begin{array}{l}\{U_{\text{AB}}^{2}\cos[(k-2)\omega t]-U_{ \text{AB}}U_{\text{\text{so}}}\cos[(k-2)\omega t+\varphi]+U_{\text{\text{AB}}} ^{2}\cos[(k+2)\omega t]\\ -U_{\text{AB}}U_{\text{\text{so}}}\cos[(k+2)\omega t+\varphi]\}\end{array}